737:
996:
969:
1096:
857:
1066:
869:
895:
8368:
3999:
7693:
3757:
5310:
3964:
750:
9250:
6978:
7326:
5756:
2732:
8044:
5079:
3002:
8340:
1408:
6442:
9023:
7688:{\displaystyle T={\frac {1}{2}}m_{1}\mathbf {v} _{1}\cdot \mathbf {v} _{1}+{\frac {1}{2}}m_{2}\mathbf {v} _{2}\cdot \mathbf {v} _{2}={\frac {1}{2}}(m_{1}+m_{2})L_{1}^{2}{\dot {\theta }}_{1}^{2}+{\frac {1}{2}}m_{2}L_{2}^{2}{\dot {\theta }}_{2}^{2}+m_{2}L_{1}L_{2}\cos(\theta _{2}-\theta _{1}){\dot {\theta }}_{1}{\dot {\theta }}_{2}.}
6701:
9012:
2241:
2519:
9727:
Some authors set the constraint equations to a constant for convenience with some constraint equations (e.g. pendulums), others set it to zero. It makes no difference because the constant can be subtracted to give zero on one side of the equation. Also, in
Lagrange's equations of the first kind, only
7721:
5305:{\displaystyle {\frac {d}{dt}}{\frac {\partial T}{\partial {\dot {x}}}}-{\frac {\partial T}{\partial x}}=F_{x}+\lambda {\frac {\partial f}{\partial x}},\quad {\frac {d}{dt}}{\frac {\partial T}{\partial {\dot {y}}}}-{\frac {\partial T}{\partial y}}=F_{y}+\lambda {\frac {\partial f}{\partial y}}.}
5057:
2791:
8055:
9535:
3491:
6135:
2037:
3741:
9390:
5589:
1188:
7142:
6198:
9245:{\displaystyle \delta W=\left(\sum _{j=1}^{m}\mathbf {F} _{j}\cdot {\frac {\partial \mathbf {r} _{j}}{\partial q_{1}}}\right)\delta {q}_{1}+\ldots +\left(\sum _{j=1}^{m}\mathbf {F} _{j}\cdot {\frac {\partial \mathbf {r} _{j}}{\partial q_{n}}}\right)\delta {q}_{n}.}
7315:
6973:{\displaystyle \delta \mathbf {r} _{1}=(L_{1}\cos \theta _{1},L_{1}\sin \theta _{1})\delta \theta _{1},\quad \delta \mathbf {r} _{2}=(L_{1}\cos \theta _{1},L_{1}\sin \theta _{1})\delta \theta _{1}+(L_{2}\cos \theta _{2},L_{2}\sin \theta _{2})\delta \theta _{2}}
6673:
2482:
3307:
8868:
5464:
2078:
4600:
4891:
5951:
2727:{\displaystyle {\dot {\mathbf {r} }}_{k}\cdot {\dot {\mathbf {r} }}_{k}=\sum _{i,j=1}^{n}\left({\frac {\partial \mathbf {r} _{k}}{\partial q_{i}}}\cdot {\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}}\right){\dot {q}}_{i}{\dot {q}}_{j},}
3150:
8039:{\displaystyle (m_{1}+m_{2})L_{1}^{2}{\ddot {\theta }}_{1}+m_{2}L_{1}L_{2}{\ddot {\theta }}_{2}\cos(\theta _{2}-\theta _{1})+m_{2}L_{1}L_{2}{\dot {\theta _{2}}}^{2}\sin(\theta _{1}-\theta _{2})=-(m_{1}+m_{2})gL_{1}\sin \theta _{1},}
8803:
8578:
6555:
4902:
2997:{\displaystyle ds_{k}^{2}=d\mathbf {r} _{k}\cdot d\mathbf {r} _{k}=\sum _{i,j=1}^{n}\left({\frac {\partial \mathbf {r} _{k}}{\partial q_{i}}}\cdot {\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}}\right)dq_{i}dq_{j}\,,}
2350:
An example of such a constraint is a rolling wheel or knife-edge that constrains the direction of the velocity vector. Non-holonomic constraints can also involve next-order derivatives such as generalized accelerations.
4006:
The relationship between the use of generalized coordinates and
Cartesian coordinates to characterize the movement of a mechanical system can be illustrated by considering the constrained dynamics of a simple pendulum.
1734:
8335:{\displaystyle m_{2}L_{2}^{2}{\ddot {\theta }}_{2}+m_{2}L_{1}L_{2}{\ddot {\theta }}_{1}\cos(\theta _{2}-\theta _{1})+m_{2}L_{1}L_{2}{\dot {\theta _{1}}}^{2}\sin(\theta _{2}-\theta _{1})=-m_{2}gL_{2}\sin \theta _{2}.}
9643:
3942:
4686:
1852:
3586:
9401:
4339:
3337:
2279:
A mechanical system can involve constraints on both the generalized coordinates and their derivatives. Constraints of this type are known as non-holonomic. First-order non-holonomic constraints have the form
920:
One generalized coordinate (one degree of freedom) on paths in 2D. Only one generalized coordinate is needed to uniquely specify positions on the curve. In these examples, that variable is either arc length
5962:
5727:
4481:
3837:; either one is determined from the other. The constraint force is the reaction force the wire exerts on the bead to keep it on the wire, and the non-constraint applied force is gravity acting on the bead.
2246:
and so generally depends on the generalized velocities and coordinates. Since we are free to specify the initial values of the generalized coordinates and velocities separately, the generalized coordinates
1872:
8477:
4235:
3616:
2345:
1403:{\displaystyle {\begin{aligned}&\mathbf {r} _{1}=(x_{1},y_{1},z_{1}),\\&\mathbf {r} _{2}=(x_{2},y_{2},z_{2}),\\&\qquad \qquad \vdots \\&\mathbf {r} _{N}=(x_{N},y_{N},z_{N})\end{aligned}}}
9265:
5664:
5494:
1590:). It is ideal to use the minimum number of coordinates needed to define the configuration of the entire system, while taking advantage of the constraints on the system. These quantities are known as
91:
115:
6437:{\displaystyle \mathbf {r} _{1}=(L_{1}\sin \theta _{1},-L_{1}\cos \theta _{1}),\quad \mathbf {r} _{2}=(L_{1}\sin \theta _{1},-L_{1}\cos \theta _{1})+(L_{2}\sin \theta _{2},-L_{2}\cos \theta _{2}).}
6989:
1193:
821:
An example of a generalized coordinate would be to describe the position of a pendulum using the angle of the pendulum relative to vertical, rather than by the x and y position of the pendulum.
7153:
4772:
6586:
4133:
1761:. A degree of freedom corresponds to one quantity that changes the configuration of the system, for example the angle of a pendulum, or the arc length traversed by a bead along a wire.
1498:
2374:
1040:
Generalized coordinates are usually selected to provide the minimum number of independent coordinates that define the configuration of a system, which simplifies the formulation of
9007:{\displaystyle \delta \mathbf {r} _{j}={\frac {\partial \mathbf {r} _{j}}{\partial q_{1}}}\delta {q}_{1}+\ldots +{\frac {\partial \mathbf {r} _{j}}{\partial q_{n}}}\delta {q}_{n},}
8621:. Only two coordinates are needed instead of three, because the position of the bob can be parameterized by two numbers, and the constraint equation connects the three coordinates
4405:
3180:
824:
Although there may be many possible choices for generalized coordinates for a physical system, they are generally selected to simplify calculations, such as the solution of the
1764:
If it is possible to find from the constraints as many independent variables as there are degrees of freedom, these can be used as generalized coordinates. The position vector
1540:
constraint equations. There is not necessarily one constraint equation for each particle, and if there are no constraints on the system then there are no constraint equations.
5321:
2236:{\displaystyle \mathbf {v} _{k}={\dot {\mathbf {r} }}_{k}={\frac {d\mathbf {r} _{k}}{dt}}=\sum _{j=1}^{n}{\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}}{\dot {q}}_{j}\,.}
4511:
8667:. When formulated in terms of generalized coordinates, this is equivalent to the requirement that the generalized forces for any virtual displacement are zero, that is
4787:
5820:
814:
of the generalized coordinates of the system. The adjective "generalized" distinguishes these parameters from the traditional use of the term "coordinate" to refer to
8652:
states that if a system is in static equilibrium, the virtual work of the applied forces is zero for all virtual movements of the system from this state, that is,
3045:
10113:
781:
9395:
are the generalized forces acting on the system. Kane shows that these generalized forces can also be formulated in terms of the ratio of time derivatives,
2737:
which illustrates the kinetic energy is in general a function of the generalized velocities, coordinates, and time if the constraints also vary with time, so
8728:
370:
2770:
In the case the constraints on the particles are time-independent, then all partial derivatives with respect to time are zero, and the kinetic energy is a
8488:
1754:
5052:{\displaystyle T={\frac {1}{2}}m\mathbf {v} \cdot \mathbf {v} ={\frac {1}{2}}m({\dot {x}}^{2}+{\dot {y}}^{2})={\frac {1}{2}}mL^{2}{\dot {\theta }}^{2}.}
489:
1507:
will appear explicitly in the constraint equations. At any instant of time, any one coordinate will be determined from the other coordinates, e.g. if
6453:
462:
3017:. Thus for time-independent constraints it is sufficient to know the line element to quickly obtain the kinetic energy of particles and hence the
1503:
which connects all the 3 spatial coordinates of that particle together, so they are not independent. The constraint may change with time, so time
1745:
of the system. They are all independent of one other, and each is a function of time. Geometrically they can be lengths along straight lines, or
828:
for the system. If the coordinates are independent of one another, the number of independent generalized coordinates is defined by the number of
1621:
9571:
3846:
736:
10007:
9805:
9530:{\displaystyle F_{i}=\sum _{j=1}^{m}\mathbf {F} _{j}\cdot {\frac {\partial \mathbf {v} _{j}}{\partial {\dot {q}}_{i}}},\quad i=1,\ldots ,n,}
4611:
1789:
3522:
3486:{\displaystyle \left({\frac {ds}{dt}}\right)^{2}={\dot {r}}^{2}+r^{2}{\dot {\theta }}^{2}+r^{2}\sin ^{2}\theta \,{\dot {\varphi }}^{2}\,.}
6130:{\displaystyle f_{2}(x_{1},y_{1},x_{2},y_{2})=(\mathbf {r} _{2}-\mathbf {r} _{1})\cdot (\mathbf {r} _{2}-\mathbf {r} _{1})-L_{2}^{2}=0.}
4262:
444:
2032:{\displaystyle {\dot {\mathbf {q} }}={\frac {d\mathbf {q} }{dt}}=({\dot {q}}_{1}(t),\ {\dot {q}}_{2}(t),\ \ldots ,\ {\dot {q}}_{n}(t))}
10097:
10052:
9565:
In order for the virtual work to be zero for an arbitrary virtual displacement, each of the generalized forces must be zero, that is
5675:
4441:
3840:
Suppose the wire changes its shape with time, by flexing. Then the constraint equation and position of the particle are respectively
3736:{\displaystyle {\frac {\partial L}{\partial q_{i}}}={\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}_{i}}}={\dot {p}}_{i}=0\,.}
10130:
10033:
9929:
9776:
3776:
For a bead sliding on a frictionless wire subject only to gravity in 2d space, the constraint on the bead can be stated in the form
774:
9385:{\displaystyle F_{i}=\sum _{j=1}^{m}\mathbf {F} _{j}\cdot {\frac {\partial \mathbf {r} _{j}}{\partial q_{i}}},\quad i=1,\ldots ,n,}
8389:
5584:{\displaystyle {\frac {d}{dt}}{\frac {\partial T}{\partial {\dot {\theta }}}}-{\frac {\partial T}{\partial \theta }}=F_{\theta },}
4148:
2286:
995:
5600:
110:
47:
7137:{\displaystyle \delta W=-(m_{1}+m_{2})gL_{1}\sin \theta _{1}\delta \theta _{1}-m_{2}gL_{2}\sin \theta _{2}\delta \theta _{2},}
6564:
is the acceleration due to gravity. Therefore, the virtual work of gravity on the two masses as they follow the trajectories
968:
10164:
10078:
829:
105:
365:
1095:
856:
10149:
1741:
803:
6192:
that define the angular position of each mass of the double pendulum from the vertical direction. In this case, we have
3024:
It is instructive to see the various cases of polar coordinates in 2D and 3D, owing to their frequent appearance. In 2D
360:
8383:
free to swing in any angular direction subject to gravity, the constraint on the pendulum bob can be stated in the form
7310:{\displaystyle F_{\theta _{1}}=-(m_{1}+m_{2})gL_{1}\sin \theta _{1},\quad F_{\theta _{2}}=-m_{2}gL_{2}\sin \theta _{2}.}
1065:
868:
767:
754:
515:
438:
7698:
203:
3821:
along the curve from some point on the wire. This is a suitable choice of generalized coordinate for the system. Only
1554:
coordinates can be eliminated, one coordinate from each constraint equation. The number of independent coordinates is
434:
235:
6668:{\displaystyle \delta W=\mathbf {F} _{1}\cdot \delta \mathbf {r} _{1}+\mathbf {F} _{2}\cdot \delta \mathbf {r} _{2}.}
10154:
9999:
4723:
4415:
654:
543:
469:
329:
262:
3951:
due to the changing coordinates as the wire changes its shape. Notice time appears implicitly via the coordinates
894:
4055:
408:
1026:
along the curve is a legitimate generalized coordinate since the position is uniquely determined, but the angle
6140:
The formulation of
Lagrange's equations for this system yields six equations in the four Cartesian coordinates
2477:{\displaystyle T={\frac {1}{2}}\sum _{k=1}^{N}m_{k}{\dot {\mathbf {r} }}_{k}\cdot {\dot {\mathbf {r} }}_{k}\,,}
1049:
684:
528:
8367:
3998:
1453:
9674:
3302:{\displaystyle \left({\frac {ds}{dt}}\right)^{2}={\dot {r}}^{2}+r^{2}{\dot {\theta }}^{2}+{\dot {z}}^{2}\,,}
3156:
674:
634:
398:
4361:
806:. These parameters must uniquely define the configuration of the system relative to a reference state. The
10070:
9669:
3967:
Simple pendulum. Since the rod is rigid, the position of the bob is constrained according to the equation
3825:
coordinate is needed instead of two, because the position of the bead can be parameterized by one number,
3606:, then it follows from the Euler–Lagrange equations that the corresponding generalized momentum will be a
1750:
679:
496:
4348:
to define the configuration of this system avoids the constraint provided by the equation of the circle.
10159:
9659:
9654:
8596:
5459:{\displaystyle m{\ddot {x}}=\lambda (2x),\quad m{\ddot {y}}=-mg+\lambda (2y),\quad x^{2}+y^{2}-L^{2}=0,}
3506:
3313:
1179:
1167:
1138:
836:
815:
689:
664:
350:
168:
10021:
8722:, then the virtual work generated by a virtual displacement from the equilibrium position is given by
1137:
are needed to specify the points on the curve, one possibility is shown for each case. The full three
5732:
This formulation yields one equation because there is a single parameter and no constraint equation.
3018:
2771:
1436:
1041:
795:
709:
704:
669:
577:
573:
565:
555:
345:
338:
94:
4595:{\displaystyle \delta \mathbf {r} =(\delta x,\delta y)=(L\cos \theta ,L\sin \theta )\delta \theta .}
4886:{\displaystyle \mathbf {v} =({\dot {x}},{\dot {y}})=(L\cos \theta ,L\sin \theta ){\dot {\theta }},}
825:
484:
425:
403:
148:
143:
138:
38:
10107:
9689:
8376:
5946:{\displaystyle f_{1}(x_{1},y_{1},x_{2},y_{2})=\mathbf {r} _{1}\cdot \mathbf {r} _{1}-L_{1}^{2}=0}
3607:
614:
355:
230:
198:
158:
9945:
1857:
and the generalized coordinates can be thought of as parameters associated with the constraint.
1156:
are not necessary because any two determines the third according to the equations of the curves.
1125:
Two generalized coordinates, two degrees of freedom, on curved surfaces in 3D. Only two numbers
3772:
in this case is gravity. Notice the initial position of the wire can lead to different motions.
10126:
10093:
10074:
10048:
10029:
10017:
10003:
9958:
9925:
9801:
9793:
9772:
9751:
4710:
3025:
624:
581:
538:
533:
474:
250:
240:
133:
9981:
T. R. Kane and D. A. Levinson, Dynamics: theory and applications, McGraw-Hill, New York, 1985
9764:
8359:
provides an alternative to the
Cartesian formulation of the dynamics of the double pendulum.
3145:{\displaystyle \left({\frac {ds}{dt}}\right)^{2}={\dot {r}}^{2}+r^{2}{\dot {\theta }}^{2}\,,}
9684:
2058:
719:
699:
644:
639:
585:
560:
415:
273:
218:
193:
1044:
of motion. However, it can also occur that a useful set of generalized coordinates may be
10125:. HRW Series in Mechanical Engineering. United States of America: CBS College Publishing.
5767:
5760:
5739:
is a generalized coordinate that can be used in the same way as the
Cartesian coordinates
2043:
1171:
949:
is related to the other by the equations of the curves. They can also be parameterized by
714:
659:
609:
604:
523:
3610:, because the time derivative is zero implying the momentum is a constant of the motion;
3756:
8606:
2365:
741:
649:
550:
267:
213:
9712:, as shown here, as the condition for the constraint on that particle to be holonomic.
8798:{\displaystyle \delta W=\sum _{j=1}^{m}\mathbf {F} _{j}\cdot \delta \mathbf {r} _{j}.}
17:
10143:
9708:
Some authors e.g. Hand & Finch take the form of the position vector for particle
629:
456:
8609:. A logical choice of generalized coordinates to describe the motion are the angles
8573:{\displaystyle \mathbf {r} =(x(\theta ,\phi ),y(\theta ,\phi ),z(\theta ,\phi ))\,,}
5814:
define their two trajectories. These vectors satisfy the two constraint equations,
1757:, so the number of generalized coordinates equals the number of degrees of freedom,
9664:
8648:
5062:
4422:
4018:
hanging from a pivot point so that it is constrained to move on a circle of radius
3963:
2778:
694:
619:
308:
188:
3760:
Bead constrained to move on a frictionless wire. The wire exerts a reaction force
2777:
Still for the time-independent case, this expression is equivalent to taking the
1749:
along curves, or angles; not necessarily
Cartesian coordinates or other standard
9679:
2488:
840:
6550:{\displaystyle \mathbf {F} _{1}=(0,-m_{1}g),\quad \mathbf {F} _{2}=(0,-m_{2}g)}
5766:
The benefits of generalized coordinates become apparent with the analysis of a
10062:
5755:
3815:
1746:
811:
479:
153:
501:
1729:{\displaystyle \mathbf {q} (t)=(q_{1}(t),\ q_{2}(t),\ \ldots ,\ q_{n}(t))}
4011:
420:
303:
278:
9638:{\displaystyle \delta W=0\quad \Rightarrow \quad F_{i}=0,i=1,\ldots ,n.}
4142:. This equation also provides a constraint on the velocity components,
3937:{\displaystyle f(\mathbf {r} ,t)=0\,,\quad \mathbf {r} =(x(s,t),y(s,t))}
4248:
from the vertical direction. It can be used to define the coordinates
835:
Generalized coordinates are paired with generalized momenta to provide
393:
246:
163:
4702:-component of the applied force. In the same way, the coefficient of
8605:
is measured along the suspension point to the bob, here treated as a
4681:{\displaystyle \delta W=-mg\delta y=-mgL\sin(\theta )\delta \theta .}
1847:{\displaystyle \mathbf {r} _{k}=\mathbf {r} _{k}(\mathbf {q} (t))\,,}
802:
are a set of parameters used to represent the state of a system in a
452:
298:
208:
3581:{\displaystyle p_{i}={\frac {\partial L}{\partial {\dot {q}}_{i}}}.}
5754:
4334:{\displaystyle \mathbf {r} =(x,y)=(L\sin \theta ,-L\cos \theta ).}
3997:
1613:
1175:
288:
283:
225:
4022:. The position of the mass is defined by the coordinate vector
1030:
is not since there are multiple positions for a single value of
293:
256:
8599:
because the bob moves in the surface of a sphere. The position
5722:{\displaystyle {\ddot {\theta }}+{\frac {g}{L}}\sin \theta =0.}
4476:{\displaystyle \delta W=\mathbf {F} \cdot \delta \mathbf {r} .}
2368:
of the system is the energy of the system's motion, defined as
3766:
on the bead to keep it on the wire. The non-constraint force
8829:
denote the virtual displacements of each point in the body.
8472:{\displaystyle f(\mathbf {r} )=x^{2}+y^{2}+z^{2}-l^{2}=0\,,}
4230:{\displaystyle {\dot {f}}(x,y)=2x{\dot {x}}+2y{\dot {y}}=0.}
2340:{\displaystyle g(\mathbf {q} ,{\dot {\mathbf {q} }},t)=0\,,}
9769:
Fundamentals of multibody dynamics: theory and applications
7701:
yield two equations in the unknown generalized coordinates
3829:, and the constraint equation connects the two coordinates
3988:
is the tension in the rod. Again the non-constraint force
2491:. The kinetic energy is a function only of the velocities
9752:
p. 397, §7.2.1 Selection of generalized coordinates
9551:
is the velocity of the point of application of the force
5659:{\displaystyle mL^{2}{\ddot {\theta }}=-mgL\sin \theta ,}
86:{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}
6447:
The force of gravity acting on the masses is given by,
9831:
9829:
8482:
where the position of the pendulum bob can be written
1783:
generalized coordinates (and, through them, of time),
1543:
So far, the configuration of the system is defined by
9574:
9404:
9268:
9026:
8871:
8731:
8637:
so any one of them is determined from the other two.
8491:
8392:
8058:
7724:
7329:
7156:
6992:
6704:
6589:
6456:
6201:
5965:
5823:
5678:
5603:
5497:
5324:
5082:
4905:
4790:
4777:
To complete the analysis consider the kinetic energy
4726:
4614:
4514:
4444:
4364:
4265:
4151:
4058:
3849:
3619:
3525:
3340:
3183:
3048:
2794:
2522:
2377:
2289:
2081:
1875:
1792:
1624:
1456:
1191:
50:
9798:
4351:
Notice that the force of gravity acting on the mass
2513:
themselves. By contrast an important observation is
1576:
coordinates, and the reduction by constraints means
1536:
constraints, each has an equation, so there will be
10028:(3rd ed.). San Francisco, CA: Addison Wesley.
9895:
9859:
9820:
1048:, which means that they are related by one or more
9637:
9529:
9384:
9244:
9006:
8797:
8572:
8471:
8334:
8038:
7687:
7309:
7136:
6972:
6667:
6549:
6436:
6129:
5945:
5721:
5658:
5583:
5458:
5304:
5063:D'Alembert's form of the principle of virtual work
5051:
4885:
4766:
4680:
4594:
4475:
4399:
4355:is formulated in the usual Cartesian coordinates,
4333:
4229:
4127:
3936:
3735:
3580:
3485:
3301:
3144:
2996:
2726:
2476:
2339:
2235:
2031:
1846:
1728:
1572:dimensions, the original configuration would need
1492:
1402:
85:
3007:and dividing by the square differential in time,
8702:be applied to points with Cartesian coordinates
7320:Compute the kinetic energy of this system to be
3787:, where the position of the bead can be written
9835:
4041:is in the vertical direction. The coordinates
9907:
6175:that arise from the two constraint equations.
4037:measured in the plane of the circle such that
2355:Physical quantities in generalized coordinates
5065:for the pendulum in terms of the coordinates
4767:{\displaystyle F_{\theta }=-mgL\sin \theta .}
3013:, to obtain the velocity squared of particle
989:. Multiple intersections of radius with path.
775:
8:
4493:can be computed in terms of the coordinates
9794:"§5.1 Manifolds of generalized coordinates"
4128:{\displaystyle f(x,y)=x^{2}+y^{2}-L^{2}=0,}
2774:of degree 2 in the generalized velocities.
1608:. It is convenient to collect them into an
1413:Any of the position vectors can be denoted
929:. Having both of the Cartesian coordinates
10112:: CS1 maint: location missing publisher (
8371:Spherical pendulum: angles and velocities.
6178:Now introduce the generalized coordinates
4049:are related by the equation of the circle
782:
768:
29:
10047:. Cambridge: Cambridge University Press.
9871:
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9490:
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8120:
8110:
8097:
8086:
8085:
8078:
8073:
8063:
8057:
8027:
8011:
7995:
7982:
7960:
7947:
7928:
7916:
7910:
7909:
7902:
7892:
7882:
7866:
7853:
7834:
7823:
7822:
7815:
7805:
7795:
7782:
7771:
7770:
7763:
7758:
7745:
7732:
7723:
7676:
7665:
7664:
7657:
7646:
7645:
7635:
7622:
7603:
7593:
7583:
7570:
7565:
7554:
7553:
7546:
7541:
7531:
7517:
7508:
7503:
7492:
7491:
7484:
7479:
7466:
7453:
7436:
7427:
7422:
7412:
7407:
7400:
7386:
7377:
7372:
7362:
7357:
7350:
7336:
7328:
7298:
7282:
7269:
7251:
7246:
7232:
7216:
7200:
7187:
7166:
7161:
7155:
7125:
7112:
7096:
7083:
7070:
7057:
7041:
7025:
7012:
6991:
6964:
6948:
6932:
6919:
6903:
6887:
6871:
6855:
6842:
6826:
6810:
6805:
6791:
6775:
6759:
6746:
6730:
6714:
6709:
6703:
6656:
6651:
6638:
6633:
6623:
6618:
6605:
6600:
6588:
6535:
6510:
6505:
6488:
6463:
6458:
6455:
6422:
6406:
6390:
6374:
6355:
6339:
6323:
6307:
6291:
6286:
6272:
6256:
6240:
6224:
6208:
6203:
6200:
6115:
6110:
6094:
6089:
6079:
6074:
6058:
6053:
6043:
6038:
6022:
6009:
5996:
5983:
5970:
5964:
5931:
5926:
5913:
5908:
5898:
5893:
5880:
5867:
5854:
5841:
5828:
5822:
5694:
5680:
5679:
5677:
5618:
5617:
5611:
5602:
5572:
5545:
5528:
5527:
5513:
5498:
5496:
5441:
5428:
5415:
5366:
5365:
5329:
5328:
5323:
5279:
5267:
5240:
5223:
5222:
5208:
5193:
5169:
5157:
5130:
5113:
5112:
5098:
5083:
5081:
5040:
5029:
5028:
5021:
5004:
4992:
4981:
4980:
4970:
4959:
4958:
4941:
4933:
4925:
4912:
4904:
4869:
4868:
4818:
4817:
4803:
4802:
4791:
4789:
4731:
4725:
4613:
4518:
4513:
4465:
4454:
4443:
4365:
4363:
4266:
4264:
4210:
4209:
4189:
4188:
4153:
4152:
4150:
4110:
4097:
4084:
4057:
3881:
3876:
3856:
3848:
3729:
3717:
3706:
3705:
3692:
3681:
3680:
3665:
3650:
3638:
3620:
3618:
3566:
3555:
3554:
3539:
3530:
3524:
3479:
3473:
3462:
3461:
3459:
3447:
3437:
3424:
3413:
3412:
3405:
3392:
3381:
3380:
3370:
3346:
3339:
3295:
3289:
3278:
3277:
3267:
3256:
3255:
3248:
3235:
3224:
3223:
3213:
3189:
3182:
3138:
3132:
3121:
3120:
3113:
3100:
3089:
3088:
3078:
3054:
3047:
2990:
2984:
2971:
2950:
2935:
2930:
2923:
2911:
2896:
2891:
2884:
2873:
2856:
2843:
2838:
2825:
2820:
2807:
2802:
2793:
2715:
2704:
2703:
2696:
2685:
2684:
2669:
2654:
2649:
2642:
2630:
2615:
2610:
2603:
2592:
2575:
2562:
2551:
2549:
2548:
2538:
2527:
2525:
2524:
2521:
2470:
2464:
2453:
2451:
2450:
2440:
2429:
2427:
2426:
2419:
2409:
2398:
2384:
2376:
2333:
2307:
2305:
2304:
2296:
2288:
2229:
2223:
2212:
2211:
2201:
2186:
2181:
2174:
2168:
2157:
2133:
2128:
2121:
2112:
2101:
2099:
2098:
2088:
2083:
2080:
2011:
2000:
1999:
1968:
1957:
1956:
1934:
1923:
1922:
1899:
1893:
1879:
1877:
1876:
1874:
1840:
1823:
1814:
1809:
1799:
1794:
1791:
1708:
1674:
1649:
1625:
1623:
1469:
1464:
1455:
1387:
1374:
1361:
1345:
1340:
1310:
1297:
1284:
1268:
1263:
1245:
1232:
1219:
1203:
1198:
1192:
1190:
67:
61:
52:
51:
49:
10043:Hand, Louis N.; Finch, Janet D. (1998).
9747:
8641:Generalized coordinates and virtual work
8366:
3962:
3955:explicitly in the constraint equations.
3755:
2042:(each dot over a quantity indicates one
27:System configuration relative to another
10121:Torby, Bruce (1984). "Energy Methods".
9800:(5th ed.). Springer. p. 286.
9740:
9720:
9701:
8844:depends on the generalized coordinates
8345:The use of the generalized coordinates
4244:, that defines the angular position of
2781:squared of the trajectory for particle
1493:{\displaystyle f(\mathbf {r} _{k},t)=0}
1174:of each particle can be written as a 3-
101:
37:
10105:
10088:Landau, L. D.; Lifshitz, E.M. (1976).
1860:The corresponding time derivatives of
9970:
9883:
9847:
4400:{\displaystyle \mathbf {F} =(0,-mg),}
464:Newton's law of universal gravitation
7:
1528:. One constraint equation counts as
6983:Thus, the virtual work is given by
4605:Thus, the virtual work is given by
4002:Dynamic model of a simple pendulum.
445:Mechanics of planar particle motion
53:
9474:
9457:
9338:
9321:
9203:
9186:
9097:
9080:
8970:
8953:
8910:
8893:
5556:
5548:
5524:
5516:
5290:
5282:
5251:
5243:
5219:
5211:
5180:
5172:
5141:
5133:
5109:
5101:
3676:
3668:
3631:
3623:
3550:
3542:
2943:
2926:
2904:
2887:
2662:
2645:
2623:
2606:
2194:
2177:
847:Constraints and degrees of freedom
25:
9896:Goldstein, Poole & Safko 2002
9860:Goldstein, Poole & Safko 2002
9821:Goldstein, Poole & Safko 2002
6161:and the two Lagrange multipliers
4781:of the mass, using the velocity,
9988:Bibliography of cited references
9961:, scienceworld.wolfram.com. 2007
9462:
9441:
9326:
9305:
9191:
9170:
9085:
9064:
8958:
8898:
8877:
8782:
8764:
8682:Let the forces on the system be
8493:
8400:
7423:
7408:
7373:
7358:
6806:
6710:
6652:
6634:
6619:
6601:
6506:
6459:
6287:
6204:
6090:
6075:
6054:
6039:
5909:
5894:
5488:, those equations take the form
5315:This yields the three equations
4934:
4926:
4792:
4519:
4466:
4455:
4366:
4267:
4138:that constrains the movement of
3882:
3857:
2931:
2892:
2839:
2821:
2650:
2611:
2552:
2528:
2454:
2430:
2308:
2297:
2182:
2129:
2102:
2084:
1900:
1880:
1866:are the generalized velocities,
1824:
1810:
1795:
1626:
1465:
1341:
1264:
1199:
1094:
1064:
994:
967:
893:
867:
855:
749:
748:
735:
68:
10123:Advanced Dynamics for Engineers
10069:(5th ed.). River Edge NJ:
9924:(2nd ed.). Prentice Hall.
9765:"§2.4: Generalized coordinates"
9591:
9587:
9502:
9357:
7241:
7147:and the generalized forces are
6800:
6503:
6284:
5410:
5361:
5192:
4691:Notice that the coefficient of
4501:, or in terms of the parameter
3880:
1330:
1329:
9998:(3rd ed.). Cambridge UK:
9763:Farid M. L. Amirouche (2006).
9588:
8563:
8560:
8548:
8539:
8527:
8518:
8506:
8500:
8404:
8396:
8281:
8255:
8187:
8161:
8001:
7975:
7966:
7940:
7872:
7846:
7751:
7725:
7641:
7615:
7472:
7446:
7206:
7180:
7031:
7005:
6954:
6896:
6877:
6819:
6781:
6723:
6544:
6519:
6497:
6472:
6428:
6367:
6361:
6300:
6278:
6217:
6100:
6070:
6064:
6034:
6028:
5976:
5886:
5834:
5735:This shows that the parameter
5404:
5395:
5355:
5346:
4998:
4954:
4865:
4835:
4829:
4799:
4666:
4660:
4580:
4550:
4544:
4526:
4391:
4373:
4325:
4292:
4286:
4274:
4176:
4164:
4074:
4062:
3947:which now both depend on time
3931:
3928:
3916:
3907:
3895:
3889:
3867:
3853:
2324:
2293:
2026:
2023:
2017:
1980:
1974:
1946:
1940:
1918:
1837:
1834:
1828:
1820:
1723:
1720:
1714:
1686:
1680:
1661:
1655:
1642:
1636:
1630:
1481:
1460:
1393:
1354:
1316:
1277:
1251:
1212:
1:
9920:Greenwood, Donald T. (1987).
4713:along generalized coordinate
4429:as it follows the trajectory
941:are unnecessary since either
371:Koopman–von Neumann mechanics
4240:Now introduce the parameter
1016:. Self-intersection of path.
439:Non-inertial reference frame
9994:Ginsberg, Jerry H. (2008).
9836:Kibble & Berkshire 2004
9728:the derivatives are needed.
4416:acceleration due to gravity
366:Appell's equation of motion
236:Inertial frame of reference
10181:
10092:(Third ed.). Oxford.
10065:; Berkshire, F.H. (2004).
10000:Cambridge University Press
9908:Landau & Lifshitz 1976
3599:depend on some coordinate
5747:to analyze the pendulum.
2275:Non-holonomic constraints
1779:is a function of all the
1594:in this context, denoted
1532:constraint. If there are
1444:of the form for particle
9771:. Springer. p. 46.
3994:in this case is gravity.
1753:. There is one for each
1739:which is a point in the
1434:labels the particles. A
529:Rotating reference frame
361:Hamilton–Jacobi equation
9792:Florian Scheck (2010).
9675:Curvilinear coordinates
7699:Euler–Lagrange equation
5469:in the three unknowns,
4425:of gravity on the mass
3982:, the constraint force
3157:cylindrical coordinates
2046:). The velocity vector
1592:generalized coordinates
800:generalized coordinates
470:Newton's laws of motion
330:Newton's laws of motion
10071:Imperial College Press
10024:; Safko, John (2002).
9922:Principles of Dynamics
9670:Orthogonal coordinates
9639:
9531:
9438:
9386:
9302:
9246:
9167:
9061:
9008:
8799:
8761:
8597:spherical polar angles
8574:
8473:
8372:
8336:
8040:
7689:
7311:
7138:
6974:
6695:can be computed to be
6669:
6551:
6438:
6131:
5947:
5763:
5723:
5660:
5585:
5460:
5306:
5053:
4887:
4768:
4682:
4596:
4477:
4401:
4335:
4231:
4129:
4003:
3995:
3938:
3773:
3737:
3582:
3487:
3303:
3146:
2998:
2878:
2728:
2597:
2502:, not the coordinates
2478:
2414:
2341:
2237:
2173:
2033:
1848:
1751:orthogonal coordinates
1730:
1521:are given, then so is
1494:
1404:
1101:Closed curved surface
808:generalized velocities
497:Simple harmonic motion
410:Euler's laws of motion
204:D'Alembert's principle
87:
18:Generalized coordinate
10165:Mechanical quantities
9944:Richard Fitzpatrick,
9872:Hand & Finch 1998
9660:Hamiltonian mechanics
9655:Canonical coordinates
9640:
9532:
9418:
9387:
9282:
9247:
9147:
9041:
9009:
8832:Now assume that each
8800:
8741:
8575:
8474:
8379:with constant length
8370:
8337:
8041:
7690:
7312:
7139:
6975:
6670:
6552:
6439:
6132:
5948:
5770:. For the two masses
5758:
5724:
5661:
5586:
5461:
5307:
5054:
4888:
4769:
4683:
4597:
4478:
4402:
4336:
4232:
4130:
4001:
3966:
3939:
3759:
3738:
3583:
3507:canonically conjugate
3488:
3314:spherical coordinates
3304:
3147:
2999:
2852:
2729:
2571:
2479:
2394:
2342:
2269:independent variables
2238:
2153:
2072:with respect to time
2034:
1849:
1731:
1495:
1405:
1180:Cartesian coordinates
1168:real coordinate space
1139:Cartesian coordinates
1056:Holonomic constraints
837:canonical coordinates
816:Cartesian coordinates
351:Hamiltonian mechanics
169:Statistical mechanics
88:
10150:Lagrangian mechanics
10045:Analytical mechanics
9996:Engineering dynamics
9572:
9402:
9266:
9024:
8869:
8729:
8489:
8390:
8375:For a 3D example, a
8056:
7722:
7327:
7154:
6990:
6702:
6587:
6454:
6199:
5963:
5821:
5676:
5601:
5495:
5484:Using the parameter
5322:
5080:
4903:
4788:
4724:
4612:
4512:
4442:
4362:
4263:
4149:
4056:
3847:
3814:is a parameter, the
3617:
3523:
3503:generalized momentum
3497:Generalized momentum
3338:
3181:
3046:
2792:
2772:homogeneous function
2520:
2375:
2287:
2079:
1873:
1790:
1622:
1454:
1437:holonomic constraint
1189:
1071:Open curved surface
1042:Lagrange's equations
796:analytical mechanics
574:Angular acceleration
566:Rotational frequency
346:Lagrangian mechanics
339:Analytical mechanics
95:Second law of motion
48:
10067:Classical Mechanics
10026:Classical Mechanics
9957:Eric W. Weisstein,
8083:
7768:
7575:
7551:
7513:
7489:
6120:
5936:
4014:consists of a mass
3509:to" the coordinate
2812:
1742:configuration space
1442:constraint equation
1001:Closed curved path
900:Closed curved path
826:equations of motion
804:configuration space
426:Harmonic oscillator
404:Equations of motion
39:Classical mechanics
33:Part of a series on
10018:Goldstein, Herbert
9946:Newtonian Dynamics
9690:Generalized forces
9635:
9527:
9382:
9242:
9004:
8795:
8660:for any variation
8570:
8469:
8377:spherical pendulum
8373:
8363:Spherical pendulum
8332:
8069:
8036:
7754:
7685:
7552:
7537:
7490:
7475:
7307:
7134:
6970:
6665:
6547:
6434:
6127:
6106:
5943:
5922:
5764:
5719:
5656:
5581:
5456:
5302:
5049:
4883:
4764:
4678:
4592:
4473:
4397:
4331:
4227:
4125:
4004:
3996:
3934:
3774:
3733:
3608:conserved quantity
3591:If the Lagrangian
3578:
3483:
3299:
3142:
2994:
2798:
2724:
2487:in which · is the
2474:
2337:
2267:can be treated as
2233:
2029:
1844:
1726:
1490:
1400:
1398:
862:Open straight path
830:degrees of freedom
742:Physics portal
356:Routhian mechanics
231:Frame of reference
83:
10155:Dynamical systems
10009:978-0-521-88303-0
9807:978-3-642-05369-6
9497:
9487:
9352:
9217:
9111:
8984:
8924:
8240:
8146:
8094:
7925:
7831:
7779:
7673:
7654:
7562:
7525:
7500:
7444:
7394:
7344:
5702:
5688:
5626:
5563:
5540:
5536:
5511:
5374:
5337:
5297:
5258:
5235:
5231:
5206:
5187:
5148:
5125:
5121:
5096:
5037:
5012:
4989:
4967:
4949:
4920:
4877:
4826:
4811:
4711:generalized force
4218:
4197:
4161:
3714:
3699:
3689:
3663:
3645:
3573:
3563:
3470:
3421:
3389:
3364:
3286:
3264:
3232:
3207:
3129:
3097:
3072:
3026:polar coordinates
2957:
2918:
2712:
2693:
2676:
2637:
2559:
2535:
2461:
2437:
2392:
2315:
2220:
2208:
2148:
2109:
2008:
1997:
1988:
1965:
1954:
1931:
1913:
1887:
1755:degree of freedom
1703:
1694:
1669:
974:Open curved path
874:Open curved path
792:
791:
539:Centrifugal force
534:Centripetal force
490:Euler's equations
475:Relative velocity
251:Moment of inertia
81:
55:
16:(Redirected from
10172:
10136:
10117:
10111:
10103:
10084:
10058:
10039:
10013:
9982:
9979:
9973:
9968:
9962:
9955:
9949:
9942:
9936:
9935:
9917:
9911:
9905:
9899:
9893:
9887:
9881:
9875:
9869:
9863:
9857:
9851:
9845:
9839:
9833:
9824:
9818:
9812:
9811:
9789:
9783:
9782:
9760:
9754:
9745:
9729:
9725:
9713:
9711:
9706:
9685:Stiffness matrix
9644:
9642:
9641:
9636:
9601:
9600:
9561:
9550:
9536:
9534:
9533:
9528:
9498:
9496:
9495:
9494:
9489:
9488:
9480:
9472:
9471:
9470:
9465:
9455:
9450:
9449:
9444:
9437:
9432:
9414:
9413:
9391:
9389:
9388:
9383:
9353:
9351:
9350:
9349:
9336:
9335:
9334:
9329:
9319:
9314:
9313:
9308:
9301:
9296:
9278:
9277:
9258:
9251:
9249:
9248:
9243:
9238:
9237:
9232:
9223:
9219:
9218:
9216:
9215:
9214:
9201:
9200:
9199:
9194:
9184:
9179:
9178:
9173:
9166:
9161:
9132:
9131:
9126:
9117:
9113:
9112:
9110:
9109:
9108:
9095:
9094:
9093:
9088:
9078:
9073:
9072:
9067:
9060:
9055:
9013:
9011:
9010:
9005:
9000:
8999:
8994:
8985:
8983:
8982:
8981:
8968:
8967:
8966:
8961:
8951:
8940:
8939:
8934:
8925:
8923:
8922:
8921:
8908:
8907:
8906:
8901:
8891:
8886:
8885:
8880:
8861:
8843:
8828:
8804:
8802:
8801:
8796:
8791:
8790:
8785:
8773:
8772:
8767:
8760:
8755:
8721:
8701:
8678:
8666:
8659:
8636:
8620:
8604:
8594:
8579:
8577:
8576:
8571:
8496:
8478:
8476:
8475:
8470:
8458:
8457:
8445:
8444:
8432:
8431:
8419:
8418:
8403:
8382:
8358:
8341:
8339:
8338:
8333:
8328:
8327:
8312:
8311:
8299:
8298:
8280:
8279:
8267:
8266:
8248:
8247:
8242:
8241:
8236:
8235:
8226:
8222:
8221:
8212:
8211:
8202:
8201:
8186:
8185:
8173:
8172:
8154:
8153:
8148:
8147:
8139:
8135:
8134:
8125:
8124:
8115:
8114:
8102:
8101:
8096:
8095:
8087:
8082:
8077:
8068:
8067:
8045:
8043:
8042:
8037:
8032:
8031:
8016:
8015:
8000:
7999:
7987:
7986:
7965:
7964:
7952:
7951:
7933:
7932:
7927:
7926:
7921:
7920:
7911:
7907:
7906:
7897:
7896:
7887:
7886:
7871:
7870:
7858:
7857:
7839:
7838:
7833:
7832:
7824:
7820:
7819:
7810:
7809:
7800:
7799:
7787:
7786:
7781:
7780:
7772:
7767:
7762:
7750:
7749:
7737:
7736:
7714:
7694:
7692:
7691:
7686:
7681:
7680:
7675:
7674:
7666:
7662:
7661:
7656:
7655:
7647:
7640:
7639:
7627:
7626:
7608:
7607:
7598:
7597:
7588:
7587:
7574:
7569:
7564:
7563:
7555:
7550:
7545:
7536:
7535:
7526:
7518:
7512:
7507:
7502:
7501:
7493:
7488:
7483:
7471:
7470:
7458:
7457:
7445:
7437:
7432:
7431:
7426:
7417:
7416:
7411:
7405:
7404:
7395:
7387:
7382:
7381:
7376:
7367:
7366:
7361:
7355:
7354:
7345:
7337:
7316:
7314:
7313:
7308:
7303:
7302:
7287:
7286:
7274:
7273:
7258:
7257:
7256:
7255:
7237:
7236:
7221:
7220:
7205:
7204:
7192:
7191:
7173:
7172:
7171:
7170:
7143:
7141:
7140:
7135:
7130:
7129:
7117:
7116:
7101:
7100:
7088:
7087:
7075:
7074:
7062:
7061:
7046:
7045:
7030:
7029:
7017:
7016:
6979:
6977:
6976:
6971:
6969:
6968:
6953:
6952:
6937:
6936:
6924:
6923:
6908:
6907:
6892:
6891:
6876:
6875:
6860:
6859:
6847:
6846:
6831:
6830:
6815:
6814:
6809:
6796:
6795:
6780:
6779:
6764:
6763:
6751:
6750:
6735:
6734:
6719:
6718:
6713:
6694:
6674:
6672:
6671:
6666:
6661:
6660:
6655:
6643:
6642:
6637:
6628:
6627:
6622:
6610:
6609:
6604:
6579:
6563:
6556:
6554:
6553:
6548:
6540:
6539:
6515:
6514:
6509:
6493:
6492:
6468:
6467:
6462:
6443:
6441:
6440:
6435:
6427:
6426:
6411:
6410:
6395:
6394:
6379:
6378:
6360:
6359:
6344:
6343:
6328:
6327:
6312:
6311:
6296:
6295:
6290:
6277:
6276:
6261:
6260:
6245:
6244:
6229:
6228:
6213:
6212:
6207:
6191:
6174:
6160:
6136:
6134:
6133:
6128:
6119:
6114:
6099:
6098:
6093:
6084:
6083:
6078:
6063:
6062:
6057:
6048:
6047:
6042:
6027:
6026:
6014:
6013:
6001:
6000:
5988:
5987:
5975:
5974:
5952:
5950:
5949:
5944:
5935:
5930:
5918:
5917:
5912:
5903:
5902:
5897:
5885:
5884:
5872:
5871:
5859:
5858:
5846:
5845:
5833:
5832:
5813:
5783:
5746:
5742:
5738:
5728:
5726:
5725:
5720:
5703:
5695:
5690:
5689:
5681:
5665:
5663:
5662:
5657:
5628:
5627:
5619:
5616:
5615:
5590:
5588:
5587:
5582:
5577:
5576:
5564:
5562:
5554:
5546:
5541:
5539:
5538:
5537:
5529:
5522:
5514:
5512:
5510:
5499:
5487:
5480:
5476:
5472:
5465:
5463:
5462:
5457:
5446:
5445:
5433:
5432:
5420:
5419:
5376:
5375:
5367:
5339:
5338:
5330:
5311:
5309:
5308:
5303:
5298:
5296:
5288:
5280:
5272:
5271:
5259:
5257:
5249:
5241:
5236:
5234:
5233:
5232:
5224:
5217:
5209:
5207:
5205:
5194:
5188:
5186:
5178:
5170:
5162:
5161:
5149:
5147:
5139:
5131:
5126:
5124:
5123:
5122:
5114:
5107:
5099:
5097:
5095:
5084:
5072:
5068:
5058:
5056:
5055:
5050:
5045:
5044:
5039:
5038:
5030:
5026:
5025:
5013:
5005:
4997:
4996:
4991:
4990:
4982:
4975:
4974:
4969:
4968:
4960:
4950:
4942:
4937:
4929:
4921:
4913:
4892:
4890:
4889:
4884:
4879:
4878:
4870:
4828:
4827:
4819:
4813:
4812:
4804:
4795:
4780:
4773:
4771:
4770:
4765:
4736:
4735:
4716:
4709:is known as the
4708:
4701:
4697:
4687:
4685:
4684:
4679:
4601:
4599:
4598:
4593:
4522:
4504:
4500:
4496:
4492:
4482:
4480:
4479:
4474:
4469:
4458:
4434:
4428:
4413:
4406:
4404:
4403:
4398:
4369:
4354:
4347:
4340:
4338:
4337:
4332:
4270:
4255:
4251:
4247:
4243:
4236:
4234:
4233:
4228:
4220:
4219:
4211:
4199:
4198:
4190:
4163:
4162:
4154:
4141:
4134:
4132:
4131:
4126:
4115:
4114:
4102:
4101:
4089:
4088:
4048:
4044:
4040:
4036:
4021:
4017:
3993:
3987:
3981:
3950:
3943:
3941:
3940:
3935:
3885:
3860:
3836:
3832:
3828:
3820:
3813:
3809:
3786:
3771:
3765:
3742:
3740:
3739:
3734:
3722:
3721:
3716:
3715:
3707:
3700:
3698:
3697:
3696:
3691:
3690:
3682:
3674:
3666:
3664:
3662:
3651:
3646:
3644:
3643:
3642:
3629:
3621:
3605:
3594:
3587:
3585:
3584:
3579:
3574:
3572:
3571:
3570:
3565:
3564:
3556:
3548:
3540:
3535:
3534:
3515:
3492:
3490:
3489:
3484:
3478:
3477:
3472:
3471:
3463:
3452:
3451:
3442:
3441:
3429:
3428:
3423:
3422:
3414:
3410:
3409:
3397:
3396:
3391:
3390:
3382:
3375:
3374:
3369:
3365:
3363:
3355:
3347:
3330:
3308:
3306:
3305:
3300:
3294:
3293:
3288:
3287:
3279:
3272:
3271:
3266:
3265:
3257:
3253:
3252:
3240:
3239:
3234:
3233:
3225:
3218:
3217:
3212:
3208:
3206:
3198:
3190:
3173:
3151:
3149:
3148:
3143:
3137:
3136:
3131:
3130:
3122:
3118:
3117:
3105:
3104:
3099:
3098:
3090:
3083:
3082:
3077:
3073:
3071:
3063:
3055:
3038:
3016:
3012:
3003:
3001:
3000:
2995:
2989:
2988:
2976:
2975:
2963:
2959:
2958:
2956:
2955:
2954:
2941:
2940:
2939:
2934:
2924:
2919:
2917:
2916:
2915:
2902:
2901:
2900:
2895:
2885:
2877:
2872:
2848:
2847:
2842:
2830:
2829:
2824:
2811:
2806:
2784:
2766:
2733:
2731:
2730:
2725:
2720:
2719:
2714:
2713:
2705:
2701:
2700:
2695:
2694:
2686:
2682:
2678:
2677:
2675:
2674:
2673:
2660:
2659:
2658:
2653:
2643:
2638:
2636:
2635:
2634:
2621:
2620:
2619:
2614:
2604:
2596:
2591:
2567:
2566:
2561:
2560:
2555:
2550:
2543:
2542:
2537:
2536:
2531:
2526:
2512:
2501:
2483:
2481:
2480:
2475:
2469:
2468:
2463:
2462:
2457:
2452:
2445:
2444:
2439:
2438:
2433:
2428:
2424:
2423:
2413:
2408:
2393:
2385:
2346:
2344:
2343:
2338:
2317:
2316:
2311:
2306:
2300:
2266:
2253:
2242:
2240:
2239:
2234:
2228:
2227:
2222:
2221:
2213:
2209:
2207:
2206:
2205:
2192:
2191:
2190:
2185:
2175:
2172:
2167:
2149:
2147:
2139:
2138:
2137:
2132:
2122:
2117:
2116:
2111:
2110:
2105:
2100:
2093:
2092:
2087:
2071:
2059:total derivative
2056:
2038:
2036:
2035:
2030:
2016:
2015:
2010:
2009:
2001:
1995:
1986:
1973:
1972:
1967:
1966:
1958:
1952:
1939:
1938:
1933:
1932:
1924:
1914:
1912:
1904:
1903:
1894:
1889:
1888:
1883:
1878:
1865:
1853:
1851:
1850:
1845:
1827:
1819:
1818:
1813:
1804:
1803:
1798:
1782:
1778:
1774:
1760:
1735:
1733:
1732:
1727:
1713:
1712:
1701:
1692:
1679:
1678:
1667:
1654:
1653:
1629:
1611:
1607:
1589:
1575:
1571:
1567:
1553:
1550:quantities, but
1549:
1539:
1535:
1527:
1520:
1513:
1506:
1499:
1497:
1496:
1491:
1474:
1473:
1468:
1447:
1433:
1423:
1409:
1407:
1406:
1401:
1399:
1392:
1391:
1379:
1378:
1366:
1365:
1350:
1349:
1344:
1337:
1325:
1315:
1314:
1302:
1301:
1289:
1288:
1273:
1272:
1267:
1260:
1250:
1249:
1237:
1236:
1224:
1223:
1208:
1207:
1202:
1195:
1166:particles in 3D
1165:
1162:For a system of
1155:
1136:
1119:
1098:
1089:
1068:
1033:
1029:
1025:
1015:
998:
988:
971:
956:
952:
948:
944:
940:
928:
924:
914:
897:
888:
871:
859:
784:
777:
770:
757:
752:
751:
744:
740:
739:
645:Johann Bernoulli
640:Daniel Bernoulli
561:Tangential speed
465:
441:
416:Fictitious force
411:
263:Mechanical power
253:
194:Angular momentum
92:
90:
89:
84:
82:
80:
72:
71:
62:
57:
56:
30:
21:
10180:
10179:
10175:
10174:
10173:
10171:
10170:
10169:
10140:
10139:
10133:
10120:
10104:
10100:
10087:
10081:
10061:
10055:
10042:
10036:
10016:
10010:
9993:
9990:
9985:
9980:
9976:
9969:
9965:
9959:Double Pendulum
9956:
9952:
9943:
9939:
9932:
9919:
9918:
9914:
9906:
9902:
9894:
9890:
9882:
9878:
9870:
9866:
9858:
9854:
9846:
9842:
9834:
9827:
9819:
9815:
9808:
9791:
9790:
9786:
9779:
9762:
9761:
9757:
9746:
9742:
9738:
9733:
9732:
9726:
9722:
9717:
9716:
9709:
9707:
9703:
9698:
9651:
9592:
9570:
9569:
9560:
9552:
9549:
9541:
9477:
9473:
9460:
9456:
9439:
9405:
9400:
9399:
9341:
9337:
9324:
9320:
9303:
9269:
9264:
9263:
9256:
9227:
9206:
9202:
9189:
9185:
9168:
9146:
9142:
9121:
9100:
9096:
9083:
9079:
9062:
9040:
9036:
9022:
9021:
8989:
8973:
8969:
8956:
8952:
8929:
8913:
8909:
8896:
8892:
8875:
8867:
8866:
8850:
8845:
8842:
8833:
8818:
8809:
8780:
8762:
8727:
8726:
8711:
8703:
8691:
8683:
8676:
8668:
8661:
8653:
8643:
8622:
8610:
8600:
8584:
8487:
8486:
8449:
8436:
8423:
8410:
8388:
8387:
8380:
8365:
8351:
8346:
8319:
8303:
8290:
8271:
8258:
8227:
8223:
8213:
8203:
8193:
8177:
8164:
8136:
8126:
8116:
8106:
8084:
8059:
8054:
8053:
8023:
8007:
7991:
7978:
7956:
7943:
7912:
7908:
7898:
7888:
7878:
7862:
7849:
7821:
7811:
7801:
7791:
7769:
7741:
7728:
7720:
7719:
7707:
7702:
7663:
7644:
7631:
7618:
7599:
7589:
7579:
7527:
7462:
7449:
7421:
7406:
7396:
7371:
7356:
7346:
7325:
7324:
7294:
7278:
7265:
7247:
7242:
7228:
7212:
7196:
7183:
7162:
7157:
7152:
7151:
7121:
7108:
7092:
7079:
7066:
7053:
7037:
7021:
7008:
6988:
6987:
6960:
6944:
6928:
6915:
6899:
6883:
6867:
6851:
6838:
6822:
6804:
6787:
6771:
6755:
6742:
6726:
6708:
6700:
6699:
6688:
6679:
6678:The variations
6650:
6632:
6617:
6599:
6585:
6584:
6573:
6565:
6561:
6531:
6504:
6484:
6457:
6452:
6451:
6418:
6402:
6386:
6370:
6351:
6335:
6319:
6303:
6285:
6268:
6252:
6236:
6220:
6202:
6197:
6196:
6184:
6179:
6167:
6162:
6153:
6146:
6141:
6088:
6073:
6052:
6037:
6018:
6005:
5992:
5979:
5966:
5961:
5960:
5907:
5892:
5876:
5863:
5850:
5837:
5824:
5819:
5818:
5806:
5799:
5793:
5785:
5776:
5771:
5768:double pendulum
5761:double pendulum
5753:
5751:Double pendulum
5744:
5740:
5736:
5674:
5673:
5607:
5599:
5598:
5594:which becomes,
5568:
5555:
5547:
5523:
5515:
5503:
5493:
5492:
5485:
5478:
5474:
5470:
5437:
5424:
5411:
5320:
5319:
5289:
5281:
5263:
5250:
5242:
5218:
5210:
5198:
5179:
5171:
5153:
5140:
5132:
5108:
5100:
5088:
5078:
5077:
5070:
5066:
5027:
5017:
4979:
4957:
4901:
4900:
4786:
4785:
4778:
4727:
4722:
4721:
4714:
4703:
4699:
4692:
4610:
4609:
4510:
4509:
4502:
4498:
4494:
4487:
4440:
4439:
4430:
4426:
4411:
4360:
4359:
4352:
4345:
4261:
4260:
4253:
4249:
4245:
4241:
4147:
4146:
4139:
4106:
4093:
4080:
4054:
4053:
4046:
4042:
4038:
4023:
4019:
4015:
3989:
3983:
3968:
3961:
3959:Simple pendulum
3948:
3845:
3844:
3834:
3830:
3826:
3818:
3811:
3788:
3777:
3767:
3761:
3754:
3749:
3704:
3679:
3675:
3667:
3655:
3634:
3630:
3622:
3615:
3614:
3604:
3600:
3592:
3553:
3549:
3541:
3526:
3521:
3520:
3514:
3510:
3499:
3460:
3443:
3433:
3411:
3401:
3379:
3356:
3348:
3342:
3341:
3336:
3335:
3316:
3276:
3254:
3244:
3222:
3199:
3191:
3185:
3184:
3179:
3178:
3159:
3119:
3109:
3087:
3064:
3056:
3050:
3049:
3044:
3043:
3028:
3014:
3008:
2980:
2967:
2946:
2942:
2929:
2925:
2907:
2903:
2890:
2886:
2883:
2879:
2837:
2819:
2790:
2789:
2782:
2738:
2702:
2683:
2665:
2661:
2648:
2644:
2626:
2622:
2609:
2605:
2602:
2598:
2547:
2523:
2518:
2517:
2511:
2503:
2500:
2492:
2449:
2425:
2415:
2373:
2372:
2362:
2357:
2285:
2284:
2277:
2260:
2255:
2254:and velocities
2252:
2248:
2210:
2197:
2193:
2180:
2176:
2140:
2127:
2123:
2097:
2082:
2077:
2076:
2070:
2062:
2055:
2047:
2044:time derivative
1998:
1955:
1921:
1905:
1895:
1871:
1870:
1861:
1808:
1793:
1788:
1787:
1780:
1776:
1773:
1765:
1758:
1704:
1670:
1645:
1620:
1619:
1609:
1600:
1595:
1577:
1573:
1569:
1555:
1551:
1544:
1537:
1533:
1526:
1522:
1519:
1515:
1512:
1508:
1504:
1463:
1452:
1451:
1445:
1425:
1422:
1414:
1397:
1396:
1383:
1370:
1357:
1339:
1335:
1334:
1323:
1322:
1306:
1293:
1280:
1262:
1258:
1257:
1241:
1228:
1215:
1197:
1187:
1186:
1172:position vector
1163:
1160:
1159:
1158:
1157:
1141:
1126:
1122:
1121:
1120:
1102:
1099:
1091:
1090:
1072:
1069:
1058:
1038:
1037:
1036:
1035:
1031:
1027:
1023:
1022:The arc length
1019:
1018:
1017:
1002:
999:
991:
990:
975:
972:
961:
960:
959:
958:
954:
950:
946:
942:
930:
926:
922:
917:
916:
915:
901:
898:
890:
889:
875:
872:
864:
863:
860:
849:
832:of the system.
788:
747:
734:
733:
726:
725:
724:
599:
591:
590:
570:
524:Circular motion
518:
508:
507:
506:
463:
433:
430:
409:
388:
380:
379:
376:
375:
333:
323:
315:
314:
313:
272:
268:Mechanical work
261:
245:
183:
175:
174:
173:
128:
120:
97:
73:
63:
46:
45:
28:
23:
22:
15:
12:
11:
5:
10178:
10176:
10168:
10167:
10162:
10157:
10152:
10142:
10141:
10138:
10137:
10131:
10118:
10099:978-0750628969
10098:
10085:
10079:
10059:
10054:978-0521575720
10053:
10040:
10034:
10022:Poole, Charles
10014:
10008:
9989:
9986:
9984:
9983:
9974:
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9950:
9937:
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9912:
9900:
9888:
9876:
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9852:
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9806:
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9777:
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8766:
8759:
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8751:
8748:
8744:
8740:
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8734:
8707:
8687:
8672:
8642:
8639:
8607:point particle
8581:
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8119:
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7306:
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7268:
7264:
7261:
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7240:
7235:
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7227:
7224:
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7211:
7208:
7203:
7199:
7195:
7190:
7186:
7182:
7179:
7176:
7169:
7165:
7160:
7145:
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7133:
7128:
7124:
7120:
7115:
7111:
7107:
7104:
7099:
7095:
7091:
7086:
7082:
7078:
7073:
7069:
7065:
7060:
7056:
7052:
7049:
7044:
7040:
7036:
7033:
7028:
7024:
7020:
7015:
7011:
7007:
7004:
7001:
6998:
6995:
6981:
6980:
6967:
6963:
6959:
6956:
6951:
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6943:
6940:
6935:
6931:
6927:
6922:
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6911:
6906:
6902:
6898:
6895:
6890:
6886:
6882:
6879:
6874:
6870:
6866:
6863:
6858:
6854:
6850:
6845:
6841:
6837:
6834:
6829:
6825:
6821:
6818:
6813:
6808:
6803:
6799:
6794:
6790:
6786:
6783:
6778:
6774:
6770:
6767:
6762:
6758:
6754:
6749:
6745:
6741:
6738:
6733:
6729:
6725:
6722:
6717:
6712:
6707:
6684:
6676:
6675:
6664:
6659:
6654:
6649:
6646:
6641:
6636:
6631:
6626:
6621:
6616:
6613:
6608:
6603:
6598:
6595:
6592:
6569:
6558:
6557:
6546:
6543:
6538:
6534:
6530:
6527:
6524:
6521:
6518:
6513:
6508:
6502:
6499:
6496:
6491:
6487:
6483:
6480:
6477:
6474:
6471:
6466:
6461:
6445:
6444:
6433:
6430:
6425:
6421:
6417:
6414:
6409:
6405:
6401:
6398:
6393:
6389:
6385:
6382:
6377:
6373:
6369:
6366:
6363:
6358:
6354:
6350:
6347:
6342:
6338:
6334:
6331:
6326:
6322:
6318:
6315:
6310:
6306:
6302:
6299:
6294:
6289:
6283:
6280:
6275:
6271:
6267:
6264:
6259:
6255:
6251:
6248:
6243:
6239:
6235:
6232:
6227:
6223:
6219:
6216:
6211:
6206:
6182:
6165:
6151:
6144:
6138:
6137:
6126:
6123:
6118:
6113:
6109:
6105:
6102:
6097:
6092:
6087:
6082:
6077:
6072:
6069:
6066:
6061:
6056:
6051:
6046:
6041:
6036:
6033:
6030:
6025:
6021:
6017:
6012:
6008:
6004:
5999:
5995:
5991:
5986:
5982:
5978:
5973:
5969:
5954:
5953:
5942:
5939:
5934:
5929:
5925:
5921:
5916:
5911:
5906:
5901:
5896:
5891:
5888:
5883:
5879:
5875:
5870:
5866:
5862:
5857:
5853:
5849:
5844:
5840:
5836:
5831:
5827:
5804:
5797:
5789:
5774:
5752:
5749:
5730:
5729:
5718:
5715:
5712:
5709:
5706:
5701:
5698:
5693:
5687:
5684:
5667:
5666:
5655:
5652:
5649:
5646:
5643:
5640:
5637:
5634:
5631:
5625:
5622:
5614:
5610:
5606:
5592:
5591:
5580:
5575:
5571:
5567:
5561:
5558:
5553:
5550:
5544:
5535:
5532:
5526:
5521:
5518:
5509:
5506:
5502:
5467:
5466:
5455:
5452:
5449:
5444:
5440:
5436:
5431:
5427:
5423:
5418:
5414:
5409:
5406:
5403:
5400:
5397:
5394:
5391:
5388:
5385:
5382:
5379:
5373:
5370:
5364:
5360:
5357:
5354:
5351:
5348:
5345:
5342:
5336:
5333:
5327:
5313:
5312:
5301:
5295:
5292:
5287:
5284:
5278:
5275:
5270:
5266:
5262:
5256:
5253:
5248:
5245:
5239:
5230:
5227:
5221:
5216:
5213:
5204:
5201:
5197:
5191:
5185:
5182:
5177:
5174:
5168:
5165:
5160:
5156:
5152:
5146:
5143:
5138:
5135:
5129:
5120:
5117:
5111:
5106:
5103:
5094:
5091:
5087:
5073:are given by,
5060:
5059:
5048:
5043:
5036:
5033:
5024:
5020:
5016:
5011:
5008:
5003:
5000:
4995:
4988:
4985:
4978:
4973:
4966:
4963:
4956:
4953:
4948:
4945:
4940:
4936:
4932:
4928:
4924:
4919:
4916:
4911:
4908:
4894:
4893:
4882:
4876:
4873:
4867:
4864:
4861:
4858:
4855:
4852:
4849:
4846:
4843:
4840:
4837:
4834:
4831:
4825:
4822:
4816:
4810:
4807:
4801:
4798:
4794:
4775:
4774:
4763:
4760:
4757:
4754:
4751:
4748:
4745:
4742:
4739:
4734:
4730:
4689:
4688:
4677:
4674:
4671:
4668:
4665:
4662:
4659:
4656:
4653:
4650:
4647:
4644:
4641:
4638:
4635:
4632:
4629:
4626:
4623:
4620:
4617:
4603:
4602:
4591:
4588:
4585:
4582:
4579:
4576:
4573:
4570:
4567:
4564:
4561:
4558:
4555:
4552:
4549:
4546:
4543:
4540:
4537:
4534:
4531:
4528:
4525:
4521:
4517:
4486:The variation
4484:
4483:
4472:
4468:
4464:
4461:
4457:
4453:
4450:
4447:
4408:
4407:
4396:
4393:
4390:
4387:
4384:
4381:
4378:
4375:
4372:
4368:
4342:
4341:
4330:
4327:
4324:
4321:
4318:
4315:
4312:
4309:
4306:
4303:
4300:
4297:
4294:
4291:
4288:
4285:
4282:
4279:
4276:
4273:
4269:
4238:
4237:
4226:
4223:
4217:
4214:
4208:
4205:
4202:
4196:
4193:
4187:
4184:
4181:
4178:
4175:
4172:
4169:
4166:
4160:
4157:
4136:
4135:
4124:
4121:
4118:
4113:
4109:
4105:
4100:
4096:
4092:
4087:
4083:
4079:
4076:
4073:
4070:
4067:
4064:
4061:
3960:
3957:
3945:
3944:
3933:
3930:
3927:
3924:
3921:
3918:
3915:
3912:
3909:
3906:
3903:
3900:
3897:
3894:
3891:
3888:
3884:
3879:
3875:
3872:
3869:
3866:
3863:
3859:
3855:
3852:
3753:
3752:Bead on a wire
3750:
3748:
3745:
3744:
3743:
3732:
3728:
3725:
3720:
3713:
3710:
3703:
3695:
3688:
3685:
3678:
3673:
3670:
3661:
3658:
3654:
3649:
3641:
3637:
3633:
3628:
3625:
3602:
3589:
3588:
3577:
3569:
3562:
3559:
3552:
3547:
3544:
3538:
3533:
3529:
3516:is defined by
3512:
3498:
3495:
3494:
3493:
3482:
3476:
3469:
3466:
3458:
3455:
3450:
3446:
3440:
3436:
3432:
3427:
3420:
3417:
3408:
3404:
3400:
3395:
3388:
3385:
3378:
3373:
3368:
3362:
3359:
3354:
3351:
3345:
3310:
3309:
3298:
3292:
3285:
3282:
3275:
3270:
3263:
3260:
3251:
3247:
3243:
3238:
3231:
3228:
3221:
3216:
3211:
3205:
3202:
3197:
3194:
3188:
3153:
3152:
3141:
3135:
3128:
3125:
3116:
3112:
3108:
3103:
3096:
3093:
3086:
3081:
3076:
3070:
3067:
3062:
3059:
3053:
3005:
3004:
2993:
2987:
2983:
2979:
2974:
2970:
2966:
2962:
2953:
2949:
2945:
2938:
2933:
2928:
2922:
2914:
2910:
2906:
2899:
2894:
2889:
2882:
2876:
2871:
2868:
2865:
2862:
2859:
2855:
2851:
2846:
2841:
2836:
2833:
2828:
2823:
2818:
2815:
2810:
2805:
2801:
2797:
2735:
2734:
2723:
2718:
2711:
2708:
2699:
2692:
2689:
2681:
2672:
2668:
2664:
2657:
2652:
2647:
2641:
2633:
2629:
2625:
2618:
2613:
2608:
2601:
2595:
2590:
2587:
2584:
2581:
2578:
2574:
2570:
2565:
2558:
2554:
2546:
2541:
2534:
2530:
2507:
2496:
2485:
2484:
2473:
2467:
2460:
2456:
2448:
2443:
2436:
2432:
2422:
2418:
2412:
2407:
2404:
2401:
2397:
2391:
2388:
2383:
2380:
2366:kinetic energy
2361:
2360:Kinetic energy
2358:
2356:
2353:
2348:
2347:
2336:
2332:
2329:
2326:
2323:
2320:
2314:
2310:
2303:
2299:
2295:
2292:
2276:
2273:
2258:
2250:
2244:
2243:
2232:
2226:
2219:
2216:
2204:
2200:
2196:
2189:
2184:
2179:
2171:
2166:
2163:
2160:
2156:
2152:
2146:
2143:
2136:
2131:
2126:
2120:
2115:
2108:
2104:
2096:
2091:
2086:
2066:
2051:
2040:
2039:
2028:
2025:
2022:
2019:
2014:
2007:
2004:
1994:
1991:
1985:
1982:
1979:
1976:
1971:
1964:
1961:
1951:
1948:
1945:
1942:
1937:
1930:
1927:
1920:
1917:
1911:
1908:
1902:
1898:
1892:
1886:
1882:
1855:
1854:
1843:
1839:
1836:
1833:
1830:
1826:
1822:
1817:
1812:
1807:
1802:
1797:
1769:
1737:
1736:
1725:
1722:
1719:
1716:
1711:
1707:
1700:
1697:
1691:
1688:
1685:
1682:
1677:
1673:
1666:
1663:
1660:
1657:
1652:
1648:
1644:
1641:
1638:
1635:
1632:
1628:
1598:
1524:
1517:
1510:
1501:
1500:
1489:
1486:
1483:
1480:
1477:
1472:
1467:
1462:
1459:
1418:
1411:
1410:
1395:
1390:
1386:
1382:
1377:
1373:
1369:
1364:
1360:
1356:
1353:
1348:
1343:
1338:
1336:
1333:
1328:
1326:
1324:
1321:
1318:
1313:
1309:
1305:
1300:
1296:
1292:
1287:
1283:
1279:
1276:
1271:
1266:
1261:
1259:
1256:
1253:
1248:
1244:
1240:
1235:
1231:
1227:
1222:
1218:
1214:
1211:
1206:
1201:
1196:
1194:
1124:
1123:
1100:
1093:
1092:
1070:
1063:
1062:
1061:
1060:
1059:
1057:
1054:
1021:
1020:
1000:
993:
992:
973:
966:
965:
964:
963:
962:
919:
918:
899:
892:
891:
873:
866:
865:
861:
854:
853:
852:
851:
850:
848:
845:
790:
789:
787:
786:
779:
772:
764:
761:
760:
759:
758:
745:
728:
727:
723:
722:
717:
712:
707:
702:
697:
692:
687:
682:
677:
672:
667:
662:
657:
652:
647:
642:
637:
632:
627:
622:
617:
612:
607:
601:
600:
597:
596:
593:
592:
589:
588:
569:
568:
563:
558:
553:
551:Coriolis force
548:
547:
546:
536:
531:
526:
520:
519:
514:
513:
510:
509:
505:
504:
499:
494:
493:
492:
487:
477:
472:
467:
460:
449:
448:
447:
442:
429:
428:
423:
418:
413:
406:
401:
396:
390:
389:
386:
385:
382:
381:
378:
377:
374:
373:
368:
363:
358:
353:
348:
342:
336:
334:
327:
324:
321:
320:
317:
316:
312:
311:
306:
301:
296:
291:
286:
281:
276:
270:
265:
259:
254:
243:
238:
233:
228:
223:
222:
221:
216:
206:
201:
196:
191:
185:
184:
181:
180:
177:
176:
172:
171:
166:
161:
156:
151:
146:
141:
136:
130:
129:
126:
125:
122:
121:
119:
118:
113:
108:
102:
99:
98:
93:
79:
76:
70:
66:
60:
42:
41:
35:
34:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
10177:
10166:
10163:
10161:
10158:
10156:
10153:
10151:
10148:
10147:
10145:
10134:
10132:0-03-063366-4
10128:
10124:
10119:
10115:
10109:
10101:
10095:
10091:
10086:
10082:
10076:
10072:
10068:
10064:
10063:Kibble, T.W.B
10060:
10056:
10050:
10046:
10041:
10037:
10035:0-201-65702-3
10031:
10027:
10023:
10019:
10015:
10011:
10005:
10001:
9997:
9992:
9991:
9987:
9978:
9975:
9972:
9967:
9964:
9960:
9954:
9951:
9947:
9941:
9938:
9933:
9931:0-13-709981-9
9927:
9923:
9916:
9913:
9909:
9904:
9901:
9897:
9892:
9889:
9886:, p. 269
9885:
9880:
9877:
9873:
9868:
9865:
9861:
9856:
9853:
9850:, p. 260
9849:
9844:
9841:
9838:, p. 232
9837:
9832:
9830:
9826:
9822:
9817:
9814:
9809:
9803:
9799:
9795:
9788:
9785:
9780:
9778:0-8176-4236-6
9774:
9770:
9766:
9759:
9756:
9753:
9749:
9748:Ginsberg 2008
9744:
9741:
9735:
9724:
9721:
9705:
9702:
9695:
9691:
9688:
9686:
9683:
9681:
9678:
9676:
9673:
9671:
9668:
9666:
9663:
9661:
9658:
9656:
9653:
9652:
9648:
9632:
9629:
9626:
9623:
9620:
9617:
9614:
9611:
9608:
9605:
9602:
9597:
9593:
9584:
9581:
9578:
9575:
9568:
9567:
9566:
9563:
9559:
9555:
9548:
9544:
9524:
9521:
9518:
9515:
9512:
9509:
9506:
9503:
9499:
9491:
9484:
9481:
9467:
9451:
9446:
9434:
9429:
9426:
9423:
9419:
9415:
9410:
9406:
9398:
9397:
9396:
9379:
9376:
9373:
9370:
9367:
9364:
9361:
9358:
9354:
9346:
9342:
9331:
9315:
9310:
9298:
9293:
9290:
9287:
9283:
9279:
9274:
9270:
9262:
9261:
9260:
9239:
9234:
9229:
9224:
9220:
9211:
9207:
9196:
9180:
9175:
9163:
9158:
9155:
9152:
9148:
9143:
9139:
9136:
9133:
9128:
9123:
9118:
9114:
9105:
9101:
9090:
9074:
9069:
9057:
9052:
9049:
9046:
9042:
9037:
9033:
9030:
9027:
9020:
9019:
9018:
9001:
8996:
8991:
8986:
8978:
8974:
8963:
8947:
8944:
8941:
8936:
8931:
8926:
8918:
8914:
8903:
8887:
8882:
8872:
8865:
8864:
8863:
8859:
8855:
8851:
8841:
8837:
8830:
8826:
8822:
8817:
8813:
8792:
8787:
8777:
8774:
8769:
8757:
8752:
8749:
8746:
8742:
8738:
8735:
8732:
8725:
8724:
8723:
8719:
8715:
8710:
8706:
8699:
8695:
8690:
8686:
8680:
8675:
8671:
8665:
8657:
8651:
8650:
8647:principle of
8640:
8638:
8634:
8630:
8626:
8618:
8614:
8608:
8603:
8598:
8592:
8588:
8567:
8557:
8554:
8551:
8545:
8542:
8536:
8533:
8530:
8524:
8521:
8515:
8512:
8509:
8503:
8497:
8485:
8484:
8483:
8466:
8462:
8459:
8454:
8450:
8446:
8441:
8437:
8433:
8428:
8424:
8420:
8415:
8411:
8407:
8393:
8386:
8385:
8384:
8378:
8369:
8362:
8360:
8356:
8352:
8329:
8324:
8320:
8316:
8313:
8308:
8304:
8300:
8295:
8291:
8287:
8284:
8276:
8272:
8268:
8263:
8259:
8252:
8249:
8244:
8237:
8232:
8228:
8218:
8214:
8208:
8204:
8198:
8194:
8190:
8182:
8178:
8174:
8169:
8165:
8158:
8155:
8150:
8143:
8140:
8131:
8127:
8121:
8117:
8111:
8107:
8103:
8098:
8091:
8088:
8079:
8074:
8070:
8064:
8060:
8052:
8051:
8050:
8033:
8028:
8024:
8020:
8017:
8012:
8008:
8004:
7996:
7992:
7988:
7983:
7979:
7972:
7969:
7961:
7957:
7953:
7948:
7944:
7937:
7934:
7929:
7922:
7917:
7913:
7903:
7899:
7893:
7889:
7883:
7879:
7875:
7867:
7863:
7859:
7854:
7850:
7843:
7840:
7835:
7828:
7825:
7816:
7812:
7806:
7802:
7796:
7792:
7788:
7783:
7776:
7773:
7764:
7759:
7755:
7746:
7742:
7738:
7733:
7729:
7718:
7717:
7716:
7712:
7708:
7700:
7682:
7677:
7670:
7667:
7658:
7651:
7648:
7636:
7632:
7628:
7623:
7619:
7612:
7609:
7604:
7600:
7594:
7590:
7584:
7580:
7576:
7571:
7566:
7559:
7556:
7547:
7542:
7538:
7532:
7528:
7522:
7519:
7514:
7509:
7504:
7497:
7494:
7485:
7480:
7476:
7467:
7463:
7459:
7454:
7450:
7441:
7438:
7433:
7428:
7418:
7413:
7401:
7397:
7391:
7388:
7383:
7378:
7368:
7363:
7351:
7347:
7341:
7338:
7333:
7330:
7323:
7322:
7321:
7304:
7299:
7295:
7291:
7288:
7283:
7279:
7275:
7270:
7266:
7262:
7259:
7252:
7248:
7243:
7238:
7233:
7229:
7225:
7222:
7217:
7213:
7209:
7201:
7197:
7193:
7188:
7184:
7177:
7174:
7167:
7163:
7158:
7150:
7149:
7148:
7131:
7126:
7122:
7118:
7113:
7109:
7105:
7102:
7097:
7093:
7089:
7084:
7080:
7076:
7071:
7067:
7063:
7058:
7054:
7050:
7047:
7042:
7038:
7034:
7026:
7022:
7018:
7013:
7009:
7002:
6999:
6996:
6993:
6986:
6985:
6984:
6965:
6961:
6957:
6949:
6945:
6941:
6938:
6933:
6929:
6925:
6920:
6916:
6912:
6909:
6904:
6900:
6893:
6888:
6884:
6880:
6872:
6868:
6864:
6861:
6856:
6852:
6848:
6843:
6839:
6835:
6832:
6827:
6823:
6816:
6811:
6801:
6797:
6792:
6788:
6784:
6776:
6772:
6768:
6765:
6760:
6756:
6752:
6747:
6743:
6739:
6736:
6731:
6727:
6720:
6715:
6705:
6698:
6697:
6696:
6692:
6687:
6683:
6662:
6657:
6647:
6644:
6639:
6629:
6624:
6614:
6611:
6606:
6596:
6593:
6590:
6583:
6582:
6581:
6577:
6572:
6568:
6541:
6536:
6532:
6528:
6525:
6522:
6516:
6511:
6500:
6494:
6489:
6485:
6481:
6478:
6475:
6469:
6464:
6450:
6449:
6448:
6431:
6423:
6419:
6415:
6412:
6407:
6403:
6399:
6396:
6391:
6387:
6383:
6380:
6375:
6371:
6364:
6356:
6352:
6348:
6345:
6340:
6336:
6332:
6329:
6324:
6320:
6316:
6313:
6308:
6304:
6297:
6292:
6281:
6273:
6269:
6265:
6262:
6257:
6253:
6249:
6246:
6241:
6237:
6233:
6230:
6225:
6221:
6214:
6209:
6195:
6194:
6193:
6189:
6185:
6176:
6172:
6168:
6158:
6154:
6147:
6124:
6121:
6116:
6111:
6107:
6103:
6095:
6085:
6080:
6067:
6059:
6049:
6044:
6031:
6023:
6019:
6015:
6010:
6006:
6002:
5997:
5993:
5989:
5984:
5980:
5971:
5967:
5959:
5958:
5957:
5940:
5937:
5932:
5927:
5923:
5919:
5914:
5904:
5899:
5889:
5881:
5877:
5873:
5868:
5864:
5860:
5855:
5851:
5847:
5842:
5838:
5829:
5825:
5817:
5816:
5815:
5811:
5807:
5800:
5792:
5788:
5781:
5777:
5769:
5762:
5757:
5750:
5748:
5733:
5716:
5713:
5710:
5707:
5704:
5699:
5696:
5691:
5685:
5682:
5672:
5671:
5670:
5653:
5650:
5647:
5644:
5641:
5638:
5635:
5632:
5629:
5623:
5620:
5612:
5608:
5604:
5597:
5596:
5595:
5578:
5573:
5569:
5565:
5559:
5551:
5542:
5533:
5530:
5519:
5507:
5504:
5500:
5491:
5490:
5489:
5482:
5453:
5450:
5447:
5442:
5438:
5434:
5429:
5425:
5421:
5416:
5412:
5407:
5401:
5398:
5392:
5389:
5386:
5383:
5380:
5377:
5371:
5368:
5362:
5358:
5352:
5349:
5343:
5340:
5334:
5331:
5325:
5318:
5317:
5316:
5299:
5293:
5285:
5276:
5273:
5268:
5264:
5260:
5254:
5246:
5237:
5228:
5225:
5214:
5202:
5199:
5195:
5189:
5183:
5175:
5166:
5163:
5158:
5154:
5150:
5144:
5136:
5127:
5118:
5115:
5104:
5092:
5089:
5085:
5076:
5075:
5074:
5064:
5046:
5041:
5034:
5031:
5022:
5018:
5014:
5009:
5006:
5001:
4993:
4986:
4983:
4976:
4971:
4964:
4961:
4951:
4946:
4943:
4938:
4930:
4922:
4917:
4914:
4909:
4906:
4899:
4898:
4897:
4880:
4874:
4871:
4862:
4859:
4856:
4853:
4850:
4847:
4844:
4841:
4838:
4832:
4823:
4820:
4814:
4808:
4805:
4796:
4784:
4783:
4782:
4761:
4758:
4755:
4752:
4749:
4746:
4743:
4740:
4737:
4732:
4728:
4720:
4719:
4718:
4712:
4707:
4696:
4675:
4672:
4669:
4663:
4657:
4654:
4651:
4648:
4645:
4642:
4639:
4636:
4633:
4630:
4627:
4624:
4621:
4618:
4615:
4608:
4607:
4606:
4589:
4586:
4583:
4577:
4574:
4571:
4568:
4565:
4562:
4559:
4556:
4553:
4547:
4541:
4538:
4535:
4532:
4529:
4523:
4515:
4508:
4507:
4506:
4491:
4470:
4462:
4459:
4451:
4448:
4445:
4438:
4437:
4436:
4433:
4424:
4419:
4417:
4394:
4388:
4385:
4382:
4379:
4376:
4370:
4358:
4357:
4356:
4349:
4328:
4322:
4319:
4316:
4313:
4310:
4307:
4304:
4301:
4298:
4295:
4289:
4283:
4280:
4277:
4271:
4259:
4258:
4257:
4224:
4221:
4215:
4212:
4206:
4203:
4200:
4194:
4191:
4185:
4182:
4179:
4173:
4170:
4167:
4158:
4155:
4145:
4144:
4143:
4122:
4119:
4116:
4111:
4107:
4103:
4098:
4094:
4090:
4085:
4081:
4077:
4071:
4068:
4065:
4059:
4052:
4051:
4050:
4034:
4030:
4026:
4013:
4008:
4000:
3992:
3986:
3979:
3975:
3971:
3965:
3958:
3956:
3954:
3925:
3922:
3919:
3913:
3910:
3904:
3901:
3898:
3892:
3886:
3877:
3873:
3870:
3864:
3861:
3850:
3843:
3842:
3841:
3838:
3824:
3817:
3807:
3803:
3799:
3795:
3791:
3784:
3780:
3770:
3764:
3758:
3751:
3746:
3730:
3726:
3723:
3718:
3711:
3708:
3701:
3693:
3686:
3683:
3671:
3659:
3656:
3652:
3647:
3639:
3635:
3626:
3613:
3612:
3611:
3609:
3598:
3575:
3567:
3560:
3557:
3545:
3536:
3531:
3527:
3519:
3518:
3517:
3508:
3504:
3496:
3480:
3474:
3467:
3464:
3456:
3453:
3448:
3444:
3438:
3434:
3430:
3425:
3418:
3415:
3406:
3402:
3398:
3393:
3386:
3383:
3376:
3371:
3366:
3360:
3357:
3352:
3349:
3343:
3334:
3333:
3332:
3328:
3324:
3320:
3315:
3296:
3290:
3283:
3280:
3273:
3268:
3261:
3258:
3249:
3245:
3241:
3236:
3229:
3226:
3219:
3214:
3209:
3203:
3200:
3195:
3192:
3186:
3177:
3176:
3175:
3171:
3167:
3163:
3158:
3139:
3133:
3126:
3123:
3114:
3110:
3106:
3101:
3094:
3091:
3084:
3079:
3074:
3068:
3065:
3060:
3057:
3051:
3042:
3041:
3040:
3036:
3032:
3027:
3022:
3020:
3011:
2991:
2985:
2981:
2977:
2972:
2968:
2964:
2960:
2951:
2947:
2936:
2920:
2912:
2908:
2897:
2880:
2874:
2869:
2866:
2863:
2860:
2857:
2853:
2849:
2844:
2834:
2831:
2826:
2816:
2813:
2808:
2803:
2799:
2795:
2788:
2787:
2786:
2780:
2775:
2773:
2768:
2764:
2760:
2756:
2753:
2749:
2745:
2741:
2721:
2716:
2709:
2706:
2697:
2690:
2687:
2679:
2670:
2666:
2655:
2639:
2631:
2627:
2616:
2599:
2593:
2588:
2585:
2582:
2579:
2576:
2572:
2568:
2563:
2556:
2544:
2539:
2532:
2516:
2515:
2514:
2510:
2506:
2499:
2495:
2490:
2471:
2465:
2458:
2446:
2441:
2434:
2420:
2416:
2410:
2405:
2402:
2399:
2395:
2389:
2386:
2381:
2378:
2371:
2370:
2369:
2367:
2359:
2354:
2352:
2334:
2330:
2327:
2321:
2318:
2312:
2301:
2290:
2283:
2282:
2281:
2274:
2272:
2270:
2265:
2261:
2230:
2224:
2217:
2214:
2202:
2198:
2187:
2169:
2164:
2161:
2158:
2154:
2150:
2144:
2141:
2134:
2124:
2118:
2113:
2106:
2094:
2089:
2075:
2074:
2073:
2069:
2065:
2060:
2054:
2050:
2045:
2020:
2012:
2005:
2002:
1992:
1989:
1983:
1977:
1969:
1962:
1959:
1949:
1943:
1935:
1928:
1925:
1915:
1909:
1906:
1896:
1890:
1884:
1869:
1868:
1867:
1864:
1858:
1841:
1831:
1815:
1805:
1800:
1786:
1785:
1784:
1772:
1768:
1762:
1756:
1752:
1748:
1744:
1743:
1717:
1709:
1705:
1698:
1695:
1689:
1683:
1675:
1671:
1664:
1658:
1650:
1646:
1639:
1633:
1618:
1617:
1616:
1615:
1605:
1601:
1593:
1588:
1584:
1580:
1566:
1562:
1558:
1548:
1541:
1531:
1487:
1484:
1478:
1475:
1470:
1457:
1450:
1449:
1448:
1443:
1439:
1438:
1432:
1428:
1421:
1417:
1388:
1384:
1380:
1375:
1371:
1367:
1362:
1358:
1351:
1346:
1331:
1327:
1319:
1311:
1307:
1303:
1298:
1294:
1290:
1285:
1281:
1274:
1269:
1254:
1246:
1242:
1238:
1233:
1229:
1225:
1220:
1216:
1209:
1204:
1185:
1184:
1183:
1181:
1177:
1173:
1169:
1153:
1149:
1145:
1140:
1134:
1130:
1117:
1113:
1109:
1105:
1097:
1087:
1083:
1079:
1075:
1067:
1055:
1053:
1051:
1047:
1043:
1013:
1009:
1005:
997:
986:
982:
978:
970:
938:
934:
912:
908:
904:
896:
886:
882:
878:
870:
858:
846:
844:
842:
838:
833:
831:
827:
822:
819:
817:
813:
810:are the time
809:
805:
801:
797:
785:
780:
778:
773:
771:
766:
765:
763:
762:
756:
746:
743:
738:
732:
731:
730:
729:
721:
718:
716:
713:
711:
708:
706:
703:
701:
698:
696:
693:
691:
688:
686:
683:
681:
678:
676:
673:
671:
668:
666:
663:
661:
658:
656:
653:
651:
648:
646:
643:
641:
638:
636:
633:
631:
628:
626:
623:
621:
618:
616:
613:
611:
608:
606:
603:
602:
595:
594:
587:
583:
579:
575:
572:
571:
567:
564:
562:
559:
557:
554:
552:
549:
545:
542:
541:
540:
537:
535:
532:
530:
527:
525:
522:
521:
517:
512:
511:
503:
500:
498:
495:
491:
488:
486:
483:
482:
481:
478:
476:
473:
471:
468:
466:
461:
458:
454:
451:
450:
446:
443:
440:
436:
432:
431:
427:
424:
422:
419:
417:
414:
412:
407:
405:
402:
400:
397:
395:
392:
391:
384:
383:
372:
369:
367:
364:
362:
359:
357:
354:
352:
349:
347:
344:
343:
341:
340:
335:
332:
331:
326:
325:
319:
318:
310:
307:
305:
302:
300:
297:
295:
292:
290:
287:
285:
282:
280:
277:
275:
271:
269:
266:
264:
260:
258:
255:
252:
248:
244:
242:
239:
237:
234:
232:
229:
227:
224:
220:
217:
215:
212:
211:
210:
207:
205:
202:
200:
197:
195:
192:
190:
187:
186:
179:
178:
170:
167:
165:
162:
160:
157:
155:
152:
150:
147:
145:
142:
140:
137:
135:
132:
131:
124:
123:
117:
114:
112:
109:
107:
104:
103:
100:
96:
77:
74:
64:
58:
44:
43:
40:
36:
32:
31:
19:
10160:Rigid bodies
10122:
10089:
10066:
10044:
10025:
9995:
9977:
9966:
9953:
9940:
9921:
9915:
9903:
9898:, p. 25
9891:
9879:
9874:, p. 15
9867:
9862:, p. 13
9855:
9843:
9823:, p. 12
9816:
9797:
9787:
9768:
9758:
9743:
9723:
9704:
9665:Virtual work
9564:
9557:
9553:
9546:
9542:
9539:
9394:
9254:
9016:
8857:
8853:
8846:
8839:
8835:
8831:
8824:
8820:
8815:
8811:
8807:
8717:
8713:
8708:
8704:
8697:
8693:
8688:
8684:
8681:
8673:
8669:
8663:
8655:
8649:virtual work
8646:
8644:
8632:
8628:
8624:
8616:
8612:
8601:
8590:
8586:
8582:
8481:
8374:
8354:
8347:
8344:
8048:
7710:
7703:
7697:
7319:
7146:
6982:
6690:
6685:
6681:
6677:
6580:is given by
6575:
6570:
6566:
6559:
6446:
6187:
6180:
6177:
6170:
6163:
6156:
6149:
6142:
6139:
5955:
5809:
5802:
5795:
5790:
5786:
5779:
5772:
5765:
5734:
5731:
5668:
5593:
5483:
5468:
5314:
5061:
4895:
4776:
4705:
4694:
4690:
4604:
4489:
4485:
4435:is given by
4431:
4423:virtual work
4420:
4409:
4350:
4343:
4256:, such that
4239:
4137:
4032:
4028:
4024:
4009:
4005:
3990:
3984:
3977:
3973:
3969:
3952:
3946:
3839:
3822:
3805:
3801:
3797:
3793:
3789:
3782:
3778:
3775:
3768:
3762:
3596:
3590:
3502:
3500:
3326:
3322:
3318:
3311:
3169:
3165:
3161:
3154:
3034:
3030:
3023:
3009:
3006:
2779:line element
2776:
2769:
2762:
2758:
2754:
2751:
2747:
2743:
2739:
2736:
2508:
2504:
2497:
2493:
2486:
2363:
2349:
2278:
2268:
2263:
2256:
2245:
2067:
2063:
2052:
2048:
2041:
1862:
1859:
1856:
1775:of particle
1770:
1766:
1763:
1740:
1738:
1603:
1596:
1591:
1586:
1582:
1578:
1564:
1560:
1556:
1546:
1542:
1529:
1502:
1441:
1435:
1430:
1426:
1419:
1415:
1412:
1161:
1151:
1147:
1143:
1132:
1128:
1115:
1111:
1107:
1103:
1085:
1081:
1077:
1073:
1045:
1039:
1011:
1007:
1003:
984:
980:
976:
936:
932:
910:
906:
902:
884:
880:
876:
834:
823:
820:
807:
799:
793:
584: /
580: /
578:displacement
576: /
437: /
399:Displacement
337:
328:
322:Formulations
309:Virtual work
249: /
189:Acceleration
182:Fundamentals
9910:, p. 8
9680:Mass matrix
8856:= 1, 2, …,
8823:= 1, 2, …,
8716:= 1, 2, …,
8696:= 1, 2, …,
4717:, given by
4344:The use of
3810:, in which
2489:dot product
1747:arc lengths
1429:= 1, 2, …,
1052:equations.
841:phase space
812:derivatives
720:von Neumann
387:Core topics
10144:Categories
10080:1860944248
9971:Torby 1984
9884:Torby 1984
9848:Torby 1984
9736:References
3816:arc length
3019:Lagrangian
2364:The total
1050:constraint
655:d'Alembert
635:Maupertuis
598:Scientists
480:Rigid body
154:Kinematics
10108:cite book
10090:Mechanics
9624:…
9589:⇒
9576:δ
9516:…
9485:˙
9475:∂
9458:∂
9452:⋅
9420:∑
9371:…
9339:∂
9322:∂
9316:⋅
9284:∑
9225:δ
9204:∂
9187:∂
9181:⋅
9149:∑
9137:…
9119:δ
9098:∂
9081:∂
9075:⋅
9043:∑
9028:δ
8987:δ
8971:∂
8954:∂
8945:…
8927:δ
8911:∂
8894:∂
8873:δ
8778:δ
8775:⋅
8743:∑
8733:δ
8583:in which
8558:ϕ
8552:θ
8537:ϕ
8531:θ
8516:ϕ
8510:θ
8447:−
8321:θ
8317:
8288:−
8273:θ
8269:−
8260:θ
8253:
8238:˙
8229:θ
8179:θ
8175:−
8166:θ
8159:
8144:¨
8141:θ
8092:¨
8089:θ
8025:θ
8021:
7973:−
7958:θ
7954:−
7945:θ
7938:
7923:˙
7914:θ
7864:θ
7860:−
7851:θ
7844:
7829:¨
7826:θ
7777:¨
7774:θ
7715:given by
7671:˙
7668:θ
7652:˙
7649:θ
7633:θ
7629:−
7620:θ
7613:
7560:˙
7557:θ
7498:˙
7495:θ
7419:⋅
7369:⋅
7296:θ
7292:
7263:−
7249:θ
7230:θ
7226:
7178:−
7164:θ
7123:θ
7119:δ
7110:θ
7106:
7077:−
7068:θ
7064:δ
7055:θ
7051:
7003:−
6994:δ
6962:θ
6958:δ
6946:θ
6942:
6917:θ
6913:
6885:θ
6881:δ
6869:θ
6865:
6840:θ
6836:
6802:δ
6789:θ
6785:δ
6773:θ
6769:
6744:θ
6740:
6706:δ
6648:δ
6645:⋅
6615:δ
6612:⋅
6591:δ
6529:−
6482:−
6420:θ
6416:
6400:−
6388:θ
6384:
6353:θ
6349:
6333:−
6321:θ
6317:
6270:θ
6266:
6250:−
6238:θ
6234:
6104:−
6086:−
6068:⋅
6050:−
5920:−
5905:⋅
5711:θ
5708:
5686:¨
5683:θ
5651:θ
5648:
5633:−
5624:¨
5621:θ
5574:θ
5560:θ
5557:∂
5549:∂
5543:−
5534:˙
5531:θ
5525:∂
5517:∂
5435:−
5393:λ
5381:−
5372:¨
5344:λ
5335:¨
5291:∂
5283:∂
5277:λ
5252:∂
5244:∂
5238:−
5229:˙
5220:∂
5212:∂
5181:∂
5173:∂
5167:λ
5142:∂
5134:∂
5128:−
5119:˙
5110:∂
5102:∂
5035:˙
5032:θ
4987:˙
4965:˙
4931:⋅
4875:˙
4872:θ
4863:θ
4860:
4848:θ
4845:
4824:˙
4809:˙
4759:θ
4756:
4741:−
4733:θ
4673:θ
4670:δ
4664:θ
4658:
4643:−
4634:δ
4625:−
4616:δ
4587:θ
4584:δ
4578:θ
4575:
4563:θ
4560:
4539:δ
4530:δ
4516:δ
4463:δ
4460:⋅
4446:δ
4383:−
4323:θ
4320:
4311:−
4305:θ
4302:
4216:˙
4195:˙
4159:˙
4104:−
4010:A simple
3712:˙
3687:˙
3677:∂
3669:∂
3632:∂
3624:∂
3561:˙
3551:∂
3543:∂
3468:˙
3465:φ
3457:θ
3454:
3419:˙
3416:θ
3387:˙
3284:˙
3262:˙
3259:θ
3230:˙
3127:˙
3124:θ
3095:˙
2944:∂
2927:∂
2921:⋅
2905:∂
2888:∂
2854:∑
2832:⋅
2710:˙
2691:˙
2663:∂
2646:∂
2640:⋅
2624:∂
2607:∂
2573:∑
2557:˙
2545:⋅
2533:˙
2459:˙
2447:⋅
2435:˙
2396:∑
2313:˙
2218:˙
2195:∂
2178:∂
2155:∑
2107:˙
2006:˙
1990:…
1963:˙
1929:˙
1885:˙
1696:…
1332:⋮
1046:dependent
925:or angle
700:Liouville
582:frequency
502:Vibration
219:potential
144:Continuum
139:Celestial
116:Textbooks
9649:See also
8595:are the
4012:pendulum
3747:Examples
755:Category
680:Hamilton
665:Lagrange
660:Clairaut
625:Horrocks
586:velocity
556:Pendulum
544:reactive
516:Rotation
485:dynamics
435:Inertial
421:Friction
304:Velocity
279:Momentum
159:Kinetics
149:Dynamics
127:Branches
111:Timeline
8357:= 1, 2)
7713:= 1, 2)
6693:= 1, 2)
6578:= 1, 2)
6190:= 1, 2)
6173:= 1, 2)
6159:= 1, 2)
5782:= 1, 2)
4698:is the
4414:is the
2057:is the
715:Koopman
675:Poisson
670:Laplace
615:Huygens
610:Galileo
455: (
394:Damping
247:Inertia
241:Impulse
214:kinetic
164:Statics
134:Applied
106:History
10129:
10096:
10077:
10051:
10032:
10006:
9928:
9804:
9775:
9540:where
9259:terms
8808:where
6560:where
5812:= 1, 2
5784:, let
4410:where
3312:in 3D
3155:in 3D
1996:
1987:
1953:
1702:
1693:
1668:
1568:. (In
1424:where
1170:, the
753:
705:Appell
690:Cauchy
685:Jacobi
630:Halley
620:Newton
605:Kepler
457:linear
453:Motion
299:Torque
274:Moment
209:Energy
199:Couple
9696:Notes
8862:then
3980:) = 0
3785:) = 0
3595:does
1614:tuple
1440:is a
1176:tuple
1118:) = 0
1088:) = 0
1014:) = 0
987:) = 0
913:) = 0
887:) = 0
710:Gibbs
695:Routh
650:Euler
289:Speed
284:Space
226:Force
10127:ISBN
10114:link
10094:ISBN
10075:ISBN
10049:ISBN
10030:ISBN
10004:ISBN
9926:ISBN
9802:ISBN
9773:ISBN
9255:The
9017:and
8645:The
8049:and
5956:and
5743:and
5477:and
5069:and
4896:so,
4497:and
4421:The
4252:and
4045:and
3833:and
3501:The
1514:and
294:Time
257:Mass
8677:= 0
8658:= 0
8314:sin
8250:sin
8156:cos
8018:sin
7935:sin
7841:cos
7610:cos
7289:sin
7223:sin
7103:sin
7048:sin
6939:sin
6910:cos
6862:sin
6833:cos
6766:sin
6737:cos
6413:cos
6381:sin
6346:cos
6314:sin
6263:cos
6231:sin
5808:),
5794:= (
5705:sin
5669:or
5645:sin
4857:sin
4842:cos
4753:sin
4655:sin
4572:sin
4557:cos
4317:cos
4299:sin
4027:= (
3953:and
3823:one
3800:),
3792:= (
3597:not
3445:sin
2061:of
1559:= 3
1530:one
1178:in
953:or
945:or
839:on
794:In
10146::
10110:}}
10106:{{
10073:.
10020:;
10002:.
9828:^
9796:.
9767:.
9750:,
9562:.
8679:.
8631:,
8627:,
8615:,
8589:,
6148:,
6125:0.
5801:,
5759:A
5717:0.
5481:.
5473:,
4505:,
4418:.
4225:0.
4031:,
3976:,
3808:))
3331:,
3325:,
3321:,
3174:,
3168:,
3164:,
3039:,
3033:,
3021:.
3010:dt
2785:,
2767:.
2761:,
2759:dt
2750:,
2742:=
2271:.
2264:dt
2257:dq
1585:−
1583:ND
1581:=
1574:ND
1563:−
1182::
1150:,
1146:,
1131:,
1114:,
1110:,
1084:,
1080:,
1010:,
983:,
935:,
909:,
883:,
843:.
818:.
798:,
10135:.
10116:)
10102:.
10083:.
10057:.
10038:.
10012:.
9948:.
9934:.
9810:.
9781:.
9710:k
9633:.
9630:n
9627:,
9621:,
9618:1
9615:=
9612:i
9609:,
9606:0
9603:=
9598:i
9594:F
9585:0
9582:=
9579:W
9558:j
9554:F
9547:j
9543:v
9525:,
9522:n
9519:,
9513:,
9510:1
9507:=
9504:i
9500:,
9492:i
9482:q
9468:j
9463:v
9447:j
9442:F
9435:m
9430:1
9427:=
9424:j
9416:=
9411:i
9407:F
9380:,
9377:n
9374:,
9368:,
9365:1
9362:=
9359:i
9355:,
9347:i
9343:q
9332:j
9327:r
9311:j
9306:F
9299:m
9294:1
9291:=
9288:j
9280:=
9275:i
9271:F
9257:n
9240:.
9235:n
9230:q
9221:)
9212:n
9208:q
9197:j
9192:r
9176:j
9171:F
9164:m
9159:1
9156:=
9153:j
9144:(
9140:+
9134:+
9129:1
9124:q
9115:)
9106:1
9102:q
9091:j
9086:r
9070:j
9065:F
9058:m
9053:1
9050:=
9047:j
9038:(
9034:=
9031:W
9002:,
8997:n
8992:q
8979:n
8975:q
8964:j
8959:r
8948:+
8942:+
8937:1
8932:q
8919:1
8915:q
8904:j
8899:r
8888:=
8883:j
8878:r
8860:)
8858:n
8854:i
8852:(
8849:i
8847:q
8840:j
8836:r
8834:δ
8827:)
8825:m
8821:j
8819:(
8816:j
8812:r
8810:δ
8793:.
8788:j
8783:r
8770:j
8765:F
8758:m
8753:1
8750:=
8747:j
8739:=
8736:W
8720:)
8718:m
8714:j
8712:(
8709:j
8705:r
8700:)
8698:m
8694:j
8692:(
8689:j
8685:F
8674:i
8670:F
8664:r
8662:δ
8656:W
8654:δ
8635:)
8633:z
8629:y
8625:x
8623:(
8619:)
8617:φ
8613:θ
8611:(
8602:r
8593:)
8591:φ
8587:θ
8585:(
8568:,
8564:)
8561:)
8555:,
8549:(
8546:z
8543:,
8540:)
8534:,
8528:(
8525:y
8522:,
8519:)
8513:,
8507:(
8504:x
8501:(
8498:=
8494:r
8467:,
8463:0
8460:=
8455:2
8451:l
8442:2
8438:z
8434:+
8429:2
8425:y
8421:+
8416:2
8412:x
8408:=
8405:)
8401:r
8397:(
8394:f
8381:l
8355:i
8353:(
8350:i
8348:θ
8330:.
8325:2
8309:2
8305:L
8301:g
8296:2
8292:m
8285:=
8282:)
8277:1
8264:2
8256:(
8245:2
8233:1
8219:2
8215:L
8209:1
8205:L
8199:2
8195:m
8191:+
8188:)
8183:1
8170:2
8162:(
8151:1
8132:2
8128:L
8122:1
8118:L
8112:2
8108:m
8104:+
8099:2
8080:2
8075:2
8071:L
8065:2
8061:m
8034:,
8029:1
8013:1
8009:L
8005:g
8002:)
7997:2
7993:m
7989:+
7984:1
7980:m
7976:(
7970:=
7967:)
7962:2
7949:1
7941:(
7930:2
7918:2
7904:2
7900:L
7894:1
7890:L
7884:2
7880:m
7876:+
7873:)
7868:1
7855:2
7847:(
7836:2
7817:2
7813:L
7807:1
7803:L
7797:2
7793:m
7789:+
7784:1
7765:2
7760:1
7756:L
7752:)
7747:2
7743:m
7739:+
7734:1
7730:m
7726:(
7711:i
7709:(
7706:i
7704:θ
7683:.
7678:2
7659:1
7642:)
7637:1
7624:2
7616:(
7605:2
7601:L
7595:1
7591:L
7585:2
7581:m
7577:+
7572:2
7567:2
7548:2
7543:2
7539:L
7533:2
7529:m
7523:2
7520:1
7515:+
7510:2
7505:1
7486:2
7481:1
7477:L
7473:)
7468:2
7464:m
7460:+
7455:1
7451:m
7447:(
7442:2
7439:1
7434:=
7429:2
7424:v
7414:2
7409:v
7402:2
7398:m
7392:2
7389:1
7384:+
7379:1
7374:v
7364:1
7359:v
7352:1
7348:m
7342:2
7339:1
7334:=
7331:T
7305:.
7300:2
7284:2
7280:L
7276:g
7271:2
7267:m
7260:=
7253:2
7244:F
7239:,
7234:1
7218:1
7214:L
7210:g
7207:)
7202:2
7198:m
7194:+
7189:1
7185:m
7181:(
7175:=
7168:1
7159:F
7132:,
7127:2
7114:2
7098:2
7094:L
7090:g
7085:2
7081:m
7072:1
7059:1
7043:1
7039:L
7035:g
7032:)
7027:2
7023:m
7019:+
7014:1
7010:m
7006:(
7000:=
6997:W
6966:2
6955:)
6950:2
6934:2
6930:L
6926:,
6921:2
6905:2
6901:L
6897:(
6894:+
6889:1
6878:)
6873:1
6857:1
6853:L
6849:,
6844:1
6828:1
6824:L
6820:(
6817:=
6812:2
6807:r
6798:,
6793:1
6782:)
6777:1
6761:1
6757:L
6753:,
6748:1
6732:1
6728:L
6724:(
6721:=
6716:1
6711:r
6691:i
6689:(
6686:i
6682:r
6680:δ
6663:.
6658:2
6653:r
6640:2
6635:F
6630:+
6625:1
6620:r
6607:1
6602:F
6597:=
6594:W
6576:i
6574:(
6571:i
6567:r
6562:g
6545:)
6542:g
6537:2
6533:m
6526:,
6523:0
6520:(
6517:=
6512:2
6507:F
6501:,
6498:)
6495:g
6490:1
6486:m
6479:,
6476:0
6473:(
6470:=
6465:1
6460:F
6432:.
6429:)
6424:2
6408:2
6404:L
6397:,
6392:2
6376:2
6372:L
6368:(
6365:+
6362:)
6357:1
6341:1
6337:L
6330:,
6325:1
6309:1
6305:L
6301:(
6298:=
6293:2
6288:r
6282:,
6279:)
6274:1
6258:1
6254:L
6247:,
6242:1
6226:1
6222:L
6218:(
6215:=
6210:1
6205:r
6188:i
6186:(
6183:i
6181:θ
6171:i
6169:(
6166:i
6164:λ
6157:i
6155:(
6152:i
6150:y
6145:i
6143:x
6122:=
6117:2
6112:2
6108:L
6101:)
6096:1
6091:r
6081:2
6076:r
6071:(
6065:)
6060:1
6055:r
6045:2
6040:r
6035:(
6032:=
6029:)
6024:2
6020:y
6016:,
6011:2
6007:x
6003:,
5998:1
5994:y
5990:,
5985:1
5981:x
5977:(
5972:2
5968:f
5941:0
5938:=
5933:2
5928:1
5924:L
5915:1
5910:r
5900:1
5895:r
5890:=
5887:)
5882:2
5878:y
5874:,
5869:2
5865:x
5861:,
5856:1
5852:y
5848:,
5843:1
5839:x
5835:(
5830:1
5826:f
5810:i
5805:i
5803:y
5798:i
5796:x
5791:i
5787:r
5780:i
5778:(
5775:i
5773:m
5745:y
5741:x
5737:θ
5714:=
5700:L
5697:g
5692:+
5654:,
5642:L
5639:g
5636:m
5630:=
5613:2
5609:L
5605:m
5579:,
5570:F
5566:=
5552:T
5520:T
5508:t
5505:d
5501:d
5486:θ
5479:λ
5475:y
5471:x
5454:,
5451:0
5448:=
5443:2
5439:L
5430:2
5426:y
5422:+
5417:2
5413:x
5408:,
5405:)
5402:y
5399:2
5396:(
5390:+
5387:g
5384:m
5378:=
5369:y
5363:m
5359:,
5356:)
5353:x
5350:2
5347:(
5341:=
5332:x
5326:m
5300:.
5294:y
5286:f
5274:+
5269:y
5265:F
5261:=
5255:y
5247:T
5226:y
5215:T
5203:t
5200:d
5196:d
5190:,
5184:x
5176:f
5164:+
5159:x
5155:F
5151:=
5145:x
5137:T
5116:x
5105:T
5093:t
5090:d
5086:d
5071:y
5067:x
5047:.
5042:2
5023:2
5019:L
5015:m
5010:2
5007:1
5002:=
4999:)
4994:2
4984:y
4977:+
4972:2
4962:x
4955:(
4952:m
4947:2
4944:1
4939:=
4935:v
4927:v
4923:m
4918:2
4915:1
4910:=
4907:T
4881:,
4866:)
4854:L
4851:,
4839:L
4836:(
4833:=
4830:)
4821:y
4815:,
4806:x
4800:(
4797:=
4793:v
4779:T
4762:.
4750:L
4747:g
4744:m
4738:=
4729:F
4715:θ
4706:θ
4704:δ
4700:y
4695:y
4693:δ
4676:.
4667:)
4661:(
4652:L
4649:g
4646:m
4640:=
4637:y
4631:g
4628:m
4622:=
4619:W
4590:.
4581:)
4569:L
4566:,
4554:L
4551:(
4548:=
4545:)
4542:y
4536:,
4533:x
4527:(
4524:=
4520:r
4503:θ
4499:y
4495:x
4490:r
4488:δ
4471:.
4467:r
4456:F
4452:=
4449:W
4432:r
4427:m
4412:g
4395:,
4392:)
4389:g
4386:m
4380:,
4377:0
4374:(
4371:=
4367:F
4353:m
4346:θ
4329:.
4326:)
4314:L
4308:,
4296:L
4293:(
4290:=
4287:)
4284:y
4281:,
4278:x
4275:(
4272:=
4268:r
4254:y
4250:x
4246:M
4242:θ
4222:=
4213:y
4207:y
4204:2
4201:+
4192:x
4186:x
4183:2
4180:=
4177:)
4174:y
4171:,
4168:x
4165:(
4156:f
4140:M
4123:,
4120:0
4117:=
4112:2
4108:L
4099:2
4095:y
4091:+
4086:2
4082:x
4078:=
4075:)
4072:y
4069:,
4066:x
4063:(
4060:f
4047:y
4043:x
4039:y
4035:)
4033:y
4029:x
4025:r
4020:L
4016:M
3991:N
3985:C
3978:y
3974:x
3972:(
3970:f
3949:t
3932:)
3929:)
3926:t
3923:,
3920:s
3917:(
3914:y
3911:,
3908:)
3905:t
3902:,
3899:s
3896:(
3893:x
3890:(
3887:=
3883:r
3878:,
3874:0
3871:=
3868:)
3865:t
3862:,
3858:r
3854:(
3851:f
3835:y
3831:x
3827:s
3819:s
3812:s
3806:s
3804:(
3802:y
3798:s
3796:(
3794:x
3790:r
3783:r
3781:(
3779:f
3769:N
3763:C
3731:.
3727:0
3724:=
3719:i
3709:p
3702:=
3694:i
3684:q
3672:L
3660:t
3657:d
3653:d
3648:=
3640:i
3636:q
3627:L
3603:i
3601:q
3593:L
3576:.
3568:i
3558:q
3546:L
3537:=
3532:i
3528:p
3513:i
3511:q
3505:"
3481:.
3475:2
3449:2
3439:2
3435:r
3431:+
3426:2
3407:2
3403:r
3399:+
3394:2
3384:r
3377:=
3372:2
3367:)
3361:t
3358:d
3353:s
3350:d
3344:(
3329:)
3327:φ
3323:θ
3319:r
3317:(
3297:,
3291:2
3281:z
3274:+
3269:2
3250:2
3246:r
3242:+
3237:2
3227:r
3220:=
3215:2
3210:)
3204:t
3201:d
3196:s
3193:d
3187:(
3172:)
3170:z
3166:θ
3162:r
3160:(
3140:,
3134:2
3115:2
3111:r
3107:+
3102:2
3092:r
3085:=
3080:2
3075:)
3069:t
3066:d
3061:s
3058:d
3052:(
3037:)
3035:θ
3031:r
3029:(
3015:k
2992:,
2986:j
2982:q
2978:d
2973:i
2969:q
2965:d
2961:)
2952:j
2948:q
2937:k
2932:r
2913:i
2909:q
2898:k
2893:r
2881:(
2875:n
2870:1
2867:=
2864:j
2861:,
2858:i
2850:=
2845:k
2840:r
2835:d
2827:k
2822:r
2817:d
2814:=
2809:2
2804:k
2800:s
2796:d
2783:k
2765:)
2763:t
2757:/
2755:q
2752:d
2748:q
2746:(
2744:T
2740:T
2722:,
2717:j
2707:q
2698:i
2688:q
2680:)
2671:j
2667:q
2656:k
2651:r
2632:i
2628:q
2617:k
2612:r
2600:(
2594:n
2589:1
2586:=
2583:j
2580:,
2577:i
2569:=
2564:k
2553:r
2540:k
2529:r
2509:k
2505:r
2498:k
2494:v
2472:,
2466:k
2455:r
2442:k
2431:r
2421:k
2417:m
2411:N
2406:1
2403:=
2400:k
2390:2
2387:1
2382:=
2379:T
2335:,
2331:0
2328:=
2325:)
2322:t
2319:,
2309:q
2302:,
2298:q
2294:(
2291:g
2262:/
2259:j
2251:j
2249:q
2231:.
2225:j
2215:q
2203:j
2199:q
2188:k
2183:r
2170:n
2165:1
2162:=
2159:j
2151:=
2145:t
2142:d
2135:k
2130:r
2125:d
2119:=
2114:k
2103:r
2095:=
2090:k
2085:v
2068:k
2064:r
2053:k
2049:v
2027:)
2024:)
2021:t
2018:(
2013:n
2003:q
1993:,
1984:,
1981:)
1978:t
1975:(
1970:2
1960:q
1950:,
1947:)
1944:t
1941:(
1936:1
1926:q
1919:(
1916:=
1910:t
1907:d
1901:q
1897:d
1891:=
1881:q
1863:q
1842:,
1838:)
1835:)
1832:t
1829:(
1825:q
1821:(
1816:k
1811:r
1806:=
1801:k
1796:r
1781:n
1777:k
1771:k
1767:r
1759:n
1724:)
1721:)
1718:t
1715:(
1710:n
1706:q
1699:,
1690:,
1687:)
1684:t
1681:(
1676:2
1672:q
1665:,
1662:)
1659:t
1656:(
1651:1
1647:q
1643:(
1640:=
1637:)
1634:t
1631:(
1627:q
1612:-
1610:n
1606:)
1604:t
1602:(
1599:j
1597:q
1587:C
1579:n
1570:D
1565:C
1561:N
1557:n
1552:C
1547:N
1545:3
1538:C
1534:C
1525:k
1523:y
1518:k
1516:z
1511:k
1509:x
1505:t
1488:0
1485:=
1482:)
1479:t
1476:,
1471:k
1466:r
1461:(
1458:f
1446:k
1431:N
1427:k
1420:k
1416:r
1394:)
1389:N
1385:z
1381:,
1376:N
1372:y
1368:,
1363:N
1359:x
1355:(
1352:=
1347:N
1342:r
1320:,
1317:)
1312:2
1308:z
1304:,
1299:2
1295:y
1291:,
1286:2
1282:x
1278:(
1275:=
1270:2
1265:r
1255:,
1252:)
1247:1
1243:z
1239:,
1234:1
1230:y
1226:,
1221:1
1217:x
1213:(
1210:=
1205:1
1200:r
1164:N
1154:)
1152:z
1148:y
1144:x
1142:(
1135:)
1133:v
1129:u
1127:(
1116:z
1112:y
1108:x
1106:(
1104:S
1086:z
1082:y
1078:x
1076:(
1074:F
1034:.
1032:θ
1028:θ
1024:s
1012:y
1008:x
1006:(
1004:C
985:y
981:x
979:(
977:F
957:.
955:θ
951:s
947:y
943:x
939:)
937:y
933:x
931:(
927:θ
923:s
911:y
907:x
905:(
903:C
885:y
881:x
879:(
877:F
783:e
776:t
769:v
459:)
78:t
75:d
69:p
65:d
59:=
54:F
20:)
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