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998:. In fact, they seem to be identical if F is not a function of v - in that case they only differ in the labeling of the nuisance variable v. Yet Leapfrog is a second-order method, while Euler-Cromer is first order. So something is strange here. I'm sure other readers than I may be confused, so a discussion of this might be nice to have in the article.
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by Hairer et al. on the history and multitude of names of the Newton-Verlet-Stoermer-... method. Also on how to interpret the methods in terms of operator/vector field splittings. It seems that you are right, the time-shift of the velocity is the only difference, and it leads to a global O(h) error
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The article states that the leapfrog method "conserves a (slightly modified) energy of dynamical systems". This is a widely spread misconception and is simply not true, except in very special cases - most notably linear systems (harm. osc.). The symplecticity implies phase-space area preservation,
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The existence of a conserved "shadow
Hamiltonian" which converges to the exact Hamiltonian in the limit of vanishing time step is quite general. A rather straightforward proof is presented in section 13 of chapter 3 of Tuckerman's Statistical Mechanics textbook, which I have now referenced in the
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Using RK methods for n-body gravitational simulators is not very useful as they do not conserve energy, so after a few orbits all bodies begin to spiral either inwards or outwards when N is larger than 3. Leapfrog conserves energy better, so provides much better long term simulations.
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mean that there is non-conservation. Most physical dynamical systems beyond of few particles exhibit chaotic dynamics, and this is precisely why symplectic integrators are used in dynamical simulations; they help to minimize energy drift by sticking to the shadow
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This term can also be found in the field of electronic filter realisation. A standard passive filter can be actively simulated (called functional simulation) by interchanging current and voltage in a "leapfrog" way in the equations describing the circuit.
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762:{\displaystyle {\begin{aligned}x_{i}&=x_{i-1}+v_{i-1/2}\,\Delta t,\\a_{i}&=F(x_{i})\\v_{i+1/2}&=v_{i-1/2}+a_{i}\,\Delta t,\end{aligned}}}
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yes - but not necessarily
Hamiltonian conservation. Instead, typical systems even in one dim are non-integrable, and display more or less chaos.
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Please propose a better formulation. If I read Hairer right, it should at least be true that a modified
Hamiltonian is preserved to the order
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in the velocity. The position in Euler-Cromer should still be O(h²) for situations that can be formulated as
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This method gives me different results to the midstep method even with constant timestep. Flaws?
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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Knowledge. If you would like to participate, please visit the project page, where you can join
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in the banner shell. Please resolve this conflict if possible.
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This article has been given a rating which conflicts with the
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Statistical
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