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| In [[mathematics]] '''leapfrog integration''' is a simple method for numerically integrating [[differential equation]]s of the form
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| :<math>\ddot x=F(x)</math>,
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| or equivalently of the form
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| :<math>\dot v=F(x),\;\dot x \equiv v</math>,
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| particularly in the case of a [[dynamical system]] of [[classical mechanics]]. Such problems often take the form
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| :<math>\ddot x=-\nabla V(x)</math>,
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| with energy function
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| :<math>E(x,v)=\tfrac12|v|^2+V(x)</math>,
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| where ''V'' is the [[potential energy]] of the system. The method is known by different names in different disciplines. In particular, it is similar to the '''Velocity Verlet''' method, which is a variant of [[Verlet integration]]. Leapfrog integration is equivalent to updating positions <math>x(t)</math> and velocities <math>v(t)=\dot x(t)</math> at interleaved time points, staggered in such a way that they 'leapfrog' over each other. For example, the position is updated at integer time steps and the velocity is updated at integer-plus-a-half time steps.
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| Leapfrog integration is a second order method, in contrast to [[Euler integration]], which is only first order, yet requires the same number of function evaluations per step. Unlike Euler integration, it is stable for oscillatory motion, as long as the time-step <math>\Delta t</math> is constant, and <math>\Delta t \leq 2/\omega</math>.<ref>[C. K. Birdsall and A. B. Langdon, Plasma Physics via Computer Simulations, McGraw-Hill Book Company, 1985, p. 56]</ref>
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| In leapfrog integration, the equations for updating position and velocity are
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| :<math>\begin{align}
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| x_i &= x_{i-1} + v_{i-1/2}\, \Delta t , \\[0.4em]
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| a_i &= F(x_i) \\[0.4em]
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| v_{i+1/2} &= v_{i-1/2} + a_{i}\, \Delta t ,
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| \end{align}</math>
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| where <math>x_i</math> is position at step <math>i</math>, <math>v_{i+1/2\,}</math> is the velocity, or first derivative of <math>x</math>, at step <math>i+1/2\,</math>, <math>a_{i}=F(x_i)</math> is the acceleration, or second derivative of <math>x</math>, at step <math>i</math> and <math>\Delta t</math> is the size of each time step. These equations can be expressed in a form which gives velocity at integer steps as well.<ref>[http://www.artcompsci.org/vol_1/v1_web/node34.html 4.1 Two Ways to Write the Leapfrog]</ref> However, even in this synchronized form, the time-step <math>\Delta t</math> must be constant to maintain stability.<ref>[Skeel, R. D., "Variable Step Size Destabilizes the Stömer/Leapfrog/Verlet Method," BIT Numerical Mathematics, Vol. 33, 1993, pp. 172-175.]</ref>
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| :<math>\begin{align}
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| x_{i+1} &= x_i + v_i\, \Delta t + \tfrac{1}{2}\,a_i\, \Delta t^{\,2} , \\[0.4em]
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| v_{i+1} &= v_i + \tfrac{1}{2}\,(a_i + a_{i+1})\,\Delta t .
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| \end{align}</math>
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| One use of this equation is in gravity simulations, since in that case the acceleration depends only on the positions of the gravitating masses, although higher order integrators (such as [[Runge–Kutta methods]]) are more frequently used.
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| There are two primary strengths to Leapfrog integration when applied to mechanics problems. The first is the [[time-reversibility]] of the Leapfrog method. One can integrate forward ''n'' steps, and then reverse the direction of integration and integrate backwards ''n'' steps to arrive at the same starting position. The second strength of Leapfrog integration is its [[Symplectic integrator|symplectic]] nature, which implies that it conserves the (slightly modified) energy of dynamical systems. This is especially useful when computing orbital dynamics, as many other integration schemes, such as the (order 4) Runge-Kutta method, do not conserve energy and allow the system to drift substantially over time.
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| Because of its time-reversibility, and because it is a [[symplectic integrator]], leapfrog integration is also used in [[Hybrid Monte Carlo|Hamiltonian Monte Carlo]], a method for drawing random samples from a probability distribution whose overall normalization is unknown.<ref>{{cite book | last=Bishop | first=Christopher | title=Pattern Recognition and Machine Learning | year=2006 | publisher=[[Springer-Verlag]] | location=New York | isbn=978-0-387-31073-2 | pages=548–554}}</ref>
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| ==See also==
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| *[[Numerical ordinary differential equations]]
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| *[[Symplectic integrator|Symplectic integration]]
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| *[[Euler integration]]
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| *[[Verlet integration]]
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| *[[Runge–Kutta methods|Runge–Kutta integration]]
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| ==References==
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| {{Reflist}}
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| ==External links==
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| *[http://einstein.drexel.edu/courses/Comp_Phys/Integrators/leapfrog/ The Leapfrog Integrator], Drexel University Physics
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| {{Numerical integrators}}
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| [[Category:Numerical differential equations]]
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