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  "body": "\n## Summary\n\nA variational integrator is a numerical method for Lagrangian mechanics obtained by discretising Hamilton's principle of stationary action instead of the equations of motion.\nOne approximates the action over a single time step by a discrete Lagrangian.\nAsking the sum of these to be stationary gives the discrete Euler–Lagrange equations.\nThese methods are automatically symplectic and keep the momenta that come from symmetries, by a discrete Noether theorem, so they are [[geometric integrators|VQ2BTYFST4]].\n\n## Explanations\n\n### L1 human\n\nIn mechanics, the true motion makes a quantity called the action stationary.\nA variational integrator copies this principle on a time grid: it looks for a list of positions that makes a discrete action stationary.\nBecause it inherits the principle and not only the equations, it inherits the structure that comes with it, such as conservation of angular momentum for a rotation-invariant system.\nThe Störmer–Verlet method used in molecular dynamics is the simplest example.\nIt comes from the discrete action with kinetic energy $\\frac12((q_{k+1} - q_k)/h)^2$ and the potential averaged at the two ends of each step.\n",
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