---
tag: 4KZ458T2QA
kind: concept
name: Variational integrator
msc: [65P10, 37M15]
specializes: [VQ2BTYFST4]
sources:
  - url: https://en.wikipedia.org/wiki/Variational_integrator
same_as:
  wikidata: Q7915797
  wikipedia: Variational integrator
---

## Summary

A variational integrator is a numerical method for Lagrangian mechanics obtained by discretising Hamilton's principle of stationary action instead of the equations of motion.
One approximates the action over a single time step by a discrete Lagrangian.
Asking the sum of these to be stationary gives the discrete Euler–Lagrange equations.
These methods are automatically symplectic and keep the momenta that come from symmetries, by a discrete Noether theorem, so they are [[geometric integrators|VQ2BTYFST4]].

## Explanations

### L1 human

In mechanics, the true motion makes a quantity called the action stationary.
A variational integrator copies this principle on a time grid: it looks for a list of positions that makes a discrete action stationary.
Because 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.
The Störmer–Verlet method used in molecular dynamics is the simplest example.
It 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.