Variational integrator

Definition

Let $L \colon \R^d \times \R^d \to \R$, $(q, \dot q) \mapsto L(q, \dot q)$, be a smooth Lagrangian, and fix a step size $h > 0$. A discrete Lagrangian is a smooth function $L_h \colon \R^d \times \R^d \to \R$ that approximates the action: $$L_h(q_0, q_1) \approx \int_0^h L(q(t), \dot q(t))\, dt.$$ Here $q$ is the solution of the Euler–Lagrange equation with $q(0) = q_0$ and $q(h) = q_1$. The discrete action of a sequence $q_0, \ldots, q_N$ is $S_h = \sum_{k=0}^{N-1} L_h(q_k, q_{k+1})$. The variational integrator defined by $L_h$ is the map $(q_{k-1}, q_k) \mapsto (q_k, q_{k+1})$ given by the discrete Euler–Lagrange equations: $$D_2 L_h(q_{k-1}, q_k) + D_1 L_h(q_k, q_{k+1}) = 0.$$ Here $D_1$ and $D_2$ denote the gradients with respect to the first and the second argument. These are the equations $\partial S_h / \partial q_k = 0$ for $1 \le k \le N - 1$, with $q_0$ and $q_N$ fixed.