Hamiltonian mechanics

Definition

Let $U \subseteq \R^n$ be open, let $I \subseteq \R$ be an open interval, and let $L \colon U \times \R^n \times I \to \R$, $L = L(q, \dot q, t)$, be a Lagrangian of class $C^2$ in generalized coordinates $q$. The generalized momenta are $p_i = \partial L / \partial \dot q_i (q, \dot q, t)$ for $i = 1, \ldots, n$. Assume that $L$ is hyperregular: for each $(q, t)$ the map $\dot q \mapsto p$ is a bijection of $\R^n$ onto an open set. Assume also that its inverse $\dot q = v(q, p, t)$ is of class $C^1$ jointly in $(q, p, t)$. The Hamiltonian of $L$ is $$H(q, p, t) = \sum_{i=1}^n p_i \, v_i(q, p, t) - L\big(q, v(q, p, t), t\big).$$ Hamiltonian mechanics describes the motion by curves $t \mapsto (q(t), p(t))$ in the phase space of the variables $(q, p)$ that satisfy the Hamiltonian system $$\dot q_i = \frac{\partial H}{\partial p_i}, \qquad \dot p_i = -\frac{\partial H}{\partial q_i}, \qquad i = 1, \ldots, n.$$