Physlib.ClassicalMechanics.HamiltonsEquations

Given a Hamiltonian function $H: \text{Time} \to X \to X \to \mathbb{R}$, a momentum trajectory $p: \text{Time} \to X$, and a position trajectory $q: \text{Time} \to X$, the Hamilton equation operator is defined as the function mapping each time $t$ to the pair: $$\left( \frac{dq}{dt}(t) - \nabla_p H(t, p(t), q(t)), \ -\frac{dp}{dt}(t) - \nabla_q H(t, p(t), q(t)) \right)$$ In this expression, $\frac{dq}{dt}$ and $\frac{dp}{dt}$ denote the time derivatives of the trajectories, $\nabla_p H$ denotes the gradient of the Hamiltonian with respect to the momentum (the first $X$ argument), and $\nabla_q H$ denotes the gradient with respect to the position (the second $X$ argument). The vanishing of this operator is equivalent to the classical Hamilton's equations $\dot{q} = \frac{\partial H}{\partial p}$ and $\dot{p} = -\frac{\partial H}{\partial q}$.

Let $H: \text{Time} \to X \to X \to \mathbb{R}$ be a Hamiltonian function, $p: \text{Time} \to X$ be a momentum trajectory, and $q: \text{Time} \to X$ be a position trajectory. The Hamilton equation operator $\text{hamiltonEqOp}(H, p, q)$ is the function mapping each time $t$ to the pair: $$\left( \frac{dq}{dt}(t) - \nabla_p H(t, p(t), q(t)), \ -\frac{dp}{dt}(t) - \nabla_q H(t, p(t), q(t)) \right)$$ where $\frac{dq}{dt}$ and $\frac{dp}{dt}$ denote the time derivatives of the trajectories, and $\nabla_p H$ and $\nabla_q H$ denote the gradients of the Hamiltonian with respect to the momentum (the first $X$ argument) and the position (the second $X$ argument), respectively.

Let $H: \text{Time} \to X \to X \to \mathbb{R}$ be a Hamiltonian function, where the first $X$ argument corresponds to momentum and the second to position. For a momentum trajectory $p: \text{Time} \to X$ and a position trajectory $q: \text{Time} \to X$, the Hamilton equation operator $\text{hamiltonEqOp}(H, p, q)$ vanishes if and only if the trajectories satisfy Hamilton's canonical equations for all $t$: $$\frac{dq}{dt}(t) = \nabla_p H(t, p(t), q(t)) \quad \text{and} \quad \frac{dp}{dt}(t) = -\nabla_q H(t, p(t), q(t))$$ where $\nabla_p H$ and $\nabla_q H$ denote the gradients of the Hamiltonian with respect to its momentum and position arguments, respectively.

Let $H : \text{Time} \to X \to X \to \mathbb{R}$ be a $C^\infty$ smooth Hamiltonian function, where the two $X$ arguments represent momentum and position respectively. Let $pq : \text{Time} \to X \times X$ be a $C^\infty$ smooth trajectory in phase space, denoted as $pq(t) = (p(t), q(t))$. The variational gradient (functional derivative) of the action functional $$S[p, q] = \int \left( \langle p(t), \dot{q}(t) \rangle - H(t, p(t), q(t)) \right) dt$$ evaluated at the trajectory $pq$ is equal to the Hamilton equation operator $\text{hamiltonEqOp}(H, p, q)$, which maps each time $t$ to the pair: $$\left( \dot{q}(t) - \nabla_p H(t, p(t), q(t)), \ -\dot{p}(t) - \nabla_q H(t, p(t), q(t)) \right)$$ where $\dot{q}$ and $\dot{p}$ are time derivatives, and $\nabla_p H$ and $\nabla_q H$ are the gradients of the Hamiltonian with respect to momentum and position respectively.