Physlib.ClassicalMechanics.HamiltonianMechanics.PoissonBracket

For two phase-space functions $f, g: X \to X \to \mathbb{R}$ defined on a real inner product space $X$, the Poisson bracket evaluated at momentum $p \in X$ and position $q \in X$ is defined as: $$\{f, g\}(p, q) = \left\langle \nabla_q f(p, q), \nabla_p g(p, q) \right\rangle - \left\langle \nabla_p f(p, q), \nabla_q g(p, q) \right\rangle$$ where $\nabla_q f$ denotes the gradient of $f$ with respect to the position variable $q$, $\nabla_p g$ denotes the gradient of $g$ with respect to the momentum variable $p$, and $\langle \cdot, \cdot \rangle$ is the inner product on $X$.

Let $X$ be a real inner product space. For any two phase-space functions $f, g : X \to X \to \mathbb{R}$ and any momentum $p \in X$ and position $q \in X$, the Poisson bracket is antisymmetric: $$\{f, g\}(p, q) = -\{g, f\}(p, q).$$

For any phase-space functions $f_1, f_2, g: X \to X \to \mathbb{R}$ defined on a real inner product space $X$, and for any momentum $p \in X$ and position $q \in X$, if $f_1$ and $f_2$ are differentiable with respect to both the momentum and position variables at the point $(p, q)$, then the Poisson bracket is additive in its first argument: $$\{f_1 + f_2, g\}(p, q) = \{f_1, g\}(p, q) + \{f_2, g\}(p, q)$$

Let $X$ be a real inner product space. Let $f, g : X \to X \to \mathbb{R}$ be two phase-space functions, where the first argument represents momentum $p \in X$ and the second represents position $q \in X$. Let $c \in \mathbb{R}$ be a constant. If $f$ is differentiable with respect to $p$ at the point $(p, q)$ and differentiable with respect to $q$ at the point $(p, q)$, then the Poisson bracket satisfies: $$\{cf, g\}(p, q) = c \{f, g\}(p, q)$$ where the Poisson bracket is defined as $\{f, g\}(p, q) = \left\langle \nabla_q f(p, q), \nabla_p g(p, q) \right\rangle - \left\langle \nabla_p f(p, q), \nabla_q g(p, q) \right\rangle$.

Let $X$ be a real inner product space. For phase-space functions $f, g_1, g_2: X \to X \to \mathbb{R}$ and points $p, q \in X$, suppose that both $g_1$ and $g_2$ are differentiable with respect to their first argument at $p$ and with respect to their second argument at $q$. Then the Poisson bracket is additive in its second argument: $$\{f, g_1 + g_2\}(p, q) = \{f, g_1\}(p, q) + \{f, g_2\}(p, q)$$ where $\{f, g\}$ denotes the Poisson bracket.

Let $X$ be a real inner product space. For any scalar $c \in \mathbb{R}$ and phase-space functions $f, g : X \to X \to \mathbb{R}$, if $g$ is differentiable with respect to its first argument at $p \in X$ and its second argument at $q \in X$, then the Poisson bracket is homogeneous of degree one in its second argument, satisfying: $$\{f, c g\}(p, q) = c \{f, g\}(p, q)$$ where $\{f, g\}$ denotes the Poisson bracket evaluated at momentum $p$ and position $q$.

Let $X$ be a real inner product space. For a phase-space function $f : X \times X \to \mathbb{R}$ and trajectories $p, q : \text{Time} \to X$, suppose that $p$ and $q$ are differentiable at time $t$, and $f$ is differentiable at the point $(p(t), q(t))$. Then the time derivative of the observable $f$ along the trajectory $(p(t), q(t))$ is given by the sum of the inner products of its gradients with the respective velocities: $$\partial_t (f(p(t), q(t))) = \langle \nabla_p f(p(t), q(t)), \partial_t p(t) \rangle + \langle \nabla_q f(p(t), q(t)), \partial_t q(t) \rangle$$ where $\nabla_p f$ and $\nabla_q f$ denote the gradients of $f$ with respect to its first (momentum) and second (position) arguments, respectively, and $\partial_t$ denotes the time derivative.

Let $X$ be a real inner product space. Consider a time-dependent Hamiltonian $H : \text{Time} \to X \to X \to \mathbb{R}$, trajectories $p, q : \text{Time} \to X$, and a phase-space observable $f : X \to X \to \mathbb{R}$. Suppose that at time $t$, the trajectories $p$ and $q$ are differentiable, $f$ is differentiable at the point $(p(t), q(t))$, and the trajectories satisfy Hamilton's equations (i.e., $\text{hamiltonEqOp}(H, p, q) = 0$). Then the time derivative of the observable $f$ along the trajectory is given by its Poisson bracket with $H$ at time $t$: $$\frac{d}{dt} (f(p(t), q(t))) = \{f, H(t)\}(p(t), q(t))$$ where $\{\cdot, \cdot\}$ denotes the Poisson bracket on phase space.