[proofplan] We show that the Hamiltonian $H$ is conserved along its own flow by applying the evolution formula $\dot{f} = \{f, H\}$ with $f = H$, then using the antisymmetry of the Poisson bracket to conclude $\{H, H\} = 0$. [/proofplan] [step:Apply the evolution formula with $f = H$] Let $(q(t), p(t))$ be a solution of Hamilton's equations with Hamiltonian $H: \mathbb{R}^{2n} \to \mathbb{R}$. By the [Evolution Via Poisson Bracket](/theorems/1334) theorem applied with $f = H$, the time derivative of the Hamiltonian along the trajectory is \begin{align*} \frac{dH}{dt} = \{H, H\}. \end{align*} The hypothesis of the evolution theorem requires that $H$ is a smooth function of $(q, p)$ with no explicit time dependence, which is satisfied since $H$ is the (autonomous) Hamiltonian of the system. [/step] [step:Conclude conservation from antisymmetry of the Poisson bracket] By the antisymmetry property of the Poisson bracket (established in the [Properties of the Poisson Bracket](/theorems/1333), property (2)): \begin{align*} \{H, H\} = -\{H, H\}. \end{align*} Adding $\{H, H\}$ to both sides gives $2\{H, H\} = 0$, so $\{H, H\} = 0$. Substituting into the evolution equation: \begin{align*} \frac{dH}{dt} = 0. \end{align*} Therefore $H(q(t), p(t))$ is constant along every solution of Hamilton's equations. That is, the total energy $H$ is a conserved quantity (first integral) of the Hamiltonian system. [guided] This two-line argument is deceptively simple, but it encodes a deep structural fact: conservation of energy in Hamiltonian mechanics is not a dynamical accident but a consequence of the algebraic structure of the Poisson bracket. Any quantity $f$ satisfying $\{f, H\} = 0$ is conserved, and the Hamiltonian is always such a quantity because the Poisson bracket is antisymmetric. The same argument shows more generally that $\{F, F\} = 0$ for any smooth function $F$, so every Hamiltonian system conserves its own Hamiltonian. This is the classical-mechanical statement that "every symmetry generator is conserved under its own flow." [/guided] [/step]