Analytical mechanics

Hamilton's equations, phase space

Hamiltonian mechanics describes dynamics with canonical coordinate–momentum pairs; phase-space trajectories obey first-order differential equations.

For each degree of freedom the state is represented by (q,p). The Hamiltonian H(q,p,t) generates motion through the canonical equations.

Harmonic oscillator

For H=p²/(2m)+mω²q²/2 with no explicit time dependence, energy H is conserved. Constant-energy trajectories are ellipses in (q,p); the sign of p determines the direction of passage.

Canonical coordinates and canonical transformations

A change of variables $(q,p)\mapsto(Q,P)$ is canonical if it preserves the symplectic structure, so Hamilton’s equations retain their form in the new variables. For example, the shear $Q=q,\ P=p+\alpha q$ has unit Jacobian determinant and is canonical. It deforms a phase curve while preserving oriented area.

Quick check

For two phase-space functions, the Poisson bracket is $\{F,G\}=\sum_i(\partial F/\partial q_i\,\partial G/\partial p_i-\partial F/\partial p_i\,\partial G/\partial q_i)$. Along a Hamiltonian trajectory, $dF/dt=\partial F/\partial t+\{F,H\}$. Thus a quantity with no explicit time dependence is conserved when its Poisson bracket with $H$ vanishes. This structure also explains why canonical transformations preserve the form of the equations and provides the classical framework that motivates quantization.

For an oscillator with $H=p^2/(2m)+m\omega^2q^2/2$, Hamilton’s equations give $\dot q=p/m$ and $\dot p=-m\omega^2q$. Eliminating $p$ recovers $\ddot q+\omega^2q=0$. The pair of first-order equations describes the same motion as the second-order Lagrangian equation while exposing its phase-space state structure.