Hamiltonian Mechanics

Manual: General · Subject: Classical Mechanics

Phase space, canonical equations, and conservation laws

The Hamiltonian and Phase Space

From Lagrangian to Hamiltonian

The generalised momentum: $p_i = \partial L / \partial \dot{q}_i$ . The Hamiltonian is the Legendre transform: $H = \sum_i p_i \dot{q}_i - L$ For conservative systems $H = T + V$ (total energy).

Hamilton's equations:<br> $\dot{q}_i = \frac{\partial H}{\partial p_i}, \qquad \dot{p}_i = -\frac{\partial H}{\partial q_i}$

Poisson Brackets and Liouville's Theorem

The Poisson bracket: $\{f,g\} = \sum_i \left(\frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i}\right)$ .

Liouville's theorem: the phase-space volume element is conserved under Hamiltonian flow ( $d\rho/dt = 0$ ). This underpins statistical mechanics.

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Phase Space Trajectories

In Hamiltonian mechanics, the state of a system is a point in 2n-dimensional phase space $(q_1,...,q_n, p_1,...,p_n)$ . Trajectories never cross since evolution is deterministic.

For a 1D harmonic oscillator $H = p^2/2m + kx^2/2$ , Hamilton's equations give:

\dot{x} = p/m

and

\dot{p} = -kx

\dot{x} = -kx

and

\dot{p} = p/m

\dot{x} = kx

and

\dot{p} = p/m

\dot{p} = 0

Liouville's theorem implies that in phase space:

The volume of a region is preserved under time evolution Trajectories expand over time The density decreases with time Energy is always conserved

If $H$ has no explicit time dependence ( $\partial H/\partial t = 0$ ), what does this imply?

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Hamiltonian Mechanics

The Hamiltonian and Phase Space

From Lagrangian to Hamiltonian

Poisson Brackets and Liouville's Theorem