Hamiltonian mechanics

Hamiltonian mechanics replaces Newton’s second-order equations with two coupled first-order canonical equations. For generalized coordinates qⁱ and conjugate momenta p_i, the Hamiltonian H(q,p,t) generates motion according to dqⁱ/dt = ∂H/∂p_i and dp_i/dt = −∂H/∂qⁱ.¹ These equations define a trajectory in phase space, whose points specify both configuration and momentum.

For many systems, the Hamiltonian equals the total energy, H=T+V, but this identification is not universal. Time-dependent constraints, electromagnetic potentials, velocity-dependent interactions, and non-Cartesian coordinates can alter the relationship between the Hamiltonian and mechanical energy.²

The Hamiltonian is commonly obtained from a Lagrangian by a Legendre transformation. Starting with L(q, q̇, t), one defines the canonical momentum p_i = ∂L/∂q̇ⁱ and then H = Σ p_iq̇ⁱ − L, with velocities expressed in terms of coordinates and momenta when the transformation is nonsingular.³

This construction reveals a geometric structure: phase space carries a symplectic form, commonly written ω = Σ dqⁱ ∧ dp_i. Hamiltonian flow preserves this form and therefore preserves phase-space volume, a result associated with Liouville’s theorem.⁴ Canonical transformations preserve the same structure and can convert a difficult Hamiltonian into a simpler one.

Poisson brackets provide the natural language for Hamiltonian evolution and conserved quantities. For functions f and g on phase space, their Poisson bracket determines how they interact under the symplectic structure; an observable evolves as df/dt = {f,H} + ∂f/∂t. If a quantity has a vanishing bracket with the Hamiltonian and no explicit time dependence, it is conserved.

Continuous symmetries therefore lead to constants of motion through Noether’s theorem. In integrable systems, action-angle variables can reduce the motion to uniform evolution on invariant tori. The Hamilton–Jacobi equation offers another powerful reformulation: solving a single first-order partial differential equation can generate canonical transformations and, in favorable cases, the complete solution of the dynamics.

Hamiltonian mechanics remains useful when the Hamiltonian is not simply kinetic plus potential energy. A charged particle in an electromagnetic field uses canonical momentum that differs from mechanical momentum, while constrained systems may require singular Legendre transformations and the constraint analysis developed by Dirac.⁵

The formalism also underlies modern numerical mechanics. Symplectic integrators preserve the geometric structure of phase space and often control long-term energy errors better than generic methods, making them valuable in celestial mechanics and molecular dynamics. In statistical mechanics, Hamiltonian phase space supplies the foundation for ensembles; in quantum mechanics, canonical quantization promotes coordinates and momenta to operators with commutation relations modeled on Poisson brackets.⁶

Hamiltonian

A function of generalized coordinates, conjugate momenta, and possibly time that generates canonical evolution.

Canonical coordinates

Paired phase-space variables, conventionally denoted q and p, related by the symplectic structure.

Poisson bracket

A bilinear operation on phase-space functions that expresses canonical evolution and conservation relations.

Symplectic form

A nondegenerate antisymmetric two-form that supplies phase space with its geometric structure.

Canonical transformation

A transformation of phase-space variables that preserves the symplectic structure.