Hamiltonian Mechanics — Mathematics Knowledge System

Overview

Hamiltonian mechanics reformulates classical mechanics by replacing generalised velocities with generalised momenta, switching from configuration space to phase space. The Hamiltonian function H(q, p, t) — the Legendre transform of the Lagrangian — governs evolution through Hamilton's equations, a symmetric system of first-order ODEs. The phase-space picture reveals deep geometric structure: Liouville's theorem, Poisson brackets, canonical transformations, and the connection to quantum mechanics via canonical quantisation.

Intuition

In Lagrangian mechanics the state is (q, q̇) — position and velocity. Hamiltonian mechanics replaces velocity with momentum p = ∂L/∂q̇, moving to phase space (q, p). Hamilton's equations are symmetric first-order ODEs: q̇ = ∂H/∂p and ṗ = −∂H/∂q. The geometric structure of phase space (symplectic structure) is preserved by time evolution — this is Liouville's theorem — and underpins both classical statistical mechanics and quantum mechanics.

Formal Definition

Definition

The Hamiltonian H is defined via the Legendre transform of the Lagrangian L. The canonical equations govern the evolution of position and momentum.

H(q, p, t) = \sum_{i=1}^n p_i \dot{q}_i - L(q, \dot{q}, t)

Hamiltonian via Legendre transform

\dot{q}_i = \frac{\partial H}{\partial p_i}, \qquad \dot{p}_i = -\frac{\partial H}{\partial q_i}

Hamilton's canonical equations

\{f, g\} = \sum_{i=1}^n \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)

Poisson bracket

\frac{df}{dt} = \{f, H\} + \frac{\partial f}{\partial t}

Time evolution via Poisson bracket

Notation

Notation	Meaning
$(q, p)$	Phase-space coordinates (position, momentum)
$H$	Hamiltonian function
$\{f, g\}$	Poisson bracket of phase-space functions f and g
$\omega = \sum_i dq_i \wedge dp_i$	Symplectic 2-form on phase space

Theorems

Theorem 1: Liouville's Theorem

\frac{d\rho}{dt} = \frac{\partial \rho}{\partial t} + \{\rho, H\} = 0

Theorem 2: Conservation of the Hamiltonian

\frac{dH}{dt} = \frac{\partial H}{\partial t}

Theorem 3: Canonical Transformation Preservation

\{q_i', p_j'\} = \delta_{ij},\quad \{q_i', q_j'\} = 0,\quad \{p_i', p_j'\} = 0

Theorem 4: Hamilton–Jacobi Equation

H\!\left(q, \frac{\partial S}{\partial q}, t\right) + \frac{\partial S}{\partial t} = 0

Worked Examples

1
Apply Hamilton's first equation.
$\dot{q} = \frac{\partial H}{\partial p} = \frac{p}{m}$
2
Apply Hamilton's second equation.
$\dot{p} = -\frac{\partial H}{\partial q} = -m\omega^2 q$
3
Eliminate p: differentiate the first equation and substitute the second.
$\ddot{q} = \dot{p}/m = -\omega^2 q$

✓ Answer

Hamilton's equations give q̈ + ω²q = 0 — the standard harmonic oscillator.

Practice Problems

Hardproof writing

Prove that if {f, H} = 0 and ∂f/∂t = 0, then f is a constant of the motion.

Common Mistakes

Common Mistake

The Hamiltonian always equals the total mechanical energy

H = T + V only for natural systems (time-independent constraints, no velocity-dependent potentials). In general H is the Legendre transform of L and may differ from the mechanical energy.

Common Mistake

Poisson brackets are just antisymmetric bilinear maps

Poisson brackets must also satisfy the Jacobi identity {{f,g},h} + {{g,h},f} + {{h,f},g} = 0, making them a Lie algebra structure on phase-space functions.

Quiz

Liouville's theorem states that Hamiltonian flow preserves:

Total energyPhase-space volumeKinetic energyThe Lagrangian

The Hamiltonian is obtained from the Lagrangian by:

Adding kinetic and potential energyA Fourier transformA Legendre transform in the velocitiesDifferentiating L with respect to time

Historical Background

William Rowan Hamilton developed his formulation between 1833 and 1835, seeking a unified theory for optics and mechanics. The resulting phase-space picture proved more symmetric and powerful than the Lagrangian formulation for mathematical analysis. Carl Gustav Jacobi extended the theory with the Hamilton–Jacobi equation. In the 20th century, the Hamiltonian framework became the natural language for quantum mechanics (canonical quantisation) and statistical mechanics (phase-space distributions).

1833
Hamilton presents his reformulation of mechanics using the characteristic function
William Rowan Hamilton
1835
Hamilton derives what we now call Hamilton's equations of motion
William Rowan Hamilton
1837
Jacobi reformulates the theory and introduces the Hamilton–Jacobi equation
Carl Gustav Jacobi
1838
Liouville proves volume preservation in phase space
Joseph Liouville
1925
Dirac identifies Poisson brackets with quantum commutators, enabling canonical quantisation
Paul Dirac

Summary

Hamiltonian mechanics operates in phase space (q, p); H is the Legendre transform of L in the velocities.
Hamilton's equations q̇ᵢ = ∂H/∂pᵢ and ṗᵢ = −∂H/∂qᵢ are first-order and symmetric.
Poisson brackets encode the algebraic structure: {qᵢ, pⱼ} = δᵢⱼ; time evolution: df/dt = {f, H} + ∂f/∂t.
Liouville's theorem: Hamiltonian flow preserves phase-space volume.
The Hamiltonian framework is the bridge to quantum mechanics via canonical quantisation ({·,·} → [·,·]/iℏ).

References

BookGoldstein, H., Poole, C. & Safko, J. — Classical Mechanics, 3rd ed. (2002), Addison-Wesley, Chapter 8–9
BookArnol'd, V.I. — Mathematical Methods of Classical Mechanics, 2nd ed. (1989), Springer, Chapter VIII–IX
BookLandau, L.D. & Lifshitz, E.M. — Mechanics, 3rd ed. (1976), Butterworth-Heinemann, §40–45
WebsiteWikipedia — Hamiltonian mechanics
WebsiteMathWorld — Hamiltonian