Classical Mechanics · Upper Division

Hamiltonian Mechanics

The Hamiltonian formulation replaces Newton's second-order equations with a symmetric pair of first-order equations in phase space — the natural language of statistical mechanics, quantum mechanics, and chaos theory.

Prerequisites

Lagrangian mechanics (Ch. L) \cdot Partial derivatives \cdot Phase space concept

Learning Goals

Perform the Legendre transform from the Lagrangian to the Hamiltonian and write Hamilton's equations.
Describe the harmonic oscillator phase portrait as closed ellipses at constant energy.
State Liouville's theorem and explain its consequence for statistical mechanics ensembles.
Evaluate Poisson brackets to identify conserved quantities and derive equations of motion.
Construct action-angle variables for integrable systems and apply semiclassical quantization.

H.1 From Lagrangian to Hamiltonian

Given the Lagrangian L(q, q̇, t), the generalized momentum conjugate to qᵢ is defined as:

p_i = \frac{\partial L}{\partial \dot q_i}

(H.1)

The Hamiltonian is obtained by a Legendre transform that eliminates q̇ in favor of p:

H(q,p,t) = \sum_i p_i\dot q_i - L(q,\dot q,t)

(H.2)

For a natural mechanical system (kinetic energy quadratic in velocities, no explicit time dependence), H equals the total mechanical energy: H = T + V.

Definition H.1 — Hamilton's Equations

The equations of motion in Hamiltonian mechanics are:

q̇_{i} = \partialH/\partialp_{i} \qquad ṗ_{i} = -\partialH/\partialq_{i}

These replace one second-order ODE per degree of freedom with two first-order ODEs — at the cost of doubling the number of variables but gaining perfect symmetry between q and p.

Example H.1 — Harmonic Oscillator in Phase Space

$A mass m on a spring (k): L = \frac{1}{2}mẋ^{2} - \frac{1}{2}kx^{2}. Write Hamilton's equations and describe the pha$ se portrait.

Conjugate momentum:

p = \partialL/\partialẋ = mẋ \qquad (the ordinary linear momentum

Hamiltonian:

H = pẋ - L = p^{2}/(2m) + \frac{1}{2}kx^{2} \qquad (kinetic + potential, as expected

Hamilton's equations:ẋ =

\partialH/\partialp = p/m \qquad ṗ = -\partialH/\partialx = -kx

Eliminating p:ṗ =

mẍ = -kx \to ẍ + (k/m)x = 0 ✓ (recovers Newton

Phase portrait:

H = E = const defines an ellipse in (x, p) space: x^{2}/(2E/k) + p^{2}/(2mE) = 1. Every orbit is

a closed ellipse — the system is periodic for all energies.

H.2 Phase Space and Liouville's Theorem

The phase spaceof an n-degree-of-freedom system is the 2n-dimensional space with coordinates (q₁,...,qₙ, p₁,...,pₙ). Each point represents a complete instantaneous state. Hamilton's equations define a flow in this space — every state has a unique future and past.

Theorem H.1 — Liouville's Theorem

The phase space flow generated by Hamilton's equations preserves volume. If you take a region of phase space containing many initial conditions and let the system evolve, the volume occupied is constant — even though the shape may distort arbitrarily:

dΓ/dt = 0 \qquad where Γ = \int d^n q d^n p

This is the foundation of statistical mechanics: a microcanonical ensemble occupies a constant volume in phase space. It also implies that phase space trajectories never cross.

Figure H.1. Phase portraits for three Hamiltonian systems. Harmonic oscillator: perfect ellipses (all periodic). Pendulum: libration (closed curves below the separatrix), rotation (open curves above), an

d the separatrix itself (infinite-period orbit). Double well: two stable fixed points at \pm

H.3 Poisson Brackets

For any two functions f(q, p) and g(q, p), the Poisson bracket is:

\{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)

(H.3)

Poisson brackets encode the algebraic structure of classical mechanics. Key properties:

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

(H.4)

The time evolution of any observable A is:

\frac{dA}{dt} = \{A,H\} + \frac{\partial A}{\partial t}

(H.5)

A quantity A is conserved if and only if {A, H} = 0 (and A has no explicit time dependence). The Poisson bracket structure is preserved by canonical transformations — the symplectomorphisms that are the symmetries of Hamiltonian mechanics. In quantum mechanics, Poisson brackets become commutators:{f, g} → [f̂, ĝ]/(iℏ).

Example H.2 — Angular Momentum Conservation via Poisson Brackets

$For a central force H = p^{2}/(2m) + V(r), show that Lz = xpy - ypx is conserved$ .

Compute {Lz, H}:{

Lz, p^{2}/2m

} + {Lz, V(r)}

First term:{

xpy, px^{2}/2m

} + {

xpy, py^{2}/2m

} + {

-ypx

, ...}: evaluates to py(-px/m) + (-py)(px/m)... careful algebra gives 0.

Second term:{

xpy - ypx, V(r

} =

py \partialV/\partialx - px \partialV/\partialy. Since \partialV/\partialx = (x/r)V'(r) etc., cross terms cancel exactly

Result:{Lz, H} =

0 \to Lz conserved for any central V(r). ✓ \qquad Angular momentum is conserved because H has no

dependence — Noether again.

H.4 Canonical Transformations and Action-Angle Variables

A canonical transformation(Q, P) = (Q(q,p), P(q,p)) preserves the form of Hamilton's equations. The condition is that the transformation preserves the Poisson bracket structure: {Qᵢ, Pⱼ} = δᵢⱼ.

For an integrable system (as many conserved quantities as degrees of freedom), one can always find action-angle variables (J, θ) where:

J_i = \frac{1}{2\pi}\oint p_i\,dq_i \qquad \dot\theta_i = \omega_i = \frac{\partial H}{\partial J_i} = \mathrm{constant}

(H.6)

In action-angle variables, the Hamiltonian depends only on J, not θ — every θ is cyclic, every Jᵢ is conserved, and the motion is uniform rotation on a torus in phase space. This formulation is the starting point for perturbation theory (KAM theorem) and the semi-classical quantization rule: Jᵢ = nᵢ ℏ.

Definition H.2 — Common Traps

Hamiltonian is not always total energy: it equals energy only under standard natural-system assumptions.
Phase space doubles variables: q and p are independent coordinates, not position and velocity written twice.
Canonical transformations preserve structure: not every coordinate change preserves Poisson brackets.
Liouville preserves volume, not shape: a phase-space blob can stretch and fold while keeping the same volume.
Action-angle variables require integrability: chaotic systems generally do not admit global action-angle coordinates.

Exercises — H.1–H.4 Hamiltonian Mechanics

A harmonic oscillator has spring constant k = 400 N/m and mass m = 10 kg. Find \omega_{0}

rad/s

Straightforward

State the Jacobian determinant of the time-evolution map for a Hamiltonian system, by Liouville's theorem.

Straightforward

3.Write the Hamiltonian for a simple pendulum and derive Hamilton's equations. Identify the fixed points in phase space.

Straightforward

Compute the action variable J for a harmonic oscillator. Show that E = \omegaJ, and derive the

quantization E = n\hbar\omega

Intermediate

5.Verify Liouville's theorem explicitly for the harmonic oscillator by computing the Jacobian of the time-evolution map.

Intermediate

6.Compute the Poisson bracket {Lx, Ly} for angular momentum components. Show the result equals Lz. How does this relate to quantum mechanics?

Challenging

Key Takeaways

$Hamiltonian H = \sum p_{i}q̇_{i} - L = T + V for natural systems$ .
$Hamilton's equations: q̇ = \partialH/\partialp, ṗ = -\partialH/\partialq — symmetric, first-order$ .
Phase space flow preserves volume (Liouville's theorem) — phase space trajectories never cross.
$Poisson bracket {A,H} = 0 ↔ A is conserved. Structure mirrors quantum commutators$ .
$Canonical transformations preserve {q,p} = 1 — symplectic structure$ .
$Action-angle variables: integrable systems \to uniform rotation on a torus. J = n\hbar gives sem$ iclassical quantization.