Hamiltonian Mechanics and Poisson Brackets

Why Do We Need Phase Space?

Lagrangian mechanics operates in configuration space {q} and velocities {q̇}. Hamiltonian mechanics transitions to phase space {q, p}, where p are the generalized momenta. At first glance, this appears to be merely a change of variables. In reality, it opens access to a deep symmetry of the equations of motion, creates a geometry for classical mechanics, and directly leads to quantization.

The key advantage: Hamilton’s equations are a system of first-order equations (not second order, as in Lagrange). This simplifies numerical integration and stability analysis.

Legendre Transformation and the Hamiltonian

The generalized momentum conjugate to qᵢ: pᵢ = ∂L/∂q̇ᵢ

The Hamiltonian is obtained by the Legendre transformation:

H(q, p, t) = Σᵢ pᵢ q̇ᵢ − L(q, q̇, t)

The meaning of the sum Σ pᵢ q̇ᵢ: for a point mass p = mv, q̇ = v, and pq̇ = mv² = 2T. Then H = 2T − L = 2T − (T − V) = T + V — the total mechanical energy! This is valid for conservative systems with natural kinetic energies T = (1/2)Σ mᵢ q̇ᵢ².

Hamilton’s Equations

Varying H with respect to q and p yields a system of 2n first-order equations:

q̇ᵢ = ∂H/∂pᵢ — "velocity per coordinate"
ṗᵢ = −∂H/∂qᵢ — "force in phase space"

Compare with Lagrange’s equations (n second-order equations). Hamilton’s equations are symmetric: q and p enter them on an absolutely equal footing.

On the symbols: ∂H/∂pᵢ is the partial derivative of the Hamiltonian with respect to the i-th momentum at fixed q, all other p, and t. The minus sign in ṗᵢ = −∂H/∂qᵢ shows that a decrease in H with respect to qᵢ accelerates the growth of pᵢ.

Poisson Brackets

For arbitrary functions F(q, p) and G(q, p), the Poisson bracket is defined as:

{F, G} = Σᵢ (∂F/∂qᵢ · ∂G/∂pᵢ − ∂F/∂pᵢ · ∂G/∂qᵢ)

Properties: antisymmetry {F, G} = −{G, F}; linearity; Jacobi identity {{F,G},H} + {{G,H},F} + {{H,F},G} = 0; Leibniz rule {FG, H} = F{G, H} + {F, H}G.

Fundamental brackets: {qᵢ, qⱼ} = 0, {pᵢ, pⱼ} = 0, {qᵢ, pⱼ} = δᵢⱼ (Kronecker delta: 1 if i=j, 0 otherwise).

Equation of evolution: for any observable F(q, p, t):

dF/dt = {F, H} + ∂F/∂t

If {F, H} = 0 and ∂F/∂t = 0, then F is an integral of motion (is conserved).

Example: Harmonic Oscillator

Mass m on a spring of stiffness k: H = p²/(2m) + kq²/2

Step 1. q̇ = ∂H/∂p = p/m → p = mq̇ (velocity in terms of momentum)

Step 2. ṗ = −∂H/∂q = −kq (spring force)

Step 3. Differentiating the first equation: q̈ = ṗ/m = −kq/m → q̈ + ω²q = 0, where ω = √(k/m)

Step 4. Solution: q(t) = A cos(ωt + φ), p(t) = −mωA sin(ωt + φ)

Step 5. Phase trajectories: H = p²/(2m) + kq²/2 = E = const — ellipses in the (q, p) plane with semi-axes √(2mE) and √(2E/k)

Step 6. Verification: {q, H} = {q, p²/(2m)} = p/m = q̇ ✓; {p, H} = {p, kq²/2} = −kq = ṗ ✓

Liouville’s Theorem: Conservation of Volume

The Hamiltonian flow in phase space preserves volume:

d/dt ∫∫ dq dp = 0

This is Liouville’s theorem. Geometrically: a set of initial conditions, evolving over time, changes shape but not volume. Analogy: an incompressible fluid.

Liouville’s theorem is fundamental to statistical mechanics: equal probability of microstates with the same energy.

Connection with Quantization

Upon quantization, Poisson brackets are replaced by commutators:

{F, G} → (1/iℏ)[F̂, Ĝ]

Hence: {q, p} = 1 → [q̂, p̂] = iℏ — this is precisely canonical quantization! The Heisenberg uncertainty principle Δq·Δp ≥ ℏ/2 follows from this commutator.

Real Application: Satellite Orbits

In the two-body problem (satellite-Earth), the Hamiltonian H = p²/(2m) − GMm/r (in polar coordinates). The angular momentum component Lz = r·pφ is an integral of motion, since {Lz, H} = 0. The Hamiltonian formalism allows systematically finding all integrals of motion through the structure of Poisson brackets.

Hamiltonian Mechanics and Quantum Theory

The Hamiltonian formalism turned out to be a crucial bridge from classical to quantum mechanics. The transition from Poisson brackets {A, B} to commutators [Â, B̂]/(iℏ) is the procedure of "canonical quantization": a classical function on phase space becomes an operator in Hilbert space, and the Hamiltonian itself defines the Schrödinger equation. In chaos theory, Hamiltonian dynamics is studied via the Poincaré section: a two-dimensional slice of a phase trajectory reveals the structure of tori (regular motion) and chaotic regions (stochastic motion). The KAM theorem (Kolmogorov–Arnold–Moser) describes which tori survive under small perturbations of an integrable system, which is used in calculations of the long-term stability of planetary orbits. In optics, Fermat’s principle is formally analogous to Hamilton’s principle for mechanics: optical rays are "geodesics" in a medium with a refractive index. This led Hamilton to develop Hamiltonian optics — the predecessor of modern wave optics. The phase space of symplectic geometry, preserving the two-form ω = dq∧dp, is the theoretical foundation of numerical symplectic integrators used in molecular dynamics and astrophysical simulations.

Assignment: Harmonic oscillator H = p²/2m + mω²q²/2. (a) Write down Hamilton’s equations. Solve them analytically. (b) Compute {H, q} and {H, p}. Confirm that {H, H} = 0. (c) Introduce a = √(mω/2ℏ)(q + ip/mω). Show that {a, a*} = i/ℏ. This is a precursor of the annihilation/creation operators in quantum field theory.