Hamiltonian Mechanics: Advanced Topics

10.1 Canonical Transformations

A canonical transformation from $(q, p)$ to $(Q, P)$ preserves the form of Hamilton’s equations. It is generated by a generating function $F$.

Type 1 ($F_1(q, Q, t)$): $p = \partial F_1/\partial q$, $P = -\partial F_1/\partial Q$, $K = H + \partial F_1/\partial t$.

Type 2 ($F_2(q, P, t)$): $p = \partial F_2/\partial q$, $Q = \partial F_2/\partial P$, $K = H + \partial F_2/\partial t$.

Type 3 ($F_3(p, Q, t)$): $q = -\partial F_3/\partial p$, $P = -\partial F_3/\partial Q$, $K = H + \partial F_3/\partial t$.

Type 4 ($F_4(p, P, t)$): $q = -\partial F_4/\partial p$, $Q = \partial F_4/\partial P$, $K = H + \partial F_4/\partial t$.

The transformation is canonical if and only if:

$\sum_i p_i\,dq_i - \sum_i P_i\,dQ_i = dF$

Poisson bracket invariance. Canonical transformations preserve the Poisson bracket: $\{F, G\}_{q,p} = \{F, G\}_{Q,P}$.

Example. The point transformation $Q_i = q_i$ (just relabelling coordinates) with $P_i = p_i$ is trivially canonical. A non-trivial example: the rotation $Q = q\cos\theta + p\sin\theta$, $P = -q\sin\theta + p\cos\theta$ is canonical (it preserves $dq \wedge dp = dQ \wedge dP$).

10.2 Action-Angle Variables

For a periodic system with frequency $\omega$, define the action:

$J_i = \oint p_i\,dq_i$

The conjugate angle variable $\theta_i$ evolves linearly: $\theta_i(t) = \omega_i t + \theta_i(0)$.

Properties:

• $J_i$ is adiabatically invariant (conserved under slow parameter changes).
• The motion is trivial in action-angle variables: $J_i = \text{const}$, $\dot{\theta}_i = \omega_i$.
• The transformation to action-angle variables is a canonical transformation.

Application: Kepler problem. In the Kepler problem, the action variables are: $J_1 = L$ (angular momentum), $J_2 = L + L_z$ (related to eccentricity), $J_3 = -E/\omega$ (related to energy). The angle variables give the orbital elements.

10.3 Hamilton—Jacobi Equation

The Hamilton—Jacobi equation is a reformulation of Hamiltonian mechanics as a PDE for Hamilton’s principal function $S(q, \alpha, t)$:

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

If $S$ can be found by separation of variables, the transformation to new coordinates makes all momenta constant, effectively solving the problem.

Separation for the harmonic oscillator. For $H = p^2/(2m) + m\omega^2 q^2/2$:

$S(q, E, t) = \int \sqrt{2m(E - \frac{1}{2}m\omega^2 q^2)}\,dq - Et$

The new momentum $P_1 = \alpha_1 = E$ (constant), and the new coordinate $Q_1 = \partial S/\partial E$ gives a linear function of time.

Maupertuis principle. For energy-conserving systems, the Hamilton—Jacobi equation simplifies to:

$2m\left[E - V(q)\right] = \left(\frac{\partial W}{\partial q}\right)^2$

where $W$ is the abbreviated action. This is equivalent to Fermat’s principle in optics.

10.4 Adiabatic Invariants

An adiabatic invariant is a quantity that remains constant when a parameter of the system is changed slowly compared to the period of motion.

For a harmonic oscillator with slowly varying $\omega(t)$:

$\frac{E}{\omega} = \text{const} \quad \text{(adiabatic invariant)}$

This has important applications:

• Fermi acceleration: Cosmic rays gaining energy from moving magnetic fields
• Magnetic mirrors: Charged particles trapped between magnetic mirrors conserve $\mu = mv_\perp^2/(2B)$ (the magnetic moment)
• Planetary orbits: The semi-major axis is an adiabatic invariant under slow mass loss from the Sun
• Pendulum with slowly changing length: If the length $l(t)$ changes slowly, $E/\sqrt{g/l}$ is conserved.

Derivation (sketch). For a periodic Hamiltonian $H(q, p, \lambda(t))$ with $\dot{\lambda} \ll \omega$, the action $J = \oint p\,dq$ satisfies $\dot{J} = O(\dot{\lambda}/\omega)$, so $J$ is approximately constant over many periods.

10.5 Liouville’s Theorem and Phase Space

Liouville’s theorem: The phase space distribution function $\rho(q, p, t)$ is constant along trajectories:

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

This means phase space volume is conserved: a region of phase space evolves like an incompressible fluid.

Consequences:

• The phase space density of an ensemble of systems cannot increase
• This constrains the focusing of particle beams in accelerators
• The theorem underpins the ergodic hypothesis in statistical mechanics
• Poincare recurrence: bounded Hamiltonian systems return arbitrarily close to their initial state

Proof. The flow $\phi_t$ generated by Hamilton’s equations satisfies $\det(\partial(q', p')/\partial(q, p)) = 1$ (canonical transformation), so the Jacobian is 1. By the change of variables formula, $\rho$ is conserved along trajectories. $\square$

10.6 Perturbation Theory and KAM Theorem

Perturbation theory addresses Hamiltonian systems close to integrable ones: $H = H_0 + \varepsilon H_1$.

Lindstedt—Poincare method. Expand both the solution and the frequency in powers of $\varepsilon$ to avoid secular terms.

KAM theorem (Kolmogorov—Arnold—Moser). For sufficiently small $\varepsilon$, most invariant tori of the integrable system persist under perturbation, provided the frequency vector satisfies a Diophantine condition (sufficiently irrational).

The surviving tori have positive measure for small $\varepsilon$, but the remaining phase space contains chaotic regions. This explains the stability of the solar system over billions of years (integrable Kepler problem + small planetary perturbations).

Common Pitfalls

1. Using the wrong generating function type. Each type of generating function expresses different variables as independent. Choosing the wrong type leads to implicit equations that may not be solvable.

2. Forgetting the time dependence in generating functions. If $F$ depends explicitly on $t$, the new Hamiltonian is $K = H + \partial F/\partial t$, not $K = H$.

3. Confusing canonical with point transformations. Not all coordinate changes are canonical. A transformation is canonical only if it preserves the symplectic structure $dp \wedge dq$.

4. Applying perturbation theory to non-integrable systems. The KAM theorem requires the unperturbed system to be integrable. For chaotic systems, perturbation theory fails.