Canonical Transformations & Poisson Brackets

1. Motivation for Coordinate Transformations in Phase Space

In Lagrangian mechanics, coordinate transformations are restricted to point transformations $Q_i = Q_i(q, t)$. In Hamiltonian mechanics, coordinates and conjugate momenta $(q, p)$ are treated on an equal footing. We consider broader transformations: A transformation is defined as canonical (or contact) if there exists a new Hamiltonian $K(Q, P, t)$ such that the new variables satisfy Hamilton's canonical equations:

2. Variational Condition & Generating Function

Both original and transformed trajectories must satisfy the modified Hamilton's principle: The two integrands can differ at most by the exact total time derivative of an arbitrary function $F$, called the generating function: Multiplying by $dt$:

3. The Symplectic Condition

Defining the $2n$-dimensional phase vector $\mathbf{\eta} = (q_1, \dots, q_n, p_1, \dots, p_n)^T$ and the fundamental symplectic matrix $\mathbf{J}$: Let $\mathbf{M}$ be the Jacobian matrix of the transformation $M_{ij} = \frac{\partial \zeta_i}{\partial \eta_j}$ where $\mathbf{\zeta} = (Q, P)^T$. The transformation is canonical if and only if $\mathbf{M}$ satisfies the symplectic condition: Taking the determinant yields $(\det \mathbf{M})^2 = 1 \implies \det \mathbf{M} = 1$, proving that phase space volume is strictly preserved.

1. Classification by Active Canonical Variables

Depending on which pair of old and new variables are chosen as independent variables, Legendre transformations yield four primary classes of generating functions:

1. Type 1: $F_1(q, Q, t)$
   $$dF_1 = \sum_i p_i dq_i - \sum_i P_i dQ_i + (K - H) dt$$
   Equating partial differentials:
   $$p_i = \frac{\partial F_1}{\partial q_i}, \quad P_i = -\frac{\partial F_1}{\partial Q_i}, \quad K = H + \frac{\partial F_1}{\partial t}$$
2. Type 2: $F_2(q, P, t) = F_1 + \sum_i P_i Q_i$
   $$dF_2 = \sum_i p_i dq_i + \sum_i Q_i dP_i + (K - H) dt$$
   Transformation relations:
   $$p_i = \frac{\partial F_2}{\partial q_i}, \quad Q_i = \frac{\partial F_2}{\partial P_i}, \quad K = H + \frac{\partial F_2}{\partial t}$$

   Example: The identity transformation is generated by $F_2 = \sum_i q_i P_i$, giving $p_i = P_i$ and $Q_i = q_i$.

3. Type 3: $F_3(p, Q, t) = F_1 - \sum_i p_i q_i$
   $$dF_3 = -\sum_i q_i dp_i - \sum_i P_i dQ_i + (K - H) dt$$
   Transformation relations:
   $$q_i = -\frac{\partial F_3}{\partial p_i}, \quad P_i = -\frac{\partial F_3}{\partial Q_i}, \quad K = H + \frac{\partial F_3}{\partial t}$$
4. Type 4: $F_4(p, P, t) = F_1 - \sum_i p_i q_i + \sum_i P_i Q_i$
   $$dF_4 = -\sum_i q_i dp_i + \sum_i Q_i dP_i + (K - H) dt$$
   Transformation relations:
   $$q_i = -\frac{\partial F_4}{\partial p_i}, \quad Q_i = \frac{\partial F_4}{\partial P_i}, \quad K = H + \frac{\partial F_4}{\partial t}$$

1. Definition of the Poisson Bracket

Let $u(q, p, t)$ and $v(q, p, t)$ be two continuously differentiable functions defined on phase space. The Poisson bracket $[u, v]_{q,p}$ with respect to canonical variables $(q, p)$ is:

2. Fundamental Algebraic Identities

Poisson brackets satisfy the defining axioms of a Lie algebra:

1. Anti-Symmetry: $\{u, v\} = -\{v, u\} \implies \{u, u\} = 0$.
2. Bilinearity: $\{a u + b v, w\} = a\{u, w\} + b\{v, w\}$ for scalars $a, b$.
3. Leibniz Product Rule: $\{u v, w\} = u\{v, w\} + \{u, w\}v$.
4. The Jacobi Identity:
   $$\{u, \{v, w\}\} + \{v, \{w, u\}\} + \{w, \{u, v\}\} = 0$$

3. Fundamental Canonical Poisson Brackets

Evaluating the brackets for the fundamental coordinates and conjugate momenta yields:

Canonical Invariance: A transformation $(q, p) \to (Q, P)$ is canonical if and only if it preserves the fundamental Poisson brackets: $\{Q_j, Q_k\}_{q,p} = 0$, $\{P_j, P_k\}_{q,p} = 0$, and $\{Q_j, P_k\}_{q,p} = \delta_{jk}$.

Quantum Correspondence: Paul Dirac recognized that the quantum commutator $[\hat{u}, \hat{v}]$ directly maps to the classical Poisson bracket: $[\hat{u}, \hat{v}] = i \hbar \{u, v\}$.

1. Time Evolution of an Arbitrary Phase Space Observable

Let $f(q, p, t)$ be any dynamical variable. Its total time derivative along a trajectory is: Substituting Hamilton's equations $\dot{q}_j = \frac{\partial H}{\partial p_j}$ and $\dot{p}_j = -\frac{\partial H}{\partial q_j}$: Recognizing the Poisson bracket with the Hamiltonian:

This is the master equation of classical Hamiltonian dynamics. Setting $f = q_j$ and $f = p_j$ automatically reproduces Hamilton's equations $\dot{q}_j = \{q_j, H\}$ and $\dot{p}_j = \{p_j, H\}$.

2. First Integrals & Constants of Motion

If an observable $f(q, p)$ has no explicit time dependence ($\frac{\partial f}{\partial t} = 0$), then $f$ is a constant of motion if and only if its Poisson bracket with the Hamiltonian vanishes:

3. Poisson's Theorem for Generating New Conserved Quantities

Poisson's Theorem: If $f(q, p)$ and $g(q, p)$ are two independent constants of motion (so that $\{f, H\} = 0$ and $\{g, H\} = 0$), then their Poisson bracket $\{f, g\}$ is also a constant of motion.

Proof via the Jacobi identity: Hence $\frac{d}{dt}\{f, g\} = 0$. Poisson's theorem enables systematically discovering new symmetries and conservation laws.

1. Poincaré's Integral Invariants

Henri Poincaré proved that certain differential forms integrated over closed submanifolds in phase space remain invariant under canonical transformations and Hamiltonian time evolution. The first Poincaré integral invariant of order 1 is: where $C$ is any closed circuit in phase space carried along by the Hamiltonian flow. Higher-order invariants $J_2, \dots, J_n$ correspond to integrals over $2k$-dimensional manifolds. The invariant of maximum order $2n$ represents the total volume of phase space.

2. Liouville's Phase Volume Conservation Theorem

Consider an ensemble of non-interacting identical systems represented by a cloud of phase points with density distribution $\rho(q, p, t)$ in $2n$-dimensional phase space. By the continuity equation for probability conservation: Expanding the derivatives: Using Hamilton's canonical equations: The divergence of phase velocity vanishes identically: $\nabla \cdot \vec{v}_{\text{phase}} = 0$ (the phase flow is strictly incompressible). Consequently, the convective total time derivative vanishes:

Liouville's Theorem: The phase space volume $\Gamma = \int dq dp$ and the local phase space density $\rho$ surrounding any moving system point remain strictly constant over time. Phase fluid flows like an incompressible liquid, preventing trajectories from ever crossing.