Canonical transformations are central to Hamiltonian mechanics, providing a mechanism to transform phase space coordinates without altering the physical structure of the system. This post provides detailed notes and rigorous proofs for both the Generating Function method and the Symplectic Matrix approach.
Canonical Transformations: Detailed Proofs & Applications
1. The Fundamental Principle
Hamilton's equations are derived from the variational principle \( \delta S = 0 \). For a transformation from old coordinates \( (q, p) \) to new coordinates \( (Q, P) \) to be canonical, the modified Hamilton's principle must hold in the new system.
This requires that the Lagrangian in the old system and the new system differ only by the total time derivative of a scalar function \( F \), called the Generating Function:
Multiplying by \( dt \), we obtain the fundamental differential relation used for proofs:
2. Proof of Generating Function Equations
We derive the transformation equations explicitly for Type 1 and Type 2 functions. The others follow similarly via Legendre transforms.
Proof for Type 1: \( F = F_1(q, Q, t) \)
Rearranging the fundamental relation to isolate \( dF_1 \): \[ dF_1 = p dq - P dQ + (K - H) dt \]
Since \( F_1 \) is a function of \( q, Q, \) and \( t \), we can also write its total differential mathematically using the chain rule: \[ dF_1 = \frac{\partial F_1}{\partial q} dq + \frac{\partial F_1}{\partial Q} dQ + \frac{\partial F_1}{\partial t} dt \]
Since \( q, Q, t \) are treated as independent variables here, we equate the coefficients of the differentials \( dq, dQ, \) and \( dt \) from the two equations above:
1. Coeff of \( dq \): \( p = \frac{\partial F_1}{\partial q} \)
2. Coeff of \( dQ \): \( -P = \frac{\partial F_1}{\partial Q} \)
3. Coeff of \( dt \): \( K = H + \frac{\partial F_1}{\partial t} \)
Proof for Type 2: \( F = F_2(q, P, t) \)
We perform a Legendre Transformation to change the dependence from \( Q \) to \( P \). We define: \[ F_2(q, P, t) = F_1(q, Q, t) + PQ \]
Taking the differential: \[ dF_2 = dF_1 + P dQ + Q dP \]
Substitute the fundamental relation for \( dF_1 \) (\( p dq - P dQ + (K-H)dt \)): \[ dF_2 = (p dq - P dQ + (K-H)dt) + P dQ + Q dP \] \[ dF_2 = p dq + Q dP + (K-H)dt \]
Now, expand \( dF_2 \) via the chain rule for variables \( (q, P, t) \): \[ dF_2 = \frac{\partial F_2}{\partial q} dq + \frac{\partial F_2}{\partial P} dP + \frac{\partial F_2}{\partial t} dt \]
Comparing coefficients again yields the standard Type 2 equations:
1. \( p = \frac{\partial F_2}{\partial q} \)
2. \( Q = \frac{\partial F_2}{\partial P} \)
3. \( K = H + \frac{\partial F_2}{\partial t} \)
3. Proof of the Symplectic Condition
The symplectic approach provides an algebraic condition for canonical transformations involving the Jacobian matrix.
Let \( \eta \) be the column vector of phase space coordinates: \[ \eta = \begin{pmatrix} q \\ p \end{pmatrix} \]
Hamilton's equations can be written in matrix form as: \[ \dot{\eta} = J \nabla_\eta H, \quad \text{where } J = \begin{pmatrix} 0 & I \\ -I & 0 \end{pmatrix} \]
Derivation
Let us transform to new coordinates \( \zeta = \zeta(\eta) \). The equations of motion in the new system must preserve the form: \[ \dot{\zeta} = J \nabla_\zeta K \]
Consider the time derivative \( \dot{\zeta} \) using the Jacobian matrix \( M \) where \( M_{ij} = \frac{\partial \zeta_i}{\partial \eta_j} \): \[ \dot{\zeta} = M \dot{\eta} \]
Substitute the old equation of motion \( \dot{\eta} = J \nabla_\eta H \): \[ \dot{\zeta} = M (J \nabla_\eta H) \]
We must relate \( \nabla_\eta H \) to \( \nabla_\zeta H \). By the chain rule: \( \nabla_\eta H = M^T \nabla_\zeta H \). Substituting this back: \[ \dot{\zeta} = M J M^T \nabla_\zeta H \]
For this to match the canonical form \( \dot{\zeta} = J \nabla_\zeta H \) (assuming \( K=H \) for time-independent transformations), we require:
This is the necessary and sufficient condition for a transformation to be canonical. It proves that the transformation preserves the symplectic structure of phase space.
4. Application: The Harmonic Oscillator
We solve the Harmonic Oscillator by finding a transformation that makes the Hamiltonian cyclic (independent of \( Q \)).
Old Hamiltonian: \( H = \frac{p^2}{2m} + \frac{1}{2} m \omega^2 q^2 = E \)
We propose a Type 1 Generating Function: \( F_1(q, Q) = \frac{1}{2} m \omega q^2 \cot Q \).
Applying the Transformation:
1. From \( p = \frac{\partial F_1}{\partial q} \):
\[ p = m \omega q \cot Q \]
2. From \( P = -\frac{\partial F_1}{\partial Q} \):
\[ P = -\frac{1}{2} m \omega q^2 (-\csc^2 Q) = \frac{m \omega q^2}{2 \sin^2 Q} \]
Inverting these to find \( q \) and \( p \) in terms of \( (Q, P) \): \[ q = \sqrt{\frac{2P}{m\omega}} \sin Q, \quad p = \sqrt{2m\omega P} \cos Q \]
Substituting these into the Hamiltonian \( H \): \[ H = \frac{1}{2m}(2m\omega P \cos^2 Q) + \frac{1}{2}m\omega^2 \left( \frac{2P}{m\omega} \sin^2 Q \right) \] \[ H = \omega P (\cos^2 Q + \sin^2 Q) = \omega P \]
Result: Since the new Hamiltonian \( K = \omega P \) depends only on \( P \), the equations of motion are trivial: \[ \dot{P} = 0 \Rightarrow P = \text{const}, \quad \dot{Q} = \omega \Rightarrow Q = \omega t + \alpha \]