Invariance of Poisson Brackets under Canonical Maps
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You must be comfortable with the definition of the Poisson bracket for two arbitrary functions of phase space and the four types of generating functions that define canonical transformations.
Why this matters
The Poisson bracket is the algebraic heartbeat of classical mechanics.
In Newtonian mechanics, the structure of the theory is hidden within vector force laws. In Hamiltonian mechanics, the structure is explicit: the time evolution of any observable is determined by its Poisson bracket with the Hamiltonian. If we change coordinates, we must ensure this algebraic structure remains intact. A transformation is "canonical" not merely because it preserves the form of Hamilton's equations, but because it preserves the fundamental commutation relations of the phase space. This invariance is the classical precursor to the commutation relations in quantum mechanics, making it a critical conceptual bridge between the two frameworks.
The Geometric Intuition
Imagine phase space as a landscape of coordinates. A canonical transformation is like rotating your map. The physical distances and angles between points do not change; only your description of them does. In phase space, the "angle" or "area" element defined by the symplectic form \( \mathrm{d}q \wedge \mathrm{d}p \) is the invariant quantity. The Poisson bracket is essentially a measure of how two functions change relative to each other with respect to these symplectic coordinates. If the transformation preserves the symplectic structure (i.e., it is canonical), then the "relative change" measured by the Poisson bracket must remain the same, regardless of whether you calculate it using the old coordinates \((q, p)\) or the new coordinates \((Q, P)\). This means the algebraic relationship between observables is an intrinsic property of the system, not an artifact of the coordinate choice.
Definition and Basic Properties
For two functions \( f(q, p, t) \) and \( g(q, p, t) \) of the phase space variables, the Poisson bracket is defined as:
This operation is bilinear, antisymmetric (\( \{f, g\} = -\{g, f\} \)), and satisfies the Jacobi identity. Crucially, it acts as a derivation: \( \{f, gh\} = \{f, g\}h + g\{f, h\} \). The fundamental brackets are \( \{q_i, q_j\} = 0 \), \( \{p_i, p_j\} = 0 \), and \( \{q_i, p_j\} = \delta_{ij} \). These fundamental brackets define the symplectic structure of the phase space. Any canonical transformation must preserve these specific values.
Preservation under Canonical Transformations
A transformation from \((q, p)\) to \((Q, P)\) is canonical if and only if the Poisson brackets of the new variables with respect to the old coordinates satisfy the same fundamental relations. That is, the transformation preserves the bracket if:
Here, the subscript \( \{ \cdot, \cdot \}_{q,p} \) indicates that the derivatives are taken with respect to the old variables \( q \) and \( p \). If these conditions hold, then for any two functions \( f \) and \( g \), the Poisson bracket calculated in the new coordinates is identical to that in the old coordinates:
This result is profound. It means that the algebraic structure of the observables is invariant under canonical transformations. The time evolution equation \( \frac{\mathrm{d}f}{\mathrm{d}t} = \{f, H\} + \frac{\partial f}{\partial t} \) retains its form, ensuring that the dynamics are consistent across all canonical frames.
Connection to Generating Functions
We established in the previous lecture that canonical transformations can be generated by a function \( F \). The invariance of the Poisson bracket is a direct consequence of the existence of such a generating function. If a transformation is defined by a generating function \( F_2(q, P, t) \), then the relations \( p = \frac{\partial F_2}{\partial q} \) and \( Q = \frac{\partial F_2}{\partial P} \) ensure that the one-form \( \sum p_i \mathrm{d}q_i - H \mathrm{d}t \) changes only by an exact differential, which in turn means that the symplectic form \( \sum \mathrm{d}p_i \wedge \mathrm{d}q_i \) is preserved. This preservation of the symplectic form is mathematically equivalent to the preservation of the Poisson brackets. Thus, the existence of a generating function is not just a computational tool; it is the guarantee that the algebraic structure of the theory remains intact.
The derivations
Canonical Transformations and Generating Functions
A change of phase-space variables (q,p) → (Q,P) preserves the form of Hamilton's equations if and only if there exists a generating function F such that p q̇ − H = P Q̇ − K + dF/dt. Choosing F to depend on one old and one new variable yields exactly four types — F1(q,Q,t), F2(q,P,t), F3(p,Q,t), F4(p,P,t) — whose partial derivatives supply the transformation equations and the new Hamiltonian K = H + ∂F/∂t.
F3(p,Q): q=−∂pF3, P=−∂QF3 • F4(p,P): q=−∂pF4, Q=∂PF4
and in all four cases K = H + ∂F/∂t
Poisson Brackets and the Algebra of Observables
Derives the Poisson bracket as the natural bilinear operation giving time evolution df/dt = {f,H} + df/dt-partial and the fundamental brackets of phase space.
A common pitfall is confusing the Poisson bracket with the commutator in quantum mechanics. While they are analogous, the Poisson bracket is a classical, c-number operation, whereas the commutator involves non-commuting operators. The correspondence principle states that \( \{f, g\} \) corresponds to \( \frac{1}{i\hbar}[\hat{f}, \hat{g}] \), but one must be careful not to treat classical brackets as if they were quantum operators, especially when dealing with singularities or non-smooth functions. C
Worked example
Problem. Test three transformations of one degree of freedom for canonicity using \( \{Q, P\}_{q,p} = 1 \): (a) \( Q = p,\ P = -q \); (b) \( Q = p,\ P = q \); (c) the scaling \( Q = \lambda q,\ P = p/\lambda \) with \( \lambda = 3.0 \). For the oscillator \( H = \frac{p^2}{2m} + \frac{1}{2}kq^2 \), write \( H \) in the variables of (a).
- (a) \( \{Q, P\} = \frac{\partial p}{\partial q}\frac{\partial(-q)}{\partial p} - \frac{\partial p}{\partial p}\frac{\partial(-q)}{\partial q} = 0 - (1)(-1) = 1 \): canonical. Swapping position and momentum is allowed, provided one sign is flipped.
- (b) \( \{Q, P\} = \{p, q\} = -1 \): not canonical. Hamilton's equations in \( (Q, P) \) would come out with the wrong sign (time running backwards).
- (c) \( \{Q, P\} = \lambda \cdot \frac{1}{\lambda} = 1 \) for any \( \lambda \ne 0 \), so \( \lambda = 3.0 \) is canonical: stretching \( q \) by 3 squeezes \( p \) by 3 and phase-space area is kept.
- With (a), \( p = Q \) and \( q = -P \), so \( K = \frac{Q^2}{2m} + \frac{1}{2}kP^2 \). Then \( \dot Q = \partial K/\partial P = kP = -kq = \dot p \) and \( \dot P = -\partial K/\partial Q = -Q/m = -p/m = -\dot q \), consistent with the original motion.
Units check. \( \{Q, P\} \) is dimensionless only if \( QP \) has the units of \( qp \) (an action, \( \mathrm{J\,s} \)); in (a) and (c) it does, since \( Q P = -pq \) and \( QP = qp \).
Three quick tests
- What is the value of the Poisson bracket \( \{q, q\} \)?
Answer
0 - If a transformation preserves the symplectic form, does it necessarily preserve the Poisson brackets?
Answer
Yes, preservation of the symplectic form is equivalent to preservation of the Poisson brackets. - Calculate \( \{q, p^2\} \).
Answer
\( \frac{\partial q}{\partial q}\frac{\partial p^2}{\partial p} - \frac{\partial q}{\partial p}\frac{\partial p^2}{\partial q} = 1 \cdot 2p - 0 = 2p \)
Problem set for PU-201 →
Each derivation in this unit ends with five graded problems with full solutions.
Lecture notes drafted with a local model and checked line by line by the editor; every number in the worked example was recomputed.
Derivations cited: D-135 · D-005 · Threads: force, energy, symmetry