Poisson Bracket, Poisson Theorems

Learning Objectives:

Understand the definition and meaning of a Poisson bracket in classical mechanics.
Derive and interpret Poisson’s theorems.
Use Poisson brackets to verify conservation laws and symmetries.

Key Concepts / Definitions:

Poisson Bracket: A bilinear operation defined between two functions in phase space, used extensively in Hamiltonian mechanics.
Canonical Variables: Pairs of variables like $(q_i, p_i)$ that satisfy specific Poisson bracket relations.
Poisson Theorems: Theorems that describe the properties and implications of Poisson brackets such as their antisymmetry, bilinearity, and Jacobi identity.

Theory and Explanation:

In Hamiltonian mechanics, the dynamics of a system are described by a set of generalized coordinates $q_i$ and conjugate momenta $p_i$, evolving according to Hamilton’s equations:

\[\dot{q}_i = \frac{\partial H}{\partial p_i}, \quad \dot{p}_i = -\frac{\partial H}{\partial q_i}\]

Here, $H(q_i, p_i, t)$ is the Hamiltonian of the system.

To express these equations and many other properties compactly, we define the Poisson bracket of two functions $f(q_i, p_i, t)$ and $g(q_i, p_i, t)$ as:

\[\{f, g\} = \sum_{i} \left( \frac{\partial f}{\partial q_i} \frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i} \frac{\partial g}{\partial q_i} \right)\]

The Poisson bracket has the following essential properties:

Bilinearity: $\{af + bg, h\} = a\{f, h\} + b\{g, h\}$
Antisymmetry: $\{f, g\} = -\{g, f\}$
Jacobi Identity: $\{f, \{g, h\}\} + \{g, \{h, f\}\} + \{h, \{f, g\}\} = 0$
Leibniz Rule (Product Rule): $\{fg, h\} = f\{g, h\} + g\{f, h\}$

Using Poisson brackets, Hamilton’s equations can be rewritten as:

\[\dot{f} = \{f, H\} + \frac{\partial f}{\partial t}\]

This shows that the time evolution of any observable $f$ is governed by its Poisson bracket with the Hamiltonian.

Poisson Theorems:

Theorem 1: If $u$ and $v$ are constants of motion, then ${u, v}$ is also a constant of motion.
Theorem 2: The fundamental Poisson brackets are: $\{q_i, q_j\} = 0, \quad \{p_i, p_j\} = 0, \quad \{q_i, p_j\} = \delta_{ij}$
Theorem 3: Canonical transformations preserve the form of the Poisson brackets.

Solved Examples:

Example 1:
Problem: Show that the angular momentum components satisfy the Poisson bracket relation ${L_x, L_y} = L_z$.
Solution:
Recall that:
$L_x = yp_z - zp_y, \quad L_y = zp_x - xp_z, \quad L_z = xp_y - yp_x$
Compute ${L_x, L_y}$ using the definition of the Poisson bracket:
\[\{L_x, L_y\} = \{yp_z - zp_y, zp_x - xp_z\}\]
Calculating term by term and using the fundamental brackets, we get:
\[\{L_x, L_y\} = xp_y - yp_x = L_z\]
Example 2:
Problem: Verify that $H = \frac{p^2}{2m} + V(q)$ is conserved using Poisson bracket.
Solution:
Compute $\dot{H}$: $\dot{H} = \{H, H\} + \frac{\partial H}{\partial t}$

Since ${H, H} = 0$ and if $H$ has no explicit time dependence, then:
\[\dot{H} = 0 \Rightarrow H \text{ is conserved}\]

Important Points / Summary:

Poisson brackets provide a compact and general formalism to express time evolution in Hamiltonian mechanics.
They are fundamental to understanding symmetries, conservation laws, and canonical transformations.
Poisson’s theorems play a central role in identifying constants of motion and maintaining the structure of mechanics under transformations.

Practice Questions:

Short Answer:
1. Define the Poisson bracket. What does it signify in Hamiltonian mechanics?
2. State and explain the Jacobi identity for Poisson brackets.
Numerical:
1. Given $f = q^2p$ and $g = qp^2$, compute ${f, g}$.
2. For a simple harmonic oscillator with $H = \frac{p^2}{2m} + \frac{1}{2}m\omega^2q^2$, compute ${q, H}$ and ${p, H}$.
MCQs:
1. Which of the following is a property of the Poisson bracket?
  - (A) Commutativity
  - (B) Antisymmetry
  - (C) Associativity
  - (D) Distributivity
    Answer: (B)
2. If ${f, H} = 0$, then:
  - (A) $f$ is conserved in time
  - (B) $f$ is zero
  - (C) $f$ is a function of time only
  - (D) $f$ must be the Hamiltonian
    Answer: (A)