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Part CCSIR NET June 2025a-constant-of-motion-with-no-explicit-time-dependence-has-bracket-zero-with-H

A constant of motion with no explicit time dependence has bracket zero with H

Suppose f, g are smooth functions of generalized coordinates q1,q2,…,qnq_{1}, q_{2}, \dots, q_{n}, the associated conjugate momenta p1,p2,…,pnp_{1}, p_{2}, \dots, p_{n}, and time t. Let [f, g] denote the Poisson bracket of f and g. Suppose H is a Hamiltonian of the system. Then which of the following statements are true?

  1. A.∂/∂t[f,g]=[∂f/∂t,g]+[f,∂g/∂t]\partial/\partial{}t [f, g] = [\partial{}f/\partial{}t, g] + [f, \partial{}g/\partial{}t]✓ Correct
  2. B.If f is a constant of motion, and f is independent of t, then [H, f] is a constant of motion✓ Correct
  3. C.[[H, f], g] + [[g, H], f] + [[f, g], H] = 0✓ Correct
  4. D.If f and g are constants of motion, then [f, g] is a constant of motion✓ Correct

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests standard counterexample.

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The trap it tests

Standard counterexample

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Related counterexample: The Hamiltonian always equals the total energy

More on this topic

The chapter behind this: Hamiltonian mechanics, Poisson brackets and canonical transformations — free to read

From Classical Mechanics › Hamiltonian formalism and conservation laws

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