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 $q_{1}, q_{2}, \dots, q_{n}$ , the associated conjugate momenta $p_{1}, p_{2}, \dots, p_{n}$ , 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?

A. $\partial / \partial t [f, g] = [\partial f / \partial t, g] + [f, \partial g / \partial t]$ ✓
B.If f is a constant of motion, and f is independent of t, then [H, f] is a constant of motion✓
C.[[H, f], g] + [[g, H], f] + [[f, g], H] = 0✓
D.If f and g are constants of motion, then [f, g] is a constant of motion✓

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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50 are analysed free — try those first.

The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

Drill statements like this

Related counterexample: The Hamiltonian always equals the total energy

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

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