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Part CCSIR NET December 2025integrate-the-two-type-3-relations-for-g-separately-then-use-g-of-2-comma-e-equals-0-only-to-pin-the-constant-of-integration

Integrate the two type 3 relations for g separately then use g of 2 comma e equals 0 only to pin the constant of integration

A mechanical system is described using generalized position q and generalized momentum p. Let Q and P denote new generalized position and generalized momentum variables respectively, generated by the generating function F(q,P)=q2ePF(q,P)=q^{2}e^P, and Q,P are canonical coordinates. Let G(p,Q) be a function such that G(2,e)=0, and it generates the same canonical coordinates Q,P. Then which of the following statements are true?

  1. A.G(p,Q)=−Q[1+loge(p2/(4Q))]G(p,Q) = -Q[1 + log_{e}(p^{2}/(4Q))]✓
  2. B.G(p,Q) = −pQ[1+loge(Q2/(4p))][1 + log_{e}(Q^{2}/(4p))]
  3. C.p = 2qeP,Q=q2eP^P, Q = q^{2}e^P✓
  4. D.p = −2qeP,Q=−q2eP^P, Q = -q^{2}e^P

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.

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50 are analysed free — try those first.

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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