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GATE 2025 PH – Question 52

Classical Mechanics · Hamiltonian and Hamilton's equations of motion · 2 marks · Multiple select

A bead is constrained to move along a long, massless, frictionless horizontal rod parallel to the x axis. The rod itself is moving vertically upward along the z direction against gravity with a constant speed, starting from z= 0 at t= 0, and remains horizontal. The conjugate momenta are denoted by px , py , pz and the Hamiltonian by H. Which of the following option(s) is/are correct?

  1. H is the total energy of the system and is conserved
  2. H is the total energy of the system and is not conserved
  3. H is not the total energy of the system, but it is conserved
  4. H is not the total energy of the system and is not conserved

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Correct answer: (D) H is not the total energy of the system and is not conserved

Explanation

Eliminate the moving constraint $z=ut$: $L=m(\dot x^2+u^2)/2-mgut$. Its Hamiltonian is $m\dot x^2/2-mu^2/2+mgut$, whereas energy has $+mu^2/2$. The Hamiltonian grows at rate $mgu$.