Discussion
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 \(p_{x}, p_{y}, p_{z}\) and the Hamiltonian by \(H\). Which of the following option(s) is/are correct?
Answer: D
The correct statement is \(H\) is not the total energy of the system and is not conserved.
The bead's height is fixed by the rod, \(z = vt\), a time-dependent (rheonomic) constraint; only \(x\) is free. The Lagrangian and Hamiltonian are
\[ \begin{aligned} L &= \tfrac{1}{2}m(\dot{x}^{2} + v^{2}) - mgvt \\ H &= p_{x}\dot{x} - L = \frac{p_{x}^{2}}{2m} - \tfrac{1}{2}mv^{2} + mgvt \end{aligned} \]
The total energy would be \(\frac{p_{x}^{2}}{2m} + \frac{1}{2}mv^{2} + mgvt\), so \(H\) differs from it by \(mv^{2}\): with a moving constraint the Hamiltonian is not the energy. And \(H\) depends explicitly on time through \(mgvt\), so \(\frac{dH}{dt} = \frac{\partial H}{\partial t} = mgv \ne 0\): it is not conserved (the rod does work on the bead).
(This is a multiple-select question; the key accepts this option alone.)
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