Question

The time derivative of a differentiable function \( g(q_i, t) \) is added to a Lagrangian \( L(q_i, \dot{q}_i, t) \) such that \[ L' = L(q_i, \dot{q}_i, t) + \frac{d}{dt} g(q_i, t) \] where \( q_i \), \( \dot{q}_i \), \( t \) are the generalized coordinates, generalized velocities, and time, respectively. Let \( p_i \) be the generalized momentum and \( H \) the Hamiltonian associated with \( L(q_i, \dot{q}_i, t) \). If \( p_i' \) and \( H' \) are those associated with \( L' \), then the correct option(s) is(are):

• A. Both \( L \) and \( L' \) satisfy the Euler-Lagrange's equations of motion.
• B. \( p_i' = p_i + \frac{\partial}{\partial q_i} g(q_i, t) \)
• C. If \( p_i \) is conserved, then \( p_i' \) is necessarily conserved.
• D. \( H' = H + \frac{d}{dt} g(q_i, t) \)

Hint:

Adding a time derivative of a function \( g(q_i, t) \) to the Lagrangian does not affect the Euler-Lagrange equations, but it modifies the generalized momentum and Hamiltonian.

Answer 1

The Correct Option is A, B

In the given problem, we add the time derivative of a function \( g(q_i, t) \) to the Lagrangian \( L \), resulting in a new Lagrangian \( L' \). The goal is to determine the implications of this modification on the generalized momentum \( p_i \) and the Hamiltonian \( H \).
1. Euler-Lagrange Equations: The Euler-Lagrange equations are given by: \[ \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) - \frac{\partial L}{\partial q_i} = 0 \] If we modify the Lagrangian to \( L' = L + \frac{d}{dt} g(q_i, t) \), the new Lagrangian \( L' \) still satisfies the Euler-Lagrange equations. The time derivative of \( g(q_i, t) \) does not affect the form of the equations, so both \( L \) and \( L' \) satisfy the same Euler-Lagrange equations. Thus, option (A) is correct. 2. Generalized Momentum: The generalized momentum \( p_i \) is defined as: \[ p_i = \frac{\partial L}{\partial \dot{q}_i} \] For the modified Lagrangian \( L' \), we have: \[ p_i' = \frac{\partial L'}{\partial \dot{q}_i} = \frac{\partial L}{\partial \dot{q}_i} + \frac{\partial}{\partial \dot{q}_i} \left( \frac{d}{dt} g(q_i, t) \right) \] Since \( g(q_i, t) \) depends only on \( q_i \) and \( t \), we can compute: \[ p_i' = p_i + \frac{\partial}{\partial q_i} g(q_i, t) \] Therefore, option (B) is correct. 3. Conservation of \( p_i \): If \( p_i \) is conserved (i.e., \( \frac{d}{dt} p_i = 0 \)), it does not necessarily imply that \( p_i' \) is conserved, since the function \( g(q_i, t) \) may influence the dynamics of \( p_i' \). Thus, option (C) is not correct. 4. Modified Hamiltonian: The Hamiltonian \( H \) associated with \( L \) is given by: \[ H = \sum_i p_i \dot{q}_i - L \] For the modified Lagrangian \( L' \), the new Hamiltonian \( H' \) is: \[ H' = \sum_i p_i' \dot{q}_i - L' \] Substituting \( L' = L + \frac{d}{dt} g(q_i, t) \), we get: \[ H' = H + \frac{d}{dt} g(q_i, t) \] Therefore, option (D) is correct.
Thus, the correct options are (A) and (B).