Physics Problems & Solutions: Classical Mechanics

If ∂L/∂q_n = 0, where L is the Lagrangian for a conservative system without constraints and q_n is a generalized coordinate, then the generalized momentum p_n is

A. An ignorable coordinate
B. Constant
C. Undefined
D. Equal to $\frac{d}{dt} \left (\frac{\partial L}{\partial q_n} \right )$
E. Equal to the Hamiltonian for the system

(GR9277 #43)

Solution:

Lagrangian Equation of Motion for the generalized coordinate q:

$\frac{\partial L}{\partial q}=\frac{\partial}{\partial t}\frac{\partial L}{\partial \dot q}$

Momentum,

$\begin{aligned} p&=mv=m\dot{x} \\L&=T-U=\frac{1}{2}m\dot{x}^2-mgy\\&\rightarrow \frac{\partial L}{\partial \dot{x}}=m\dot{x}=p \end{aligned}$

For the generalized coordinate q:

If $\frac{\partial L}{\partial q_n}=0 \rightarrow \frac{\partial L}{\partial \dot{q_n}}=p_n= \text{constant}$

Answer: B

Physics Problems & Solutions

Classical Mechanics - Lagrangian

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