Question

The Lagrangian of a particle of mass \( m \) and charge \( q \) moving in a uniform magnetic field of magnitude \( 2B \) that points in the \( z \)-direction, is given by: \[ L = \frac{m}{2} v^2 + qB(x v_y - y v_x) \] where \( v_x, v_y, v_z \) are the components of its velocity \( v \). If \( p_x, p_y, p_z \) denote the conjugate momenta in the \( x, y, z \)-directions and \( H \) is the Hamiltonian, which of the following option(s) is/are correct?

• A. \( \frac{d x}{d t} = \frac{1}{m} (p_x - q B y) \)
• B. \( \frac{d p_x}{d t} = \frac{q B}{m} (p_y - q B x) \)
• C. \( \frac{d p_y}{d t} = -\frac{q B}{m} (p_x + q B y) \)
• D. \( H = \frac{1}{2m} \left[ (p_x + q B y)^2 + (p_y - q B x)^2 + p_z^2 \right] \)

Hint:

In systems involving a magnetic field, the velocity components and momenta are coupled through the interaction term, and the Hamiltonian leads to equations of motion that account for the Lorentz force.

Answer 1

The Correct Option is B, C, D

1. Conjugate momenta:
The conjugate momenta are given by: \[ p_x = \frac{\partial L}{\partial v_x} = m v_x + q B y \] \[ p_y = \frac{\partial L}{\partial v_y} = m v_y - q B x \] \[ p_z = \frac{\partial L}{\partial v_z} = m v_z \] 2. Hamiltonian:
The Hamiltonian \( H \) is the Legendre transform of the Lagrangian: \[ H = p_x v_x + p_y v_y + p_z v_z - L \] Substituting for \( v_x, v_y, v_z \) and simplifying the expression, we get: \[ H = \frac{1}{2m} \left[ (p_x + q B y)^2 + (p_y - q B x)^2 + p_z^2 \right] \] which matches option (D). 
3. Equations of motion:
The equations of motion follow from the Hamiltonian dynamics. For \( \frac{d x}{d t} \), we have: \[ \frac{d x}{d t} = \frac{\partial H}{\partial p_x} = \frac{1}{m} (p_x - q B y) \] which matches option (A). Similarly, for \( \frac{d p_x}{d t} \), we get: \[ \frac{d p_x}{d t} = -\frac{\partial H}{\partial x} = \frac{q B}{m} (p_y - q B x) \] which matches option (B). For \( \frac{d p_y}{d t} \), we get: \[ \frac{d p_y}{d t} = -\frac{\partial H}{\partial y} = -\frac{q B}{m} (p_x + q B y) \] which matches option (C). Thus, the correct answers are (B), (C), and (D).