University:
Santa Fe College
Course:
PHY 2004 | Applied Physics 1
Academic year:

2024
Views:

359

Pages:

5
Author:

Jabin F.

# Poisson Bracket (PB): For any two quantities, PB is defined as: {f, g} = Σᵢ (∂f/∂qi * ∂g/∂pi - ∂f/∂pi * ∂g/∂qi) Quantities present inside PB must be written in terms of generalized coordinates and generalized momentum. Q-13: A particle is moving in a potential V = (1/2)(x^2 + y^2 + z^2). For what value of μ is the y-component of angular momentum conserved? Solution: Ly is conserved if (∂²L/∂y² + ∂²L/∂y∂t) is present in the Lagrangian and this Lagrangian is invariant under rotation about the y-axis. Therefore: → B can have any value. ⇒ y² = 0 ⇒ y = 0 only if x = 1. P-89 Q-5: The invariance of the Lagrangian L = (1/2)m(ẋ² + ẏ² + ż²) - V(x, y, z), where z = x + iy, under the transformation z → e^(iθ)z, leads to the conservation of: Solution: L' = (1/2)m(ẋ² + ẏ² + ż²) - V(x', y', z') = (1/2)m(ẋ² + ẏ² + (ẋ + iẏ)²) - V(x + iy, y, z) = (1/2)m(2ẋ² + 2ẏ²) - V(x + iy, y, z) = (1/2)m(2ẋ² + 2ẏ²) - V(x, y, z) Lz is conserved as x² + y², x² + y² is present in L and L is invariant. z = x + iy → z' = e^(iθ)z = x' + iy' ← angle is changing. Thus, there is transformation is rotation. ⇒ Invariance of L under rotation leads to conservation of angular momentum.