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

2024
Views:

100

Pages:

5
Author:

Jabin F.

#Generalised Momentum (p_i): It may be linear momentum, angular momentum or maybe something else. It is defined as: p_i = ∂L/∂q̇_i where L is the Lagrangian and q̇_i is the generalized velocity. If Lagrangian is written in terms of velocity q̇_i: eq 1: q_i = x_i, q̇_i = ẋ_i, p_i = ∂L/∂ẋ_i = linear momentum eq 2: q_i = θ_i, q̇_i = θ̇_i, p_i = ∂L/∂θ̇_i = angular momentum When generalized coordinate is angle θ, then generalized momentum represents angular momentum. p_θ = L_z = ∂L/∂θ̇ = z-component of angular momentum (because θ represents rotation around z-axis) In quantum mechanics: L_z = -iħ∂/∂θ P-72: The parabolic co-ordinate (ξ,η) are related to cartesian co-ordinate (x,y) by x = 1/2(ξ²-η²) and y = 1/2(ξη). The Lagrangian of a 2-d simple harmonic oscillator of mass m and angular frequency ω is: L = 1/2m(ẋ² + ẏ²) - 1/2mω²(x² + y²) Solution: For 2D simple harmonic oscillator: x = 1/2(ξ²-η²) => ẋ = ξξ̇ - ηη̇ y = 1/2(ξη) => ẏ = 1/2(ξη̇ + ηξ̇) For 2D simple harmonic oscillator: L = 1/2m[(ξξ̇ - ηη̇)² + 1/4(ξη̇ + ηξ̇)²] - 1/2mω²[1/4(ξ²-η²)² + 1/4(ξη)²] = 1/8m[4ξ²ξ̇² + 4η²η̇² - 4ξηξ̇η̇ + ξ²η̇² + η²ξ̇² + 2ξηξ̇η̇] - 1/8mω²[ξ⁴ + η⁴ + 2ξ²η²] = 1/8m[5ξ²ξ̇² + 5η²η̇² + 3ξ²η̇² + 3η²ξ̇²] - 1/8mω²[ξ⁴ + η⁴ + 2ξ²η²] = 1/8m[ξ²ξ̇² + ξ²η̇² + ξ²η̇² + η²ξ̇² + ξ²η̇² + η²ξ̇²] - 1/8mω²[(ξ² + η²)² + ξ²η²] = 1/8m[ξ²(ξ̇² + 2η̇²) + η²(ξ̇² + η̇²)] - ω²/8m[(ξ² + η²)² + ξ²η²] = 1/8m[(ξ² + η²)(ξ̇² + η̇²) - ω²/4(ξ² + η²)²] - 1/8m(ξ²η²)[(ξ̇² + η̇²) - ω²(ξ² + η²)]