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

2024
Views:

259

Pages:

5
Author:

Jabin F.

# Poisson's Equation of Motion: Let A be a dynamic quantity which depends on (qi, pi, t). Let us calculate total time derivative: dA/dt = (∂A/∂qi) * dq_i/dt + (∂A/∂pi) * dp_i/dt + ∂A/∂t Use Hamilton's equations: [dq_i/dt = ∂H/∂pi] [dp_i/dt = -∂H/∂qi] → dA/dt = (∂A/∂qi) * (∂H/∂pi) + (∂A/∂pi) * (-∂H/∂qi) + ∂A/∂t → dA/dt = [(∂A/∂qi) * (∂H/∂pi) - (∂A/∂pi) * (∂H/∂qi)] + ∂A/∂t This is Poisson's equation. Similar to Heisenberg equation of QM: [dA/dt = (i/ħ) * [A, H] + ∂A/∂t] Q: {f₁, f₂} = (∂f₁/∂qi) * (∂f₂/∂pi) - (∂f₁/∂pi) * (∂f₂/∂qi) Solution: {f₁, f₂} = (∂/∂qi) * (f₁ * ∂f₂/∂pi) - (∂/∂pi) * (f₁ * ∂f₂/∂qi) = (∂f₁/∂qi) * (∂f₂/∂pi) + f₁ * (∂²/∂qi∂pi) * f₂ - (∂f₁/∂pi) * (∂f₂/∂qi) - f₁ * (∂²/∂pi∂qi) * f₂ = f₁ * (∂²/∂qi∂pi - ∂²/∂pi∂qi) * f₂ = 0 P-100 Q: {f₁, f₂} is equal to zero? Solution: {f₁, f₂} = (∂f₁/∂qi) * (∂f₂/∂pi) - (∂f₁/∂pi) * (∂f₂/∂qi) = (∂/∂qi) * (f₁ * ∂f₂/∂pi) - (∂/∂pi) * (f₁ * ∂f₂/∂qi) = (∂f₁/∂qi) * (∂f₂/∂pi) + f₁ * (∂²/∂qi∂pi) * f₂ - (∂f₁/∂pi) * (∂f₂/∂qi) - f₁ * (∂²/∂pi∂qi) * f₂ = f₁ * (∂²/∂qi∂pi - ∂²/∂pi∂qi) * f₂ = 0