GF2 and GF3 - Edubirdie

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

2024
Views:

173

Pages:

5
Author:

Jabin F.

#Differential Relation for F₀ (qi, Pi,t)→ Legendre transformation from F₁ to F₂ F₂(qi, Pi, t) = F₁(qi, Pi, t) + ΣᵢqiPi Here sign will be decided by using differential relation of F₁ For F₁ differential relation have (-)ve sign, so we must use (+)ve sign here F₂ = F₁ + ΣᵢqiPi take partial derivative w.r.t. qi: ∂F₂/∂qi = ∂F₁/∂qi + Pi take partial derivative w.r.t. Pi: ∂F₂/∂Pi = ∂F₁/∂Pi + qi take partial derivative w.r.t. t: ∂F₂/∂t = ∂F₁/∂t + Σᵢ(qi∂Pi/∂t + Pi∂qi/∂t) = ∂F₁/∂t + H #Differential Relation for F₃ (Pi, Qi,t)→ Legendre transformation from F₁ to F₃ F₃(qi, bi, t) = F₁(qi, bi, t) - Σᵢbiqi Pi = -∂F₃/∂qi = -∂F₁/∂qi - bi qi = ∂F₃/∂Pi = ∂F₁/∂Pi A canonical transformation (qᵢ, pᵢ) → (bi, Pi) is made through generating function F(qi, Pi) if pᵢ = ∂F/∂qi and qi = -∂F/∂Pi The equation of motion for (bi, Pi) are: bi = ∂H/∂Pi Pi = -∂H/∂qi