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

2024
Views:

267

Pages:

5
Author:

Jabin F.

# Generalised Function (G.F.): It is a function of one old variable and one new variable. It is introduced to write CT easily. (qi, Pi) ----------> (Qi, Pi) old variable new variable Types of G.F.: F₁(q₁, Q₁) F₂(q₁, P₁, t) F₃(q₁, P₁, t) F₄(q₁, P₁, t) If time is present → unrestricted C.T. If time is not present → restricted C.T. # Differential Relation for G.F. : F₁(q₁, Q₁, t) We know that gauge transformation of Lagrangian is a C.T. L' = L + (dF₁(q₁, Q₁, t))/dt L' = L - (dF₁(q₁, Q₁, t))/dt [We are using -ve sign because in all books -ve sign has been used. (We can also use +ve sign).] Pᵢ ̇Qᵢ - H' = Pᵢ ̇Qᵢ - H - (∂F₁/∂qᵢ) ̇Qᵢ + (∂F₁/∂Qᵢ) ̇qᵢ + (∂F₁/∂t) Multiply by dt: PᵢdQᵢ - H'dt = PᵢdQᵢ - Hdt - (∂F₁/∂qᵢ)dQᵢ + (∂F₁/∂Qᵢ)dqᵢ - (∂F₁/∂t)dt Equating the coefficients of dQᵢ, dqᵢ, dt: Pᵢ = -∂F₁/∂Qᵢ H' = H + ∂F₁/∂t These are differential relations for generating functions.