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

2024
Views:

178

Pages:

5
Author:

Jabin F.

#Canonical Transformation (CT): It is a transformation of generalized coordinate and generalized momentum. Such that Hamiltonian doesn't change. New Variables: Such transformation is done for simplification. (q1, p1) -> (Q1, P1) Qi = Qi(q, p, t) Pi = Pi(q, p, t) Hamilton's EOM: q̇ᵢ = ∂H/∂pᵢ ṗᵢ = -∂H/∂qᵢ → H(q, p) can be converted into H(Q, P) using the CT relation. → When we convert (1) into (2) or vice-versa, this conversion is possible only if: {Qi, Qj} = {Pi, Pj} = 0 {Qi, Pj} = δij Mathematical condition for CT is that PB of new quantities w.r.t. old variables and vice-versa must be 1. To answer any question related to CT use {Qi, Pj}. Example of CT: Q = p + q P = q - p This is a simple CT. EOM doesn't change after CT. Identity transformation is CT: Q = q P = p Inverse (parity) transformation is CT: Q = -q P = -p Rotation in phase space is CT: Q = q*cosθ + p*sinθ P = -q*sinθ + p*cosθ Gauge Transformation of Lagrangian is CT: L' = L + dF(q, t)/dt