Invariance of Poisson bracket under canonical transformation | Classical Mechanics

Invariance of Poisson bracket under canonical transformation

• Let u and v be two functions such that u = u (q_i, p_i, t) and v = v (q_i, p_i, t)
• Let a canonical transformation is from (q_i, p_i, t) → (Q_i, P_i, t)
• Here q = q (Q, P, t) and p = p (Q, P, t)
• Corresponding to it the transformation in u and v are u (q_i, p_i, t) → u′ (Q_i, P_i, t) and v (q_i, p_i, t) → v′ (Q_i, P_i, t)
• Now we have to prove that if (q, p, t) → (Q, P, t) is canonical then [u, v]_{p, q} = [u′, v′]_{P, Q}
• It means the Poisson bracket are invariant under a canonical transformation.

Proof

• If F₁ and F₂ are generating function, then the transformation relation for the variables are F₂ = F₁ + PQ

• Thus the Poisson brackets are invariant under a canonical transformation.

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