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

2024
Views:

474

Pages:

5
Author:

Jabin F.

#Hamilton's EOM in terms of Poisson Bracket (PB)→ → If qᵢ = qi, then dqᵢ/dt = {qi, H} → If pᵢ = Pi, then dpᵢ/dt = {Pi, H} → In writing Hamilton's EOM in terms of PB, the two equations are become exactly same. → If H = H(qᵢ, pᵢ, t), then dH/dt = {H, H} + ∂H/∂t → H is only quantity for which partial and total time derivatives are always same. we know that H = Σpᵢqᵢ - L → ∂H/∂qᵢ = pᵢ, ∂H/∂pᵢ = qᵢ → Hamiltonian EOM provides a method to calculate total time derivative of any quantity. P-100 {Σ(pk + qk²), H} is equal to Solution: Using Poisson equation ∂qᵢ/∂t = {qᵢ, H} ∂t(Σ(pk + qk²)) = {Σ(pk + qk²), H} = Σ{pk, H} + Σ{qk², H} = Σ(qk) + Σ(2qk) = Σ(3qk) Method I: {Σ(pk + qk²), H} = ∂/∂qk(Σ(pk + qk²)) * ∂H/∂pk - ∂/∂pk(Σ(pk + qk²)) * ∂H/∂qk = Σ(1) * ∂H/∂pk - Σ(2pk) * ∂H/∂qk = Σ(∂H/∂pk - 2pk∂H/∂qk)