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

2024
Views:

444

Pages:

5
Author:

Jabin F.

# Conservation laws in Lagrangian & Hamiltonian formulation: Brief recap of conservation laws in Newtonian mechanics: F = 0 ⇒ E conserved d/dt (mvr) = 0 ⇒ P conserved If F is conservative & independent of time, energy is conserved P-42 In the previous question, if my variable vector, then: q = 2x p = 2y Solution: q + mv = variable q = 2x, p = 2y ⇒ q + mv = constant P-92 Hamiltonian of the system is: H = p^2/(2m) + mgl(1-cosθ) If one quantity (pθ^2/(2m)) - mgl(1-cosθ) + (1/2)mgl^2θ̇^2 is a constant of motion, then its value: Solution: Total time derivative of conserved quantity is zero. d/dt [(pθ^2/(2m)) - mgl(1-cosθ) + (1/2)mgl^2θ̇^2] = 0 (pθ/m)θ̇ - mgl sinθθ̇ + mgl^2θ̈θ̇ = 0 (pθ/m) - mgl sinθ + mgl^2θ̈ = 0 As Hamilton's EOM, p = ∂H/∂θ = pθ/m, and θ̈ = -∂H/∂p = -mgl sinθ/mgl^2. -mgLsinθ - mgLsinθ + (1/2)Pθ²sinθ -------------------------------------- = θ̈ (1/12)ml² + ml² -2mgl sinθ + (1/2)Pθ²sinθ -------------------------------------- = θ̈ (1/12)ml² + ml² -2g/l - ω² = 0 ω = √(2g/l)