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

2024
Views:

497

Pages:

5
Author:

Jabin F.

# Cyclic Coordinates: If L (Lagrangian) is not present on a velocity or momentum term, but the corresponding coordinate is not present in them, then that coordinate is called cyclic/ignorable/removable. e.g., If θ is present but not θ̇, then θ is a cyclic coordinate. # Importance of Cyclic Coordinates: Generalized momentum corresponding to a cyclic coordinate is conserved. If q_i is a cyclic coordinate, then p_i is conserved. Note: If some coordinate is not cyclic, then its generalized momentum is not conserved. Q: L = (1/2)m(r² + r²θ̇² + r²sin²θφ̇²) - mgz. Is angular momentum conserved? Solution: θ and φ both present, φ is not cyclic, so Lz is not conserved. P-31 Q: Canonical transformation (p, q) → (P, Q) is performed on the Hamiltonian H of the system. If one of the new variables, say P, is cyclic, then which of the following graphs showing function F (the generator of the transformation) depends on Q(t)? Solution: Appropriate differential equations are: ∂F/∂q = p ∂F/∂P = Q ⇒ p = ∂F/∂q ⇒ Q = ∂F/∂P H' = H + (∂F/∂t) = (1/2m) * (p² + m²ω²q²) + (1/2m) * (m²ω²q² + p²) - ωqP ⇒ Hamiltonian's EOM: ∂Q̇/∂t = ∂H'/∂P = ω ⇒ dQ/dt = ω → Q = ωt + C Compare with equation of line y = mx + c, it shows linear dependence of Q(t) on t. Now at t = 0, Q = 0 → 0 = 0 + C → C = 0 ∴ Q = ωt