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

2024
Views:

436

Pages:

5
Author:

Jabin F.

# Conservation of Energy: If L doesn't explicitly depend on time then Hamiltonian (Energy) is conserved. ∂L/∂t = 0 ⇒ ∂H/∂t = 0 ⇒ H = constant (energy conserved) Example: L₁ = 1/2 mẋ² L₂ = 1/2 mẋ² - 1/2 kx² For system whose Lagrangian L₁: energy is conserved (t is not explicitly present) For system whose Lagrangian L₂: energy is not conserved (t is explicitly present) # Related Statements: Invariance of Lagrangian under time translation leads to conservation of energy Homogeneity of time leads to conservation of energy. P-91 For Hamiltonian, H = 1/2m(ẋ² + ẏ²) + mgz + (k/2)(x² + y² - z²). Which one of the following statement is correct? Options: (a) Total energy and total momentum are conserved. (b) Only total energy is conserved. (c) Total energy and z-component of total angular momentum are conserved. (d) Total energy and total angular momentum are conserved. H = 1/2m(ẋ² + ẏ² + ż²) + mgz + (k/2)(x² + y² - z²) This Hamiltonian is invariant under rotation about the z-axis. So z-component of total angular momentum is conserved. t is not present explicitly here, so total energy is conserved.