Hamilton's equations of motion
Lagrangian and Hamiltonian formulations · Physics
Study notes
Example: Simple Harmonic Oscillator (mass m on a spring). 1. Write the Hamiltonian: H = p^2/(2m) + (1/2)kq^2, where k is the spring constant. 2. Compute partial derivatives: ∂H/∂p = p/m, ∂H/∂q = kq. 3. Apply Hamilton’s equations: dq/dt = p/m, dp/dt = -kq. 4. Solve the system: differentiate dq/dt to get d^2q/dt^2 = (1/m)dp/dt = -(k/m)q. 5. Recognize this as simple harmonic motion: q(t) = A cos(√(k/m) t) + B sin(√(k/m) t). 6. Use initial conditions to find A and B. This shows how Hamilton’s equations give the same result as Newton’s law for a spring.