Answer to Question #198551 in Classical Mechanics for TSHEPHISHO

Question

a  harmonic oscillator has Hamiltonian, H=1/2 p^2+1/2 w^2 p^2 . Determine the Hamiltonian equation and the general solution

Answer 1

H = p²/(2m) + ½kx². H is a function of x and p, while the Lagrangian is a function of x and v.

Equations of motion: dx/dt = ∂H/∂p, dp/dt = -∂H/∂x.

dx/dt = p/m, dp/dt = -kx. We have two first-order differential equations which can be combined to yield the second order differential equation d²x/dt² = -(k/m)x.

The coordinates and the Hamiltonian do not contain time explicitly, so H = E, and E = constant. The energy E is a conserved quantity.