Chapter 6.5, Problem 8E

Interpretation Introduction

Interpretation:

Write out Hamiltonian equations for the simple harmonic oscillator of mass $\text{m}$, spring constant $\text{k}$, displacement $\text{x}$, and momentum $\text{p}$ and to show that one equation gives the usual definition of momentum and other is equivalent to $\text{F = ma}$. To prove that Hamiltonian H is the total energy.

Concept Introduction:

Hamiltonian equations of motions are $\dot{\text{q}}\text{ = }\frac{\partial \text{H}}{\partial \text{p}}$, $\dot{\text{p}}\text{ = -}\frac{\partial \text{H}}{\partial \text{q}}$ .

Total energy E of the harmonic oscillator can be expressed as $\text{E =}\frac{\text{1}}{\text{2}}{\text{m}\dot{\text{x}}}^2\text{+ V(x)}$.

The potential energy of harmonic oscillator is $\text{V(x) = }\frac{\text{1}}{\text{2}}{\text{kx}}^{\text{2}}$ .