**Physics Notes - Herong's Tutorial Notes** - v3.24, by Herong Yang

Relation of Momentum and Hamiltonian

This section describes the Relation of Momentum and Hamiltonian.

Hamiltonian, H, is related to momentum, p, through the kinetic energy, T.

Let's use a single object as an example:

H = T + V (H.1) or: H = 0.5*m*v*v + V # Since T = 0.5*m*v*v or: H = 0.5*p*v + V # p = m*v applied or: H = 0.5*p*p/m + V # p = m*v applied again

If we apply the law of conservation of energy, we can have:

H = constant T + V = constant (H.2) or: dH/dt = 0 dT/dt + dV/dt = 0 dT/dt = -dV/dt (H.3) or: d(0.5*p*p/m)/dt = - dV/dt p*p'/m = -dV/dt or: v*p' = -dV/dt # p = m*v applied

Table of Contents

Introduction of Frame of Reference

Introduction of Special Relativity

Time Dilation in Special Relativity

Length Contraction in Special Relativity

The Relativity of Simultaneity

Minkowski Spacetime and Diagrams

Hamiltonian on Free Fall Motion

Hamiltonian on Simple Harmonic Motion

Hamiltonian on Simple Pendulum Motion

►Relation of Momentum and Hamiltonian

Hamiltonian in Cartesian Coordinates

Relation of Momentum and Potential Energy

Hamilton Equations in Cartesian Coordinates

Introduction of Generalized Coordinates