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

Hamiltonian on Simple Pendulum Motion

This section provides an example of calculating the Hamiltonian on a mechanical system of an single object in simple pendulum motion and applying the Law of Conservation of Energy.

**What Is Simple Pendulum Motion?**
A Simple Pendulum Motion is an object of mass m hanging on a string
from a pivot point so that it is constrained to move on a circle of a fixed radius.

If we map the space in Cartesian coordinates, the Simple Pendulum Motion is a 2-dimensional problem, both the position, r, and the speed, v, have 2 components:

r = (x, y) v = (x', y')

If we introduce an extra variable θ as the angular position of the object from the vertical line, r and v can be expressed as:

r = (l*sin(θ), -l*cos(θ)) v = (l*cos(θ)*θ', l*sin(θ)*θ')

Now the object's kinetic energy, T, can be expressed as:

T = m*|v|**2 or: T = 0.5*m*l*l*(cos(θ)**2)+sin(θ)**2)*θ'*θ' # Pythagorean Theorem applied or: T = 0.5*m*l*l*θ'*θ' # Since cos(θ)**2)+sin(θ)**2 = 1

The potential energy, V, can be expressed as:

V = m*g*y or: V = m*g*(-l*cos(θ)) or: V = -m*g*l*cos(θ)

So the Hamiltonian, H, can be expressed as:

```
H = T + V (H.1)
or:
H = 0.5*m*l*l*θ'*θ' - m*g*l*cos(θ)
# Replaced T and V with their expressions
```

Since this simple pendulum motion can be considered as an isolated conservative system, we can apply the Law of Conservation of Energy:

H = constant or: 0.5*m*l*l*θ'*θ' - m*g*l*cos(θ) = constant or: d(0.5*m*l*l*θ'*θ' - m*g*l*cos(θ))/dt = 0 # Since d(constant)/dt = 0 or: m*l*l*θ'*θ" + m*g*l*sin(θ)*θ' = 0 l*θ" + g*sin(θ) = 0 θ" = - g*sin(θ)/l

Cool. We got the simplest form of the equations for the simple pendulum motion.

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