Hamiltonian Mechanics

Phase space, Hamilton's equations, and canonical transformations

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The Phase Space Mystery

A mass on a spring oscillates back and forth. Instead of plotting position vs time, we plot position vs momentum. What does the trajectory look like?

This strange plot - called phase space - reveals hidden symmetries and conservation laws. The answer unlocks the secrets of quantum mechanics!

The Challenge: Predict the Long-Term Behavior

A frictionless mass-spring system starts at maximum displacement (maximum x, zero momentum). As it oscillates, we track both position (x) and momentum (p = mv). In this phase space diagram, what happens to the point as time goes on?

As time passes, the system's state in phase space will:

Take a guess! Phase space is the key to understanding quantum mechanics, statistical physics, and chaos theory.

The Lab

The Lab

The Lab

The Lab

Hamiltonian Mechanics

Phase space, Hamilton's equations, and canonical transformations

The Phase Space Mystery

A mass on a spring oscillates back and forth. Instead of plotting position vs time, we plot position vs momentum. What does the trajectory look like?

The Challenge: Predict the Long-Term Behavior

As time passes, the system's state in phase space will:

Hamiltonian Mechanics

Phase space, Hamilton's equations, and canonical transformations

The Phase Space Mystery

A mass on a spring oscillates back and forth. Instead of plotting position vs time, we plot position vs momentum. What does the trajectory look like?

The Challenge: Predict the Long-Term Behavior

As time passes, the system's state in phase space will: