## Introduction to Liouville's Theorem## Paths in Simple Phase Spaces: the SHO and Falling BodiesLet's first think further about paths in phase space. For example, the simple harmonic oscillator, with Hamiltonian , describes circles in phase space parameterized with the variables . (A more usual notation is to write the potential term as .)
Here's an example from Taylor of paths in phase space: four identical falling bodies are released simultaneously, see figure, measures distance vertically down. Tw ... Read more » |

## Canonical Transformations## Point TransformationsIt's clear that Lagrange's equations are correct for any reasonable choice of parameters labeling the system configuration. Let's call our first choice Now transform to a new set, maybe even time dependent, The derivation of Lagrange's equations by minimizing the action still works, so Hamilton's equations must still also be OK too. This is called a ## General and Canonical TransformationsIn the Hamiltonian approach, we're in We can have transformations that mix up position and momentum variables: where ... Read more » |

## Maupertuis' Principle: Minimum Action Path at Fixed Energy## Divine GuidanceIncredibly, Maupertuis came up with a kind of principle of least action in 1747, long before the work of Lagrange and Hamilton. Maupertuis thought a body moved along a path such that the sum of products of mass, speed and displacement taken over time was minimized, and he saw that as the hand of God at work. This didn't go over well with his skeptical fellow countrymen, such as Voltaire, and in fact his formulation wasn't quite right, but history has given him partial credit, his name on a least action principle. Suppose we are considering the motion of a particle moving around in a plane from some initial point and time to some final . Suppose its potential energy is a function of position, For example, imagine aiming for the hole on a rather bumpy putting green, but also |

## A New Way to Write the Action Integral## IntroductionFollowing Landau, we'll first find how the action integral responds to incremental changes in the ## Function of Endpoint PositionWe'll now think of varying the action in a slightly different way. ( Taking one degree of freedom (the generalization is straightfor ... Read more » |

## Time Evolution in Phase Space: Poisson Brackets and Constants of the Motion## The Poisson BracketA functionof the phase space coordinates of the system and time has total time derivative This is often written as where is called the If, for a phase space function (that is, no explicit time dependence) then is a constant of the motion, also called an In fact, the Poisson bracket can be defined for |