Liouville's Theorem in Dynamics (UMAP)

The purpose of this module is to give an elementary introduction to Liouville's theorem in dynamics. Well known to physicists, and in particular, fundamental within statistical mechanics, this theorem has many applications, including the focusing of charged particle beams by accelerators and the determination of information about galactic systems. We begin the module by providing pertinent background material on Euler's equation in the calculus of variations (Section 2) and on Hamiltonian dynamics and phase space (Section 3). We then derive Liouville's theorem in two dimensions (Section 4). An example of three falling balls is given to help visualize an important fact related to Liouville's theorem, namely, that an energy-conserving flow through phase space is incompressible (Section 5). We conclude the module with further discussion of two of the theorem's applications (Section 6).

Table of Contents:

INTRODUCTION

EULER'S EQUATION FROM THE CALCULUS OF VARIATIONS

HAMILTONIAN DYNAMICS AND PHASE SPACE

DERIVATION OF LIOUVILLE'S THEOREM IN TWO DIMENSIONS

A SIMPLE ILLUSTRATION USING THREE FREELY-FALLING BALLS

FURTHER APPLICATIONS
Charged-Particle Accelerators
Galactic Dynamics
Analyzing motion of stars
Describing macroscopic properties
Modeling the dynamics in a cluster of galaxies

SOLUTIONS TO THE EXERCISES

REFERENCES

ACKNOWLEDGMENTS

ABOUT THE AUTHORS