
Liouville’s Theorem in Dynamics (UMAP)
Paul Atsusi Isihara, Benjamin Noonan

Mathematics Topic: Calculus 
Application Areas: Physics and Astronomy 
Prerequisites: A basic knowledge of physics and of multivariable calculus. 

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 energyconserving 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 FREELYFALLING BALLS
FURTHER APPLICATIONS
ChargedParticle 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
