This book provides an introduction to Lie Theory for first year graduate students and professional physicists who may not have come across the theory in their studies. In particular, it is a summary overview of the theory of finite groups, a brief description of a manifold, and then an informal development of the theory of one-parameter Lie groups, especially as they apply to ordinary differential equations. The treatment is informal, but systematic and reasonably self-contained, as it assumes a familiarity with basic physics and applied calculus, but it does not assume additional mathematical training. Interested readers should have a fair chance of finding symmetries of a second order differential equation and should be able to use it to reduce the order of the differential equation.

### Table of Contents

1 Groups

1.1 Permutations and symmetries

1.2 Subgroups and classes

1.3 Representations,

1.4 Orthogonality

2 Lie Groups

2.1 Lie groups as manifolds

2.2 Lie groups as groups of transformations or substitutions

2.3 Infinitesimal generators

2.4 Generator example: Lorentz boost

2.5 Transformations acting in three or more dimensions

2.6 Changing coordinates

2.7 Changing variables in the generator

2.8 Invariant functions, invariant curves, and groups that permute curves in a family

2.9 Canonical coordinates for a one-parameter group

3 Ordinary Differential Equations

3.1 Prolongation of the group generator and a symmetry criterion

3.2 Reformulation of symmetry in terms of partial differential operators

3.3 Note on evaluating commutators

3.4 Tabulating DEs according to groups they admit

3.5 Lie's integrating factor

3.6 Finding symmetries of a second order DE

3.7 Classical mechanics: Nother's theorem

### About the Author(s)

**William A. Schwalm **, University of North Dakota

Dr William A Schwalm has been in the Department of Physics and Astrophysics at the University of North Dakota since 1980. His research is in condensed matter theory and application of mathematical methods to physical problems. He has taught lots of different physics courses at all levels.

Current research involves application of Lie groups to finding generating functions for the stationary states of quantum systems, and also applying them to decoupling discrete dynamical systems. Another area of active interest is in finding Green functions for certain classes of lattice problems involving electron transport, vibrations and other collective excitations.

Dr Schwalm has received two outstanding teaching awards, the University of Utah Physics Outstanding Undergraduate Instructor (1979) and the McDermott award for Excellence in Teaching, UND (1995).