Lectures on selected topics in mathematical physics : introduction to lie theory with applications
Lie groups Lie algebras Mathematical physics
IOP Publishing
2017
EISBN 9781681744490
Preface.
Introduction.
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. Symmetries in terms of A.
3.4. Note on evaluating commutators.
3.5. Symmetries of first-order DEs.
3.6. Tabulating DEs according to groups they admit.
3.7. Lie's integrating factor.
3.8. Finding symmetries of a second order.
3.9. Using a symmetry to reduce the order.
3.10. Classical mechanics: Nöther's theorem.
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. It is an overview of the theory of finite groups, a brief description of a manifold, and an informal development of the theory of one-parameter Lie groups. Interested readers should acquire a tool that is complete and that actually works to simplify or solve differential equations as well as moving them on to other topics.
Introduction.
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. Symmetries in terms of A.
3.4. Note on evaluating commutators.
3.5. Symmetries of first-order DEs.
3.6. Tabulating DEs according to groups they admit.
3.7. Lie's integrating factor.
3.8. Finding symmetries of a second order.
3.9. Using a symmetry to reduce the order.
3.10. Classical mechanics: Nöther's theorem.
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. It is an overview of the theory of finite groups, a brief description of a manifold, and an informal development of the theory of one-parameter Lie groups. Interested readers should acquire a tool that is complete and that actually works to simplify or solve differential equations as well as moving them on to other topics.
