Contents -- Dedication -- Preface -- References -- Acknowledgments -- CHAPTER 1 Introduction -- 1.1 On Geometric Symmetry and Invariance in the Sciences -- 1.2 Fock's Discovery -- 1.3 Keplerian Symmetry -- 1.4 Dynamical Symmetry -- 1.5 Dynamical Symmetries Responsible for Degeneracies and Their Physical Consequences -- 1.6 Dynamical Symmetries When Energies Can Vary -- 1.7 The Need for Critical Reexamination of Concepts of Physical Symmetry. Lie's Discoveries -- Appendix A. Historical Note -- References -- CHAPTER 2 Physical Symmetry and Geometrical Symmetry -- 2.1 Geometrical Interpretation of the Invariance Group of an Equation -- Symmetry Groups -- 2.2 On Geometric Interpretations of Equations -- 2.3 Geometric Interpretations of Some Transformations in the Euclidean Plane -- 2.4 The Group of Linear Transformations of Two Variables -- 2.5 Physical Interpretation of Rotations -- 2.6 Intrinsic Symmetry of an Equation -- 2.7 Non-Euclidean Geometries -- 2.8 Invariance Group of a Geometry -- 2.9 Symmetry in Euclidean Spaces -- 2.10 Symmetry in the Spacetime of Special Relativity -- 2.11 Geometrical Interpretations of Nonlinear Transformations: Stereographic Projections -- 2.12 Continuous Groups that Leave Euclidean and Pseudo-Euclidean Metrics Invariant -- 2.13 Geometry, Symmetry, and Invariance -- Appendix A: Stereographic Projection of Circles -- References -- CHAPTER 3 On Symmetries Associated With Hamiltonian Dynamics -- 3.1 Invariance of a Differential Equation -- 3.2 Hamilton's Equations -- 3.3 Transformations that Convert Hamilton's Equations into Hamilton's Equations -- Symplectic Groups -- 3.4 Invariance Transformations of Hamilton's Equations of Motion -- 3.5 Geometrization of Hamiltonian Mechanics -- 3.6 Symmetry in Two-Dimensional Symplectic Space -- 3.7 Symmetry in Two-Dimensional Hamiltonian Phase Space.

Definition of Two-dimensional Hamiltonian Phase-Space: -- Definition of Symmetry in Two-Dimensional Hamiltonian Phase-Space: -- 3.8 Symmetries Defined by Linear Symplectic Transformations -- 3.9 Nonlinear Transformations in Two-Dimensional Phase Space -- 3.10 Dynamical Symmetries as Intrinsic Symmetries of Differential Equations and as Geometric Symmetries -- Exercises: -- References -- CHAPTER 4 One-Parameter Transformation Groups -- 4.1 Introduction -- 4.2 Finite Transformations of a Continuous Group Define Infinitesimal Transformations and Vector Fields -- 4.3 Spaces in which Transformations will be Assumed to Act on -- 4.4 The Defining Equations of One-Parameter Groups of Infinitesimal Transformations. Group Generators -- 4.5 The Differential Equations that Define Infinitesimal Transformations Define Finite Transformation Groups -- 4.6 The Operator of Finite Transformations -- 4.7 Changing Variables in Group Generators -- 4.8 The Rectification Theorem -- 4.9 Conversion of Non-autonomous ODEs to Autonomous ODEs -- 4.10 N-th Order ODEs as Sets of First-Order ODEs -- 4.11 Conclusion -- Appendix A. Homeomorphisms, Diffeomorphisms, and Topology -- Exercises -- References -- CHAPTER 5 Everywhere-Local Invariance -- 5.1 Invariance under the Action of One-Parameter Lie Transformation Groups -- 5.2 Transformation of Infinitesimal Displacements -- 5.3 Transformations and Invariance of Work, Pfaffians, and Metrics -- 5.4 Point Transformations of Derivatives -- 5.5 Contact Transformations -- 5.6 Invariance of an Ordinary Differential Equation of First-Order under Point Transformations -- Extended Generators -- 5.7 Invariance of Second-Order Ordinary Differential Equations under Point Transformations -- Harmonic Oscillators -- 5.8 The Commutator of Two Operators -- 5.9 Invariance of Sets of ODEs. Constants of Motion -- 5.10 Conclusion.

Appendix A. Relation between Symmetries and Intergrating Factors -- Appendix B. Proof That the Commutator of Lie Generators is Invariant under Diffeomorphisms -- Appendix C. Isolating and Non-isolating Integrals of Motion -- Exercises -- References -- CHAPTER 6 Lie Transformation Groups and Lie Algebras -- 6.1 Relation of Many-Parameter Lie Transformation Groups to Lie Algebras -- 6.2 The Differential Equations That Define Many-Parameter Groups -- 6.3 Real Lie Algebras -- 6.4 Relations between Commutation Relations and the Action of Transformation Groups: Some Examples -- 6.5 Transitivity -- 6.6 Complex Lie Algebras -- 6.7 The Cartan-Killing Form -- Labeling and Shift Operators -- 6.8 Casimir Operators -- 6.9 Groups That Vary the Parameters of Transformation Groups -- 6.10 Lie Symmetries Induced from Observations -- 6.11 Conclusion -- Appendix A. Definition of Lie Groups by Partial Differential Equations -- Exercises. -- References -- CHAPTER 7 Dynamical Symmetry in Hamiltonian Mechanics -- 7.1 General Invariance Properties of Newtonian Mechanics -- 7.2 Relationship of Phase Space to Abstract Symplectic Space -- 7.3 Hamilton's Equations in PQ Space. Constants of Motion -- 7.4 Poisson Bracket Operators -- 7.5 Hamiltonian Dynamical Symmetries in PQ Space -- 7.6 Hamilton's Equations in Classical PQET Space -- Conservation Laws Arising From Galilei Invariance -- 7.7 Time-dependent Constants of Motion -- Dynamical Groups That Act Transitively -- 7.8 The Symplectic Groups Sp(2n,r) -- 7.9 Generalizations of Symplectic Groups That Have an Infinite Number of One-parameter Groups -- Appendix A. Lagrange's Equations and the Definition of Phase Space -- Appendix B. The Variable Conjugate to Time in PQET Space -- Exercises -- References -- CHAPTER 8 Symmetries of Classical Keplerian Motion -- 8.1 Newtonian Mechanics of Planetary Motion.

8.2 Hamiltonian Formulation of Keplerian Motions in Phase Space -- 8.3 Symmetry Coordinates For Keplerian Motions -- 8.4 Geometrical Symmetries of Bound Keplerian Systems in Phase Space -- 8.6 The SO(4,1) Dynamical Symmetry -- 8.7 Concluding Remarks -- Exercises: -- References -- CHAPTER 9 Dynamical Symmetry in Schrodinger Quantum Mechanics -- 9.1 Superposition Invariance -- 9.2 The Correspondence Principle -- 9.3 Correspondence Between Quantum Mechanical Operators and Functions of Classical Dynamical Variables -- 9.4 Lie Algebraic Extension of the Correspondence Principle -- 9.5 Some Properties of Invariance Transformations of Partial Differential Equations Relevant to Quantum Mechanics -- 9.6 Determination of Generators and Lie Algebra of Invariance Transformations of ((−1/2)∂2/∂x2 − i∂/∂t)ψ(x, t) = 0 -- 9.7 Eigenfunctions of the Constants of Motion of a Free-Particle -- 9.8 Dynamical Symmetries of the Schrödinger Equations of a Harmonic Oscillator -- 9.9 Use of the Oscillator Group in Pertubation Calculations -- 9.10 Concluding Observations -- Exercises -- References -- CHAPTER 10 Spectrum-Generating Lie Algebras and Groups Admitted by Schrödinger Equations -- Introduction -- 10.1 Lie Algebras That Generate Continuous Spectra -- 10.2 Lie Algebras That Generate Discrete Spectra -- 10.3 Dynamical Groups of N-Dimensional Harmonic Oscillators -- 10.4 Linearization of Energy Spectra by Time Dilatation -- Spectrum-Generating Dynamical Group of Rigid Rotators -- 10.5 The Angular Momentum Shift Algebra -- Dynamical Group of the Laplace Equation -- 10.6 Dynamical Groups of Systems with Both Discrete and Continuous Spectra -- 10.7 Dynamical Group of the Bound States of Morse Oscillators -- 10.8 Dynamical Group of the Bound States of Hydrogen-Like Atoms -- 10.9 Matrix Representations of Generators and Group Operators -- 10.10 Invariant Scalar Products.

10.11 Direct-Products: SO(3)⊗SO(3) and the Coupling of Angular Momenta -- 10.12 Degeneracy Groups of Non-interacting Systems -- Completions of Direct-Products -- 10.13 Dynamical Groups of Time-dependent Schrodinger Equations of Compound Systems -- Many-Electron Atoms -- Appendix A. References Providing Invariance Groups of Schrodinger Equations -- Appendix B. References to Work Dealing with Dynamical Symmetries in Nuclear Shell Theory -- Exercises -- References -- CHAPTER 11 Dynamical Symmetry of Regularized Hydrogen-like Atoms -- Introduction -- 11.1 Position-space Realization of the Dynamical Symmetries -- 11.2 The Momentum-space Representation -- 11.3 The Hyperspherical Harmonics Yklm -- 11.4 Bases Provided by Eigenfunctions of J12, J34, J56 -- Appendix A. Matrix Elements of SO(4,2) Generators3,16 -- Appendix B. N-Shift Operators For the Hyperspherical Harmonics -- References -- CHAPTER 12 Uncovering Approximate Dynamical Symmetries. Examples From Atomic and Molecular Physics -- 12.1 Introduction -- 12.2 The Stark Effect -- One-Electron Diatomics -- 12.3 Correlation Diagrams and Level Crossings: General Remarks -- 12.4 Coupling SO(4)1 ⊗ SO(4)2 to Produce SO(4)12 -- 12.5 Coupling SO(4)1 ⊗ SO(4)2 to Produce SO(4)1−2 -- 12.6 Configuration Mixing in Doubly Excited States of Helium-like Atoms -- 12.7 Configuration Mixing Arising From Interactions Within Valence Shells of Second and Third Row Atoms -- 12.8 Origin of the Period-Doubling Displayed in Periodic Charts -- 12.9 Molecular Orbitals in Momentum-Space -- The Hyperspherical Basis -- 12.10 The Sturmian Ansatz of Avery, Aquilanti and Goscinski -- Exercises -- References -- CHAPTER 13 Rovibronic Systems -- 13.1 Introduction -- 13.2 Algebraic Treatment of Anharmonic Oscillators With a Finite Number of Bound States -- 13.3 U(2) ⊗ U(2) Model of Vibron Coupling.

13.4 Spectrum Generating Groups of Rigid Body Rotations.