Dynamical symmetry

Whenever systems are governed by continuous chains of causes and effects, their behavior exhibits the consequences of dynamical symmetries, many of them far from obvious. Dynamical Symmetry introduces the reader to Sophus Lie's discoveries of the connections between differential equations and continuous groups that underlie this observation. It develops and applies the mathematical relations between dynamics and geometry that result. Systematic methods for uncovering dynamical symmetries are described, and put to use. Much material in the book is new and some has only recently appeared in research journals. Though Lie groups play a key role in elementary particle physics, their connection with differential equations is more often exploited in applied mathematics and engineering. Dynamical Symmetry bridges this gap in a novel manner designed to help readers establish new connections in their own areas of interest. Emphasis is placed on applications to physics and chemistry. Applications to many of the other sciences illustrate both general principles and the ubiquitousness of dynamical symmetries.

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 -- 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 -- 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. 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 -- 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. 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 -- 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 -- 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 -- 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.

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. 8.5. Symmetries of Keplerian systems with non-negative energies. 8.6. The SO(4,1) dynamical symmetry. 8.7. Concluding remarks -- 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 [symbol] = 0. 9.7. Eigenfunctions of the constants of motion of a free-particle. 9.8. Dynamical symmetries of the Schrodinger equations of a harmonic oscillator. 9.9. Use of the oscillator group in pertubation calculations. 9.10. Concluding observations -- 10. Spectrum-generating Lie algebras and groups admitted by Schroedinger equations. 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)[symbol]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 -- 11. Dynamical symmetry of regularized hydrogen-like atoms. 11.1. Position-space realization of the dynamical symmetries. 11.2. The momentum-space representation. 11.3. The hyperspherical harmonics Y[symbol]. 11.4. Bases provided by Eigenfunctions of J[symbol], J[symbol], J[symbol] -- 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)[symbol]SO(4)[symbol] to produce SO(4)[symbol]. 12.5. Coupling SO(4)[symbol]SO(4)[symbol] to produce SO(4)[symbol]. 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 -- 13. Rovibronic systems. 13.1. Introduction. 13.2. Algebraic treatment of anharmonic oscillators with a finite number of bound states. 13.3. U(2)[symbol]U(2) model of vibron coupling. 13.4. Spectrum generating groups of rigid body rotations. 13.5. The U(4) vibron Model of rotating vibrating diatomics. 13.6. The U(4)[symbol]U(4) model of rotating vibrating triatomics. 13.7. Concluding remarks -- 14. Dynamical symmetry of Maxwell's equations. 14.1. The Poincare symmetry of Maxwell's equations. 14.2. The conformal and inversion symmetries of Maxwell's equations; their physical interpretation. 14.3. Alteration of wavelengths and frequencies by a special conformal transformation : Interpretation of Doppler shifts in stellar spectra.

Includes bibliographical references and index.