The SL(5R) Lie invariance transformation group for the 3-dimensional classical Kepler problem : a preparation, and induced group structure algorithm derivation

Recently,1 an algorithm has been derived for the explicit determination of an induced SL (n+2,R) Lie invariance transformation group for a completely integrable 2n - dimensional dynamical system defined on IR2n from that known for a free particle system with n degree of freedom. 2 In particular, the universal transitive Lie invariance transformation group for both the isotropic harmonic oscillator3 and the anharmonic oscillator4 (quartic potential) has been obtained by this algorithm. Further,5 it has been shown in theory and by example that a complete set of functionally independent constants of motion corresponds to an abelian subalgebra of the induced SL (n+2,R) group. In this work, preparations necessary to apply this algorithm to the 3-dimensional classical Kepler problem have been made. A brief explanation of the algorithm and its relation to the Kepler problem in given in Chapter I. The preparations including the identification of a suitable parametric form unifying the solution completely and simplify are given in subsequent chapters and Appendix I. They are followed in Appendix II by a paper6 containing the actually application involving the extension of the algorithm to arbitrarily reparameterized system. We should mention that only conservative Hamiltonian systems are treated in this thesis