Related integral theorem. II. A method for obtaining quadratic constants of the motion for conservative dynamical systems admitting symmetries

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Abstract
By use of Lie derivatives, symmetry mappings for conservative dynamical systems are formulated in terms of continuous groups of infinitesimal transformations within the configuration space. Such symmetry transformations, called trajectory collineations, may be interpreted as point mappings which drag along coordinates and geometric objects as they map trajectories into trajectories. It is shown that if a conservative dynamical system admits a trajectory collineation, then in general a new quadratic (in the velocity) constant of the motion will result from the deformation of a given quadratic constant of the motion under such a symmetry mapping. The theory is applied to obtain the group of symmetry transformations and concomitant constants of the motion associated with the deformations of the energy integral for the Kepler problem and for the isotropic simple harmonic oscillator. The Runge‐Lenz vector of the Kepler problem and the symmetric tensor constant of the motion for the three‐dimensional oscillator are readily obtained by this method. The trajectory collineation group for the Kepler problem is a seven‐parameter projective collineation group which is isomorphic to the similitude group in three‐dimensional Euclidean space. For the oscillator the trajectory collineation group is the nine‐parameter affine group. This dynamical symmetry group contains an eight‐parameter subgroup which is shown to have the structure of SU3.