Symmetry mappings of constrained dynamical systems and an associated related integral theorem

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Abstract
By use of Lie derivatives symmetry mappings of constrained conservative dynamical systems are formulated in terms of continuous groups of infinitesimal transformations within the configuration space. Such symmetries are called ``natural trajectory collineations'' in that the total energy has the same fixed value along each trajectory of the natural family, this value being preserved by the symmetry. It is found that these natural trajectory collineations must be conformal motions subject to an additional restriction dependent upon the potential. The corresponding groups of natural trajectory collineations are obtained for a flat configuration space with potential energy functions with rotational invariance about a point. A specialization of the theory to an indefinite Riemannian space‐time shows that homothetic transformations are necessary and sufficient to map a natural family of time (space)‐like geodesics into itself. A related integral theorem for constrained dynamical systems admitting linear or quadratic constants of the motion is obtained and illustrated. This theorem shows that in general a new constant of the motion will be obtained by deformation of an existing constant of the motion under a natural trajectory collineation.