A class of generalized symmetries of smooth flows

Pages: 183 - 195, Volume 3, Issue 2, June 2004

doi:10.3934/cpaa.2004.3.183       Abstract        Full Text (199.1K)       Related Articles

L. Bakker - Department of Mathematics, Brigham Young University, Provo, UT 84602, United States (email)
G. Conner - Department of Mathematics, Brigham Young University, Provo, UT 84602, United States (email)

Abstract: A class of generalized space-time symmetries is defined by extending the notions of classical symmetries and reversing symmetries for a smooth flow to arbitrary constant reparameterizations of time. This class is shown to be the group-theoretic normalizer of the abelian group of diffeomorphisms generated by the flow. Also, when the flow is nontrivial, this class is shown to be a nontrivial subgroup of the group of diffeomorphisms of the manifold, and to have a one-dimensional linear representation in which the image of a generalized symmetry is its unique constant reparameterization of time. This group of generalized symmetries and several groups derived from it (among which are the multiplier group and the reversing symmetry group) are shown to be nontrivial but incomplete invariants of the smooth conjugacy class of a smooth flow. Several examples are given throughout to illustrate the theory.

Keywords:  Smooth flow, generalized symmetry, incomplete invariant.
Mathematics Subject Classification:  37C15, 34C14

Received: June 2003;      Revised: December 2003;      Published: March 2004.