Symmetries and Constants of Motion of Dynamical Systems

摘要

The great success of the notion of Lie symmetry and of general symmetry methods in the study of ODE's and PDE's suffers a partial exception in the context of dynamical systems (DS), i.e. the systems of n first-order ODE's, where indeed their application appears to be more critical or problematic. The present contribution is devoted to study precisely some aspects of the role of symmetry methods in DS's. Symmetries are strictly related to the notion of first integrals (or constants of motion: the DS's describe typically time-evolution problems): an old result by Ovsjannikov [1] shows that if one knows n functionally independent constants of motion, then it is possible to deduce all symmetries admitted by the DS. Another result, based on the notion of Liouville symmetry vector fields, concerns the possibility of expressing symmetries in terms of n-1 (suitably chosen) constants of motion [2]. Both results, however, are of little use in the problem of explicitly finding the symmetries of the given DS. Indeed, finding the constants of motion of a DS is almost as difficult as directly solving the DS, or finding all its symmetries in some systematic way. I try here to partially reverse this point of view. Indeed it often happens that one is able to see, by direct inspection or by chance, just one symmetry of the given DS, or the DS belongs to particular classes of examples where some symmetry is well known (see also below). I then show how a single symmetry can provide a convenient and easy way to deduce one or more constants of motion of the DS. This is essentially based on the introduction of suitable "symmetry adapted" coordinates. The main result is that one obtains constants of motion which are also invariant under the symmetry, i.e. simultaneously invariant under the dynamical and the symmetry flows. Apart from other simple cases, two important classes of DS's are provided for which an explicit symmetry can be obtained, and examples are given where exactly n functionally independent constants of motion are obtained. One of these classes is also related to ODE's of order > 1.