Abstract
We derive the Lie and the Noether conditions for the equations of motion of a dynamical system in an n-dimensional Riemannian space. We solve these conditions in the sense that we express the symmetry generating vectors in terms of the collineation vectors of the space. More specifically we give two theorems which contain all the necessary conditions which allow one to determine the Lie and the Noether point symmetries of a specific dynamical system evolving in a given Riemannian space in terms of the projective and the homothetic collineations of space. We apply these theorems to various interesting situations in Newtonian Physics.
Highlights
In theoretical Physics one has two main tools to study the properties of evolution of dynamical systems: (a) Symmetries of the equations of motion; and (b) Collineations of the background space where evolution takes place
One may state the question as follows: To what degree and how does space modulate the evolution of dynamical systems in it? Is a dynamical system free to evolve at will in a given space or is it constrained to do so by the very symmetry structure of space? This question was answered many years ago by the Theory of Relativity with the Equivalence Principle, that is, the requirement that free motion in a given gravitational field occurs along the geodesics of space
The Noether point symmetries of all Newtonian dynamical systems follow from the elements of the first two rows of Table 2
Summary
In theoretical Physics one has two main tools to study the properties of evolution of dynamical systems: (a) Symmetries of the equations of motion; and (b) Collineations (symmetries) of the background space where evolution takes place. It is reasonable to expect that the Lie point symmetries of the system of geodesic equations of a metric will be closely related to the collineations of the metric That such a relation exists is easy to see by the following simple example. In the following we assume that F is a function of xi only As it will be shown in this case the Lie symmetries of (1) are a subalgebra of the special projective algebra of the space. In accordance to the above this implies that the Noether point symmetries will be related with a subalgebra of the special projection algebra of the space where ‘motion’ occurs. B. For the case that the function F depends only on the coordinates, i.e. F (xi) we give two theorems which establish the exact relation between the (special) projective/homothetic algebra of the space and the Lie/Noether symmetry algebra of (1) respectively;. We use the above general results to generalize the Ermakov- Pinney dynamical system to n dimensional Riemannian space
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