This paper presents two ways of comprehending the Noether symmetry for the Lagrange system.One is based on invariance of the Lagrangian, the other is based on invariance of the action. This paper proves that these two comprehensions are different from each other. We give the condition under which the invariance of the Lagrangian can become the invariance of the action,and the condition under which the invariance of the action can become the invariance of the Lagrangian is obtained. It is suitable that the Noether symmetry is comprehended as the invariance of the action.

The Noether symmetry and Lie symmetry of the Lagrange system subjected to gyroscopic forces are studied. The condition that the system, under gyroscopic forces , can keep its Noether symmetry and Noether conserved quantity is given. And the condition that the system subjected to gyroscopic forces can keep its Lie symme try and Hojman conserved quantity is also given. Finally, two examples are given to illustrate the application of the results.

First, the method presented by Hojman for finding the conserved quantity of the system of second order differential equations is generalized and applied to the system of first order differential equations, particularly to the odd-dimensional system for finding the integral in this paper. Next, it is proved that the Hojman theorem is a special case of the theorem given in this paper. Finally, an example is given to illustrate the application of the result.

This paper studies the Lagrange symmetry of a dynamical system of relative motion. The definition and the criterion of the symmetry of the system are established. The condition under which there exists a conserved quantity and the form of the conserved quantity are obtained.

A gradient representation of the first-order Lagrange system is studied. A condition under which the first-order Lagrange system can be considered as a gradient system is obtained. The stability of the system is discussed by using the property of the gradient system. Some examples are given to illustrate the application of the result.