A new type of conserved quantity of Lie symmetry for the Lagrange system

Abstract

This paper studies a new type of conserved quantity which is directly induced by Lie symmetry of the Lagrange system. Firstly, the criterion of Lie symmetry for the Lagrange system is given. Secondly, the conditions of existence of the new conserved quantity as well as its forms are proposed. Lastly, an example is given to illustrate the application of the result.

Lie symmetry and its generation of conserved quantity of Appell equation in a dynamical system of the relative motion with Chetaev-type nonholonomic constraints

“…Proposition 3 For a holonomic system (7) corresponding to a dynamical system of the relative motion with nonholonomic constraints of Chetaev's type, if the generators ξ 0 and ξ s under the special infinitesimal transformations satisfy Eq. (13), restriction equation 14, and special additional restriction equation (15), and there exists a function µ = µ (t, ,˙) satisfying Eq. (16), then the generalized Hojman conserved quantities (17) can be deduced from the strong Lie symmetry of a dynamical system of the relative motion with nonholonomic constraints of Chetaev's type.…”

“…This kind of equation is not only applicable in holonomic systems, but also in nonholonomic systems, and it is not only applied in generalized coordinates, but also applied in quasi coordinates. The modern symmetry theories in a constrained mechanical system are mainly of three types: Noether symmetry, [1][2][3][4][5][6] Lie symmetry, [7][8][9][10][11][12][13][14][15][16] and Mei symmetry. [17][18][19][20][21][22][23][24][25][26] In recent decades, Chinese scholars have achieved important results in studying symmetry theories of Appell systems and their applications.…”