Lie Symmetry and Approximate Hojman Conserved Quantity of Lagrange Equations for a Weakly Nonholonomic System

Abstract

The Lie symmetry and Hojman conserved quantity of Lagrange equations for a weakly nonholonomic system and its first-degree approximate holonomic system are studied. The differential equations of motion for the system are established. Under the special infinitesimal transformations of group in which the time is invariable, the definition of the Lie symmetry for the weakly nonholonomic system and its first-degree approximate holonomic system are given, and the exact and approximate Hojman conserved quantities deduced directly from the Lie symmetry are obtained. Finally, an example is given to study the exact and approximate Hojman conserved quantity for the system.

A special feature of employment of Lagrange’s equations to the scleronomic mechanical systems

“…As a rule, equations (14) and (15) are derived using a direct substitution of the expression of the kinetic energy in Lagrange's equations in their general form (1). Deriving equation 15, one can notice at its lefthand side an expression m 2 l _ y _ ' sin ', which appears firstly as positive and secondly as a negative expression because…”

“…There are many potentials of the fruitful application of the Lagrangian dynamics. [1][2][3][4] A general form of equations of motion known as Lagrange's equations of second type is as follows…”

A Study on Time Scale Non-Shifted Hamiltonian Dynamics and Noether's Theorems

“…Time scale theory has been widely applied and achieved many achievements in various fields [4][5][6][7][8][9][10] . In last two decades, some new advances have emerged on the study of time-scale dynamics and its symmetries, such as kinetic equations [11][12][13] , optimal control problems [14,15] , fractional variational problems [16][17][18] , Noether theorems [18][19][20][21][22] , Lie symmetries [23][24][25] , Mei symmetries [26,27] , canonical transformation and Hamilton-Jacobi method [28,29] , time-delay dynamics [30] , Herglotz variational problems [31] , higher-order delta derivatives [32] , etc. However, Ref.…”

“…Recently, three symmetries and conservation laws for non-shifted time-scales dynamic systems were studied in Refs. [25,27,[35][36][37]. The study of non-shifted variational problems on time scales is a new but important research direction of analytical mechanics.…”

Accurate conserved quantity and approximate conserved quantity deduced from Noether symmetry for a weakly Chetaev nonholonomic system

“…Since then, the symmetry of mechanical system with constraints and the theory of conservation were rapidly developed and the fruitful results have been achieved [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. Scholars have made some achievements for nonholonomic mechanical system which is one research direction of mechanical system with constraints [17][18][19][20][21][22][23][24][25][26][27][28][29][30]. Besides, there is a special nonholonomic mechanical system in which a small parameter is contained in constraint equation, which has a small difference from the holonomic system and is defined as the weakly nonholonomic system.…”