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

2013 ◽  
Vol 30 (1) ◽  
pp. 21-27 ◽  
Author(s):  
Y.-L. Han ◽  
X.-X. Wang ◽  
M.-L. Zhang ◽  
L.-Q. Jia
Keyword(s):  
Nonholonomic System ◽  
Equations Of Motion ◽  
Lie Symmetry ◽  
Conserved Quantity ◽  
Holonomic System ◽  
Lagrange Equations ◽  
Definition Of ◽  
Weakly Nonholonomic System ◽  
Infinitesimal Transformations ◽  
Hojman Conserved Quantity

ABSTRACTThe 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.

Special Lie Symmetry and Hojman Conserved Quantity of Appell Equations for a Holonomic System

2009 ◽  
Vol 26 (3) ◽  
pp. 030303 ◽  
Author(s):  
Jia Li-Qun ◽  
Cui Jin-Chao ◽  
Luo Shao-Kai ◽  
Yang Xin-Fang
Keyword(s):  
Lie Symmetry ◽  
Conserved Quantity ◽  
Holonomic System ◽  
Appell Equations ◽  
Hojman Conserved Quantity

Elements of Classical Field Theory

2021 ◽  
pp. 24-34
Author(s):  
J. Iliopoulos ◽  
T.N. Tomaras
Keyword(s):  
Field Theory ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Classical Field Theory ◽  
Classical Field ◽  
Lagrange Equations ◽  
Lie Group Of Transformations ◽  
Infinitesimal Transformations ◽  
Conserved Currents

The purpose of this chapter is to recall the principles of Lagrangian and Hamiltonian classical mechanics. Many results are presented without detailed proofs. We obtain the Euler–Lagrange equations of motion, and show the equivalence with Hamilton’s equations. We derive Noether’s theorem and show the connection between symmetries and conservation laws. These principles are extended to a system with an infinite number of degrees of freedom, i.e. a classical field theory. The invariance under a Lie group of transformations implies the existence of conserved currents. The corresponding charges generate, through the Poisson brackets, the infinitesimal transformations of the fields as well as the Lie algebra of the group.

The Hamiltonian dynamics of non-holonomic systems

1951 ◽  
Vol 205 (1083) ◽  
pp. 564-583 ◽  
Keyword(s):  
Nonholonomic System ◽  
Integrable Function ◽  
Hamiltonian Dynamics ◽  
Equations Of Motion ◽  
Holonomic System ◽  
Principal Function ◽  
Holonomic Constraints ◽  
Second Derivatives ◽  
General Dynamical System ◽  
Hamilton Jacobi Equation

A formalism is developed which makes it possible to express the equations of motion of a nonholonomic system in Poisson bracket form. The main difficulty which has to be overcome arises from the fact that the Lagrangian co-ordinates and their corresponding momenta do not form a canonical set. However, at each instant of time, these variables can be expressed in a unique way as functions of a canonical set called the locally free co-ordinates and momenta. Poisson brackets can be formed with respect to the locally free variables, and it is shown that these lead to the correct equations of motion for a general dynamical system subject to a given set of non-holonomic constraints. Hamilton’s principle applies to a non-holonomic system , so a principal function can be formed, and its properties are studied in the second part of this paper. In addition to the usual Hamilton-Jacobi equation, the principal function satisfies a set of equations corresponding to the set of constraints. It is shown that these equations imply an indefinite or non-integrable principal function. A non-integrable function is one for which the order of double differentiation is not reversible. A precise method is given for defining the principal function for a non-holonomic system, and it is shown how this leads to indefiniteness in its second derivatives.

Lie symmetry and Hojman conserved quantity of Nambu system

2008 ◽  
Vol 17 (12) ◽  
pp. 4361-4364 ◽  
Author(s):  
Lin Peng ◽  
Fang Jian-Hui ◽  
Pang Ting
Keyword(s):  
Lie Symmetry ◽  
Conserved Quantity ◽  
Hojman Conserved Quantity
Sign in / Sign up

Export Citation Format

Share Document