Special Lie symmetry and Hojman conserved quantity of Appell equations for a Chetaev nonholonomic system

Abstract

A special Lie symmetry and Hojman conserved quantity of the Appell equations for a Chetaev nonholonomic system are studied. The differential equations of motion and Appell equations of the Chetaev nonholonomic system are established. Under the special Lie symmetry group transformations in which the time is invariable, the determining equation of the special Lie symmetry of the Appell equations for a Chetaev nonholonomic system is given, and the expression of the Hojman conserved quantity is deduced directly from the Lie symmetry. Finally, an example is given to illustrate the application of the results.

Conformal invariance and conserved quantity of Mei symmetry for Appell equations in a nonholonomic system of Chetaev’s type

“…If the form of Appell equations (7) for the system and the form of the constraint equations (1) keep invariant when the dynamical functions S and Λ s are replaced by S * and Λ * s , respectively, under the infinitesimal transformations (10), and it requires that the generating functions ξ 0 and ξ s of the infinitesimal transformations satisfy the restriction equations (20) and the additional restriction equations (21), then the symmetry is called the strict Mei symmetry of Appell equations (7) of a holonomic system corresponding to Appell equations (1) and (4) for the nonholonomic system of Chetaev's type.…”

“…By using (43)-(46), it is easy to verify the determining equations (22) and the constraint equation (20) are tenable; substituting (40) and (34) into (21) shows that the additional restrictions equation (21) holds. Hence, from definitions 1-3, the infinitesimal transformation generators expressed by (40) are the infinitesimal transformation generators of Mei symmetry and the strict Mei symmetry for the system.…”

“…Research of symmetries and conserved quantities of constrained dynamical systems plays an important role in modern mechanical and mathematical sciences, and it is also a developing direction of modern mathematics, mechanics and physics [1][2][3]. Fruitful achievements have been gained in looking for conserved quantities by means of Noether symmetry, Lie symmetry and Mei symmetry [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. Theories of the conformal invariance are classified into the gauge field theories in 1960s and 1970s, particularly are a hot topic of gravitational gauge field [22,23].…”

Conformal invariance of Mei symmetry and conserved quantities of Lagrange equation of thin elastic rod

“…The key question to the conformal invariance of dynamics is to find out the conformal factor. Considerable progress has been made over past years in the application of conformal invariance to mechanical systems [20][21][22][23][24][25]45]. However, the application of conformal invariance to thin elastic rod has never been investigated.…”

“…The symmetry under the Lie group transformation has its inherent applicability in classifying and reducing nonlinear differential equations as well as in finding out conservation laws [4,5,[41][42][43][44][45][46]. So applying the symmetry to the elastic rod and finding out its conserved quantities via the symmetry analysis will be helpful for its research.…”