Lie symmetry and approximate Hojman conserved quantity of Appell equations for a weakly nonholonomic system

Abstract

For a weakly nonholonomic system, the Lie symmetry and approximate Hojman conserved quantity of Appell equations are studied. Based on the Appell equations for a weakly nonholonomic system under special infinitesimal transformations of a group in which the time is invariable, the definition of the Lie symmetry of the weakly nonholonomic system and its first-degree approximate holonomic system are given. With the aid of the structure equation that the gauge function satisfies, the exact and approximate Hojman conserved quantities deduced directly from the Lie symmetry are derived. Finally, an example is given to study the exact and approximate Hojman conserved quantity of the system.

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.…”

“…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].…”

Symmetries and conserved quantities of constrained mechanical systems

“…see references [36][37][38][39]. There have been some important results on the study of the Lie symmetry of mechanical systems [40][41][42][43][44][45][46][47][48].…”

Approximate Mei Symmetries and Invariants of the Hamiltonian

“…The Appell equations are specifically established for nonholonomic system for these symmetries. The Lie symmetry and approximate Hojman conserved quantity of Appell equations are investigated for a weakly nonholonomic system by Han et al [15]. Furthermore, under the infinitesimal transformation of group in which time is invariable, the Lie symmetries are found for weakly nonholonomic systems and their first degree approximate holonomic systems.…”