Noether theorem for nonholonomic nonconservative mechanical systems in phase space on time scales

Abstract

The paper focuses on studying the Noether theorem for nonholonomic nonconservative mechanical systems in phase space on time scales. First, the Hamilton equations of nonholonomic nonconservative systems on time scales are established, which is based on the Lagrange equations for nonholonomic systems on time scales. Then, based upon the quasi-invariance of Hamilton action of systems under the infinitesimal transformations with respect to the time and generalized coordinate on time scale, the Noether identity and the conserved quantity of nonholonomic nonconservative systems on time scales are obtained. Finally, an example is presented to illustrate the application of the results.

Mei Symmetry and Conservation Laws for Time-Scale Nonshifted Hamilton Equations

“…Anerot and his collaborators [33] proved Noether's theorem in the time-scale version of Lagrange systems under the framework of shifted and nonshifted delta calculus of variation, and the results are also a correction of References [31,32]. In the past decade, the study of time-scale dynamics and its symmetry has attracted extensive attention and made important progress, as shown in References [34][35][36][37][38][39][40][41][42][43][44]. However, the research is mainly limited to the following: (1) conservative system, (2) Noether symmetry, and (3) Noether-type conservation laws.…”

“…Substituting ( 40) and ( 41) into equation ( 28) and using equation (37), we can get 5 Advances in Mathematical Physics…”

Noether Symmetry Method for Hamiltonian Mechanics Involving Generalized Operators

“…For the Hamiltonian system, Noether symmetry is the invariance of the Hamilton action under the infinitesimal transformations. The Noether symmetry method points out that if the infinitesimal generators and the gauge function satisfy the Noether identity, then the conserved quantity of the system can be found [14][15][16][17][18][19][20]. The advantage of the Noether theory is that if there is a Noether symmetry, a corresponding conserved quantity can be found and vice versa.…”

Time-Scale Version of Generalized Birkhoffian Mechanics and Its Symmetries and Conserved Quantities of Noether Type

“…Based on the second Euler-Lagrange equations, they proposed another method to find the Noether conserved quantity. Afterwards, according to these two methods, many scholars have obtained some results have been obtained in the study of variational principle, dynamical equations, and Noether symmetries for the different mechanical systems, such as references [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31].…”