This paper presents fractional generalized canonical transformations for fractional Birkhoffian systems within Caputo derivatives.Firstly,based on fractional Pfaff-Birkhoff principle within Caputo derivatives,fractional Birkhoff’s equations are derived and the basic identity of constructing generalized canonical transformations is proposed.Secondly,according to the fact that the generating functions contain new and old variables,four kinds of generating functions of the fractional Birkhoffian system are proposed,and four basic forms of fractional generalized canonical transformations are deduced.Then,fractional canonical transformations for fractional Hamiltonian system are given.Some interesting examples are finally listed.
This paper summarized the recent development on Herglotz’s generalized variational principle and its symmetries and conserved quantities for nonconservative dynamical systems.Taking Lagrangian mechanics,Hamiltonian mechanics and Birkhoffian mechanics as three research frames,we introduce Herglotz’s generalized variational principle,dynamical equations of Herglotz type,Noether symmetry and conserved quantities,and their generalization to time-delay dynamics,fractional dynamics and time-scale dynamics,and put forward some problems as suggestions for future research.
In this paper, the Noether symmetries and the conserved quantities for a Hamilton system with time delay are dis-cussed. Firstly, the variational principles with time delay for the Hamilton system are given, and the Hamilton canonical equations with time delay are established. Secondly, according to the invariance of the function under the infinitesimal transformations of the group, the basic formulas for the variational of the Hamilton action with time delay are discussed, the definitions and the criteria of the Noether symmetric transformations and quasi-symmetric transformations with time delay are obtained, and the relationship between the Noether symmetry and the conserved quantity with time delay is studied. In addition, examples are given to illustrate the application of the results.
The Noether symmetry and the conserved quantity on time scales in event space are studied in this paper. Firstly, the Lagrangian of parameter forms on time scales in event space are established. The Euler-Lagrange equations and the second EulerLagrange equations of variational calculus on time scales in event space are established. Secondly, based upon the invariance of the Hamilton action on time scales in event space under the infinitesimal transformations of a group, the Noether symmetry and the conserved quantity on time scales in event space are established.Finally, an example is given to illustrate the method and results.
