Singular system has great relationship with gauge field theory, condensed matter theory and some other research areas. Based on the mixed integer Riemann-Liouville fractional derivatives, singular is studied. Firstly, constrained Hamilton equation inherent constraint are presented. Secondly, Lie symmetry conserved quantity analyzed, including determined equation, limited additional structural e...

Given the two boson representation of the conformal algebra Ŵ∞, the second Hamiltonian structure of the KP hierarchy, I construct a bi-Hamiltonian hierarchy for the two associated currents. The KP hierarchy appears as a composite of this new and simpler system. The bi-Hamiltonian structure of the new hierarchy gives naturally all the Hamiltonian structures of the KP system.

Recently, a quantum version of Painlevé equations from the point of view of their symmetries was proposed by H. Nagoya. These quantum Painlevé equations can be written as Hamiltonian systems with a (noncommutative) polynomial Hamiltonian HJ. We give a characterization of the quantum Painlevé equations by certain holomorphic properties. Namely, we introduce canonical transformations such that th...

Sometimes the dynamics of a physical system is described by non-Hamiltonian equations motion, and additionally, characterized long-range interactions. A concrete example that particles interacting with light as encountered in free-electron laser cold-atom experiments. In this work, we study relaxation to systems, more precisely, systems interactions both Hamiltonian origin. Our model consists $...