[en] The definition and the criterion of a unified symmetry for a Hamilton system are presented. The sufficient condition under which the Noether symmetry is a unified symmetry for the system is given. A new conserved quantity, as well as the Noether conserved quantity and the Hojman conserved quantity, deduced from the unified symmetry, is obtained. An example is finally given to illustrate the application of the results.

[en] We report the discovery of an envelope Hamiltonian describing the charged-particle dynamics in general linear coupled lattices.

[en] We show that for a natural polynomial Hamiltonian system the existence of a single Darboux polynomial (a partial polynomial first integral) is equivalent to the existence of an additional first integral functionally independent with the Hamiltonian function. Moreover, we show that, in a case when the degree of potential is odd, the system does not admit any proper Darboux polynomial, i.e., the only Darboux polynomials are first integrals

[en] We study the factorization of the PT symmetric Hamiltonian. The general expression for the superpotential corresponding to the PT symmetric potential is obtained and explicit examples are presented. (author)

[en] Chaotic adiabatic dynamics refers to the study of systems exhibiting chaotic evolution under slowly time-dependent equations of motion. In this dissertation the author restricts his attention to Hamiltonian chaotic adiabatic systems. The results presented are organized around a central theme, namely, that the energies of such systems evolve diffusively. He begins with a general analysis, in which he motivates and derives a Fokker-Planck equation governing this process of energy diffusion. He applies this equation to study the open-quotes goodnessclose quotes of an adiabatic invariant associated with chaotic motion. This formalism is then applied to two specific examples. The first is that of a gas of noninteracting point particles inside a hard container that deforms slowly with time. Both the two- and three-dimensional cases are considered. The results are discussed in the context of the Wall Formula for one-body dissipation in nuclear physics, and it is shown that such a gas approaches, asymptotically with time, an exponential velocity distribution. The second example involves the Fermi mechanism for the acceleration of cosmic rays. Explicit evolution equations are obtained for the distribution of cosmic ray energies within this model, and the steady-state energy distribution that arises when this equation is modified to account for the injection and removal of cosmic rays is discussed. Finally, the author re-examines the multiple-time-scale approach as applied to the study of phase space evolution under a chaotic adiabatic Hamiltonian. This leads to a more rigorous derivation of the above-mentioned Fokker-Planck equation, and also to a new term which has relevance to the problem of chaotic adiabatic reaction forces (the forces acting on slow, heavy degrees of freedom due to their coupling to light, fast chaotic degrees)

[en] Making use of the formalism of differential forms, it is shown that any second-order ODE, or any system of two first-order ODEs, with differentiable coefficients, can be expressed in the form of the Hamilton equations with the Hamiltonian function being a differentiable function that can be chosen arbitrarily. It is also shown that any nontrivial local one-parameter group of symmetries of a second-order ODE, or a system of two first-order ODEs, is associated with a first integral

[en] In the present paper, the well-known Noether's identity, which represents the connection between symmetries and first integrals of Euler-Lagrange equations, is rewritten in terms of the Hamiltonian function. This approach, based on the Hamiltonian identity, provides a simple and clear way to find first integrals of canonical Hamiltonian equations without integration. A discrete analog of the Hamiltonian identity is developed. It leads to a connection between symmetries and first integrals of difference Hamiltonian equations that can be used to conserve the structural properties of Hamiltonian equations under discretizaton. The results are illustrated by a number of examples for both continuous and difference Hamiltonian equations.

[en] The first part of the paper is devoted to the theory of master symmetries using the geometric formalism as an approach. It is shown that certain superintegrable systems are endowed with this property as a consequence of the existence of a family of master symmetries. In the second part, the properties of dynamical but non-Hamiltonian symmetries are studied. It is proved that the higher order superintegrability of the generalized Smorodinsky–Winternitz system is a consequence of the existence of symplectic symmetries not preserving the Hamiltonian function. (paper)

[en] We introduce an extension of Hamiltonian dynamics, defined on hyper-Kahler manifolds, which we call 'hyper-Hamiltonian dynamics'. We show that this has many of the attractive features of standard Hamiltonian dynamics. We also discuss the prototypical integrable hyper-Hamiltonian systems, i.e. quaternionic oscillators. (author)

[en] We show that the global stability of quasi-polynomial (QP) and Lotka-Volterra (LV) systems with the well-known logarithmic Lyapunov function is equivalent to the existence of a local generalized dissipative Hamiltonian description of the LV system with a diagonal quadratic form as a Hamiltonian function. The Hamiltonian function can be calculated and the quadratic dissipativity neighborhood of the origin can be estimated by solving linear matrix inequalities