Motion of an integral curve of a Hamiltonian dynamical system and the evolution equations in 3D

We show that all of the nonstretching curve motions specified in the Frenet–Serret frame in the literature can be described by the time evolution of an integral curve of a Hamiltonian dynamical system such that the underlying curve is a geodesic curve on a leaf of the foliation determined by the Poisson structure in three dimensions. As an illustrative example, we show that the focusing version of the nonlinear Schrödinger equation and the complex modified Korteweg–de Vries (mKdV) equation are obtained in this way.

On the Hamiltonian Flow of Brake Hamiltonian System

This paper studies the Hamiltonian flow of the brake Hamiltonian dynamical system on the symmetrical symplectic manifold. By using the transformation law of Hamiltonian diffeomorphisms and the Hamiltonian vectors, this paper describes the characteristics of the Hamiltonian flows and proves that the Hamiltonian flows are invariant under some transformations.

Study on the Correspondence of Hamiltonian Flows and Functions of Brake Hamiltonian System Based on Mechanical Mechanics

This paper studies the Hamiltonian flows and functions of the brake Hamiltonian dynamical system. By the properties of the brake Hamiltonian system and the transformation law of Hamiltonian diffeomorphisms, this paper proves the correspondence of the Hamiltonian flows and the symmetrical Hamiltonian functions under some conditions.

Constructing the time independent Hamiltonian from a time dependent one

In this paper we introduce a method for finding a time independent Hamiltonian of a given Hamiltonian dynamical system by canonoid transformation of canonical momenta. We find a condition that the system should satisfy to have an equivalent time independent formulation. We study the example of a damped harmonic oscillator and give the new time independent Hamiltonian for it, which has the property of tending to the standard Hamiltonian of the harmonic oscillator as damping goes to zero.

NAMBU BRACKETS WITH CONSTRAINT FUNCTIONALS

If a Hamiltonian dynamical system with n degrees of freedom admits m constants of motion more than 2n-1, then there exist some functional relations between the constants of motion. Among these relations the number of functionally independent ones are s = m-(2n-1). It is shown that for such a system in which the constants of motion constitute a polynomial algebra closing in Poisson bracket, the Nambu brackets can be written in terms of these s constraint functionals. The exemplification is very rich and several of them are analyzed in the text.