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.

This paper studies the nontrivial positive solutions to a semilinear elliptic system in the n dimensional Euclide space. By constructing new variational space, using the linking theorems and some embedding theorems, this paper proves the existence of positive solution to a semilinear elliptic system, and improves the results of Li and Wang.

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.

This paper studies the nontrivial positive solutions to a semilinear elliptic system with variable coefficients in the n dimensional Euclide space. By constructing a new variational space and using some linking theorems, this paper finally proves the existence of positive solution to a semilinear elliptic system.

This paper studies the periodic solutions to a superquadratic second-oder discrete type Hamiltonian system in the n dimensional Euclide space. By the variational methods and some discrete computional techniques, this paper proves the existence of solution to a new type discrete Hamiltonian system.