POINCARE—CARTAN INTEGRAL INVARIANTS OF BIRKHOFFIAN SYSTEMS

Based on modern differential geometry, the symplectic structure of a Birkhoffian system which is an extension of conservative and nonconservative systems is analyzed. An integral invariant of Poincaré_Cartan's type is constructed for Birkhoffian systems. Finally, one_dimensional damped vibration is taken as an illustrative example and an integral invariant of Poincaré's type is found.

Research on the discrete variational method for a Birkhoffian system

In this paper, we present a new integration algorithm based on the discrete Pfaff-Birkhoff principle for Birkhoffian systems. It is proved that the new algorithm can preserve the general symplectic geometric structures of Birkhoffian systems. A numerical experiment for a damping oscillator system is conducted. The result shows that the new algorithm can better simulate the energy dissipation than the R-K method, which illustrates that we can numerically solve the dynamical equations by the discrete variational method in a Birkhoffian framework for the systems with a general symplectic structure. Furthermore, it is demonstrated that the results of the numerical experiments are determined not by the constructing methods of Birkhoffian functions but by whether the numerical method can preserve the inherent nature of the dynamical system.

Symplectic feedback using Hamiltonian Lie algebra and its applications to an inverted pendulum

From the symplectic representation of an autonomous nonlinear dynamical system with holonomic con- straints, i.e., those that can be represented through a symplectic form derived from a Hamiltonian, we present a new proof on the realization of the symplectic feedback action, which has several theoretical advantages in demonstrating the uniqueness and existence of this type of solution. Also, we propose a technique based on the interpretation, construction and character- ization of the pull-back differential on the symplectic manifold as a member of a one-parameter Lie group. This allows one to synthesize the control law that governs a certain system to achieve a desired behavior; and the method developed from this is applied to a classical system such as the inverted pendulum.

A Symplectic Conservative Perturbation Series Expansion Method for Linear Hamiltonian Systems with Perturbations and Its Applications

In this paper,a novel symplectic conservative perturbation series expansion method is proposed to investigate the dynamic response of linear Hamiltonian systems accounting for perturbations,which mainly originate from parameters dispersions and measurement errors.Taking the perturbations into account,the perturbed system is regarded as a modification of the nominal system.By combining the perturbation series expansion method with the deterministic linear Hamiltonian system,the solution to the perturbed system is expressed in the form of asymptotic series by introducing a small parameter and a series of Hamiltonian canonical equations to predict the dynamic response are derived.Eventually,the response of the perturbed system can be obtained successfully by solving these Hamiltonian canonical equations using the symplectic difference schemes.The symplectic conservation of the proposed method is demonstrated mathematically indicating that the proposed method can preserve the characteristic property of the system.The performance of the proposed method is evaluated by three examples compared with the Runge-Kutta algorithm.Numerical examples illustrate the superiority of the proposed method in accuracy and stability,especially symplectic conservation for solving linear Hamiltonian systems with perturbations and the applicability in structural dynamic response estimation.