Shang Mei

SYMMETRIES AND CONSERVED QUANTITIES FOR SYSTEMS OF GENERALIZED CLASSICAL MECHANICS

In this paper, the symmetries and the conserved quantities for systems of generalized classical mechanics are studied. First, the generalized Noether’s theorem and the generalized Noether’s inverse theorem of the systems are given, which are based upon the invariant properties of the canonical action with respect to the action of the infinitesimal transformation of r-parameter finite group of transformation; second, the Lie symmetries and conserved quantities of the systems are studied in accordance with the Lie’s theory of the invariance of differential equations under the transformation of infinitesimal groups; and finally, the inner connection between the two kinds of symmetries of systems is discussed. PACC: 0320; 1110; 0220 I. INTRODUCTION The conservation laws (or first integrals) of the mechanical systems are always of mathematical importance and at the same time, they are regarded as the manifestation of some profound physical principles. The fact that the existence of a conserved quantity means that an inner dynamical symmetry in the mechanical system plays an irreplaceable role in the physical interpretation of motion. The modern methods in finding conservation laws are mainly two ways [1] with each having a concept of its own: the first one is based upon the invariant properties of Hamiltonian action with respect to the action of the infinitesimal transformation of the finite transformation group, i.e., Noether symmetries and conserved quantities; the second one is based upon the Lie’s theory of the invariance of differential equations under the action of infinitesimal groups. The differential equations of many physical problems appear to be the variational problems. The theory of the generalized classical mechanics, initiated by Ostrogradsky and Jacobi in the years from 1848 to 1858, has been gaining much development during the recent fifty years, and many important results have been made. [2 6] The Lagrange equations with high

Integrals of generalized Hamilton systems with additional terms

Two kinds of integrals of generalized Hamilton systems with additional terms are discussed. One kind is the integral deduced by Poisson method; the other is Hojman integral obtained by Lie symmetry.

Last Multiplier of Generalized Hamilton System

We study an application of the Jacobi last multiplier to a generalized Hamilton system. A partial differential equation on the last multiplier of the system is established. The last multiplier can be found by the equation. If the quantity of integrals of the system is sufficient, the solution of the system can be found by the last multiplier.

Stability with respect to partial variables for Birkhoff systems

In this paper, the stability with respect to partial variables for the Birkhoff system is studied. By transplanting the results of the partial stability for general systems to the Birkhoff system and constructing a suitable Liapunov function, the partial stability of the system can be achieved. Finally, two examples are given to illustrate the application of the results.

Conserved quantities and symmetries related to stochastic Hamiltonian systems

In this paper symmetries and conservation laws for stochastic dynamical systems are discussed in detail. Determining equations for infinitesimal approximate symmetries of Ito and Stratonovich dynamical systems are derived. It shows how to derive conserved quantities for stochastic dynamical systems by using their symmetries without recourse to Noethers theorem.

Noether's theorem and one-step corrections method for holonomic system *

In this paper, a new computational method for improving the accuracy of numerically computed solutions is introduced. The computational method is based on the one-step method and conserved quantities of holonomic systems are considered as kinematical constraints in this method.

Poisson theory of generalized Bikhoff equations

This paper presents a Poisson theory of the generalized Birkhoff equations, including the algebraic structure of the equations, the sufficient and necessary condition on the integral and the conditions under which a new integral can be deduced by a known integral as well as the form of the new integral.

Poincaré-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-Cartans type is constructed for Birkhoffian systems. Finally, one-dimensional damped vibration is taken as an illustrative example and an integral invariant of Poincarés type is found.

The mean-square exponential stability and instability of stochastic nonholonomic systems

We present a new methodology for studying the mean-square exponential stability and instability of nonlinear nonholonomic systems under disturbance of Gaussian white-noise by the first approximation. Firstly, we give the linearized equations of nonlinear nonholonomic stochastic systems; then we construct a proper stochastic Lyapunov function to investigate the mean-square exponential stability and instability of the linearized systems and thus determine the stability and instability in probability of corresponding competing systems. An example is given to illustrate the application procedures.

A Birkhoff-Noether method of solving differential equations

In this paper, a Birkhoff–Noether method of solving ordinary differential equations is presented. The differential equations can be expressed in terms of Birkhoffs equations. The first integrals for differential equations can be found by using the Noether theory for Birkhoffian systems. Two examples are given to illustrate the application of the method.