From Generalized Hamilton Principle to Generalized Schrodinger Equation

The Hamilton principle is a variation principle describing the isolated and conservative systems, its Lagrange function is the difference between kinetic energy and potential energy. By Feynman path integration, we can obtain the standard Schrodinger equation. In this paper, we have given the generalized Hamilton principle, which can describe the heat exchange system, and the nonconservative force system. On this basis, we have further given their generalized Lagrange functions and Hamilton functions. With the Feynman path integration, we have given the generalized Schrodinger equation of nonconservative force system and the heat exchange system.

Lie symmetries and conserved quantities of fractional nonconservative singular systems

In this paper,according to the fractional factor derivative method,we study the Lie symmetry theory of fractional nonconservative singular Lagrange systems in a configuration space.First,fractional calculus is calculated by using the fractional factor,and the fractional equations of motion are derived by using the differential variational principle.Second,the determining equations and the limiting equations of Lie symmetry under an infinitesimal group transformation are obtained.Furthermore,the fractional conserved quantity form of singular Lagrange systems caused by Lie symmetry is obtained by constructing a gauge-generating function that fulfills the structural equation,which conforms to the Noether criterion equation.Finally,we present an example of a calculation.The results show that the Lie symmetry condition of nonconservative singular Lagrange systems is more strict than conservative singular systems,but because of increased invariance restriction,the nonconservative forces do not change the form of conserved quantity;meanwhile,the fractional factor method has high natural consistency with the integral calculus,so the theory of integer-order singular systems can be easily extended to fractional singular Lagrange systems.

Nonconservative mechanical systems with nonholonomic constraints

A geometric setting for generally nonconservative mechanical systems on fibred manifolds is proposed. Emphasis is put on an explicit formulation of nonholonomic mechanics when an unconstrained Lagrangian system moves in a generally non-potential force field depending on time, positions and velocities, and the constraints are nonholonomic, not necessarily linear in velocities. Equations of motion, and the corresponding Harniltonian equations in intrinsic form are given. Regularity conditions are found and a nonholonomic Legendre transformation is proposed, leading to a canonical form of the nonholonomic Hamiltonian equations for nonconservative systems.