Birkhoff-Hamilton-Jacobi Theory Of Nonholonomically Constrained Systems

In classical dynamics,Hamilton-Jacobi theory is a method for solving Hamiltonian canonical equations,which explains the deep relation between first-order ordinary differential equations and a first-order partial differential equation.In general,the Hamilton-Jacobi equation is constructed by the method of canonical transformation with zero Hamiltonian for Hamilton equations.For the inverse problem of dynamics of the general non-conservative systems,the Hamiltonian system is not direct universal,but the inverse problem of dynamics of the Birkhoffian system with general symplectic structure is directly universal.Therefore,we can construct Birkhoff-Hamilton-Jacobi equation or generalized Hamilton-Jacobi equation under the framework of Birkhoffian system or generalized Birkhoffian system to solve the dynamical problems of nonholonomic or nonconservative systems.The concrete steps are following.Firstly,Birkhoff equation or generalized Birkhoff equation of nonholonomic system or nonconservative system is constructed.Secondly,the Birkhoff-Hamilton-Jacobi equation and the generalized Hamilton-Jacobi equation are constructed by the generalized canonical transformation theory.The detailed approaches are respectively as follows:(1)The Birkhoff-Hamilton-Jacobi equation is obtained directly from the Birkhoff equation or the generalized Birkhoff equation by the canonical transformation with zero Birkhoffian.(2)Firstly,the Birkhoff system is transformed into the generalized Hamilton system.And the generalized Hamilton canonical equations are obtained.Then the generalized Hamilton-Jacobi equation is obtained by the canonical transformation theory with zero generalized Hamiltonian.In this way,Birkhoff-Hamilton-Jacobi equation and generalized Hamilton-Jacobi equation can be used to solve differential equations of nonholonomic system or nonconservative system.