Analysis Of Local Geometric Structure Of Constrained Submanifolds And Its Application: In Dynamics Modeling Of Constrained Systems

Constraint exists in almost all mechanical systems,which is a restrictive condition added to the mechanical motion of particles in the system.The restriction of the constraint conditions on the motion is reflected by the constraint force,which is automatically adjusted according to the tendency of the mechanical system to violate the constraint.In addition,the number of unknowns is more than the number of equations in a system composed only by d ’Alembert principle and constraint equations,so the solution of the system cannot be given.Therefore,the constrained mechanical system is a "statically indeterminate" problem.Since the binding force cannot be determined solely by the constraint equation,a reasonable way to determine the binding force can be put forward by means of physical principles,control needs or empirical assumptions,and a way to solve the dynamics problem of the constrained mechanical system can be given.In this paper,we use the theory of differential geometry to geometrize the binding force of the constrained mechanical system,and realize the dynamic modeling of the constrained mechanical system.The idea is applied to the dynamic modeling of the two-body problem.Compared with the traditional method,the theory can also obtain the dynamic equation of the two-body system accurately and simply.Firstly,we analyze the geometric properties of Lagrange equations in the state space of constrained mechanical systems.Without the aid of ideal constraint assumption and d ’Alembert principle,we establish the second Lagrange equations of the invariant holonomic constrained mechanical systems on constrained submanifolds,which are curved Riemann Spaces.This method can omit the analysis and calculation of binding force.Secondly,as a typical constrained mechanical system,the kinetic modeling of the two-body system is realized by using the above method,and the equation of motion of the two-body system under this method is the same as that of the traditional method.Finally,based on Lie group theory,we verify 10 independent Killing vector fields of two-body system,namely the single-parameter adjoint lie group system of two-body system,including four translational transform vector fields,three rotational transform vector fields and three pseudo-rotational transform vector fields.The Noether conserved quantities of the two-body system are solved by Lie symmetry.The research in this paper provides a new way of thinking for dynamic modeling of constrained systems,which is expected to play a practical value in solving practical problems.