Research On Geometry Structure And Its Application Of Lagrange Equation Of Constrained Mechanical System

Many mechanical systems in actual engineering have complex constraints,these complex constraint problems cannot be solved with Newton mechanics,because definite constraint force cannot by obtained by only Newton's law and constraint equations,which makes the constraint problem a statically indeterminate problem.It was the analytical mechanics that used the ideal constraint assumption to determine the constraint force,and then solved the statically indeterminate problem encountered by Newtonian mechanics when dealing with the constraint system.In addition,it was the analytical mechanics that transformed the Euclidean coordinates describing the particle system into generalized coordinates with the help of constraints.Therefore,Analytical Mechanics is a mechanical theory system based on general differential manifolds that is free from the constraint of flat Euclidean space.Therefore,the analysis and discussion of the spatial geometric structure are important for the research on the theoretical system of analytical mechanics.This paper makes an analysis on the geometric properties of the Lagrange equation state space of the constraint mechanics system with geometric mechanics method.The study shows that for a complete constraint system,the state space of the first type of Lagrange equation is a flat Euclidean space,and the state space of the second type of Lagrange equation is a curved Riemann space;for a nonholonomic constraint mechanics system,the state space of the first type of Lagrange equation is a Weitzenb(?)ck space with no curvature and torsion,and the state space of the second type of Lagrange equation is a Riemann-Cartan space with curvature and torsion.The second type of Lagrange equation of the holonomic restraint system is equivalent to the near geodesic equation on the curved Riemann manifold(when the main force of the holonomic constrained system is zero,the Lagrange equation of the second type is equivalent to the geodesic equation on the Riemann manifold);The second type of Lagrange equations of nonholonomic constrained mechanical systems are equivalent to self-parallel equations on Riemann-Cartan manifolds.The thought of constraint force geometrization in analytical mechanics can also be directly applied to the problem of particle motion in the gravitational field.In general relativity,the gravitational field changes the flat Euclidean space to curved Riemann space.The gravitational potential is equivalent to the metric of Riemann space,and the metric is the most basic geometric quantity of Riemann space,which determines the geometric properties of curved Riemann space.Therefore,the physical problems in the gravitational field are also equivalent to the geometric problems in Riemann space.In addition,three forms of Lagrange equations(the first type,the second type,the Lagrange equation under the Gauss principle form)are used to analyze the trajectory of the end of the two-link robot arm.It is found that the numerical analysis results are consistent,which proves that the three methods are equivalent.