Method of simplifying Boltzmann-Hamel equation in holonomic system with frame field theory
Authors: Zhang Su-Xia  , Chen Wei-Ting  ,   Tianjin Key Laboratory of Nonlinear Dynamics and Chaos Control, School of Mechanical Engineering, Tianjin University, Tianjin 300354, China;,   School of Civil Engineering, Tianjin University, Tianjin 300354, China
Published: December 2020

Boltzmann-Hamel equation using quasi-velocities as variable quantities instead of generalized-velocities,is an extending form of the classical Lagrange equation.It is widely used for establishing the motion equations in constrained mechanical systems because of its unique structure.The classical method to solve Boltzmann-Hamel equation includes two steps.The first step is to substitute the relationship between the quasi-velocities and generalized-velocities into the equation to establish the second order equation relating to generalized-coordinates.The second step is to search for the analytical solutions using the method of separating variables or the method of Lie groups.However this method is not very effective in practice.In fact,the majority of studies only focus on the similarity between the quasi-coordinate form and the linear non-holonomic constraint form,without considering the effects of the selection of quasi-coordinates on the Boltzmann-Hamel equation.Because the quasi-coordinates in Boltzmann-Hamel equation can be selected freely,the problem of simplifying the Boltzmann-Hamel equation in holonomic system by choosing the appropriate quasi-coordinates is studied in this paper.Using the method of geometrodynamic analysis,the relationship between quasi-coordinates in the time-invariant configuration space and frame field is indicated based on the frame field theory of manifolds.The Boltzmann-Hamel equation in holonomic system is then derived from the tangle of geometric invariance.It differs from the ordinary methods,such as the action principle or d'Alembert's method.It is demonstrated that Boltzmann-Hamel equation can be simplified into an integrable form in homogenous configuration space with zero generalized-force or zero curvature configuration space with non-zero generalized-force.The process of simplifying the equation is provided in detail and the feasibility of this method is verified through two examples.The result in this paper reveals the close link between the intrinsic curvature of the time-invariant configuration space and the structure of Boltzmann-Hamel equation.The simplest form of Boltzmann-Hamel equation under the generalized-coordinate bases field (Lagrange equation) corresponds to the configuration space of zero curvature,and the simplest form of Boltzmann-Hamel equation under the frame field corresponds to the homogenous configuration space (more often,constant curvature space).For the complex motion equations,it should be transformed first into Boltzmann-Hamel equation,then the intrinsic curvature of the time-invariant configuration space will be calculated.If the conditions mentioned in this paper are satisfied, the Boltzmann-Hamel equation can be simplified into the simplest form by choosing appropriate quasi-coordinates,from which,the analytical solutions can be obtained,furthermore,this frame field derived by the appropriate quasi-coordinates can be used as a tool to study the symmetry and the conserved quantity of this holonomic mechanical system.The results in this paper provide a new way to search for the analytical solution of motion equations.