Equations of Motion in Tensor Variables and Their Application to Multibody Systems

Summary

Two basic methods for deriving motion equations are used in analytical mechanics: the Newton-Euler approach and the Lagrangian approach. A well-known advantage of the first method is the description of the system’s behavior by means of vector and tensor quantities with clear geometrical and mechanical sense such as mass center, inertia tensor, angular velocity, momentum, moment of momentum, etc. A disadvantage of this method is the necessary introduction of the constraint forces in the motion equations and then their elimination which very often turns out to be a complicated problem. On the other hand the second approach for deriving the motion equations — the Lagrangian formalism — uses as generalized parameters only scalar quantities which usually have a formal character and do not reflect directly the geometry and the dynamics of the system considered. In this chapter the Lagrangian formalism is generalized in such a way that not only scalar but also vector and tensor quantities can be chosen directly as generalized parameters. The derived tensor Lagrangian equations unify the advantages of both the basic methods and open new opportunities for investigations especially in the field of multibody systems. They have clear geometrical and mechanical meaning and build an effective base for analytical and numerical analysis of the system’s behavior.