Lagrangian and Hamiltonian formulations of asymmetric rigid body, considered as a constrained system

TL;DR

This paper systematically derives the dynamics of an asymmetric rigid body as a constrained system using variational principles, providing new insights into equations of motion and integrability without additional assumptions.

Contribution

It presents a comprehensive variational formulation for asymmetric rigid bodies, deriving equations of motion and integrals directly, and revises cases of integrability with novel properties.

Findings

Derived multiple equivalent forms of equations of motion

Revised integrability conditions for asymmetric rigid bodies

Discussed peculiar properties overlooked in traditional formulations

Abstract

This work is devoted to a systematic exposition of the dynamics of a rigid body, considered as a system with kinematic constraints. Having accepted the variational problem in accordance with this, we no longer need any additional postulates or assumptions about the behavior of the rigid body. All the basic quantities and characteristics of a rigid body, as well as the equations of motion and integrals of motion, are obtained from the variational problem by direct and unequivocal calculations within the framework of standard methods of classical mechanics. Several equivalent forms for the equations of motion of rotational degrees of freedom are deduced and discussed on this basis. Using the resulting formulation, we revise some cases of integrability, and discuss a number of peculiar properties, that are not always taken into account when formulating the laws of motion of a rigid body.