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

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.

32 pages, 8 figures, discussions added, new section added: Chetaev bracket is the Dirac bracket. Matches with published version: A. A. Deriglazov, Eur. J. Phys. https://doi.org/10.1088/1361-6404/ace80d

References in corpus (3)

• Basic notions of Poisson and symplectic geometry in local coordinates, with applications to Hamiltonian systems
• Geodesic motion on the symplectic leaf of $SO(3)$ with distorted $e(3)$ algebra and Liouville integrability of a free rigid body
• Poincaré-Chetaev equations in the Dirac's formalism of constrained systems

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