Rigid Body Dynamics

Rigid Body Dynamics

Rigid body dynamics is one of the oldest and most challenging subjects in classical mechanics. It was initiated by Leonhard Euler, who formulated the equations of motion for a general (asymmetric) torque-free body and obtained the first integrable, in quadratures, case known after his name. With the efforts of D’Alembert,
Poinsot, Lagrange and Poisson, the equations of motion of a body about a fixed point under the action of forces were written in their present form known as the Euler–Poisson equations: a system of six first-order differential equations for which three integrals are known. Lagrange found the second integrable case, the case of a
heavy axi-symmetric body, usually named as Lagrange’s top.

 In both cases of Euler and Lagrange, an integral of motion followed from general principles of mechanics, constancy of the angular momentum in the first and due to the cyclic angle of rotation about the axis of symmetry in the second. An important moment was that in both cases, the equations of motion were solved to the end and the solution expressed through elliptic functions, invented by Jacobi, and certain integrals involving them.

The search for integrable cases continued, but, although the problem attracted the attention of several eminent mathematicians, the search did not lead to any other cases. A whole century later, Sofia Kowalevski found a new integrable case of the heavy rigid body.

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