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On the mechanical prototypes of fundamental hydrodynamic invariants and slow manifolds

Arnol'd's group-theoretical concept of generalized rigid body includes the Euler equations of motion of the classical gyroscope and ideal homogeneous fluid as particular representatives. Here, this concept is extended to motion in force fields with a scalar or vector potential and in a Coriolis force field. The concepts of generalized heavy top and generalized MHD system are introduced. As particular cases, they include, on the one hand, the Euler–Poisson equations of the classical heavy top and the Kirchhoff equations of motion of a solid body in a potential flow of an ideal incompressible fluid and, on the other hand, the Oberbeck–Boussinesq equations of motion of a heavy fluid and MHD equations. On this basis, mechanical prototypes are constructed for all known fundamental hydrodynamic invariants and global geophysical flows, including a prototype of the general atmospheric circulation.