Mechanical prototypes of fundamental hydrodynamic invariants

V.I. Arnold [1-3] revealed that the Kelvin circulation theorem in fluid mec hanics is. in a certain sense, analogous to the law of conservation or top angular momentum in mechanics. On the basis of generalized rigid body (GRB) construction. i.e., the Euler equations of motion of a rigid body with a f ixed point in an arbitrary Lie group [1-3]. and its extensions to the case of motion in external force fields characterized by the scalar or vector po tential [6, 11], it is shown below that the respective mechanical prototype s of Moffat's invariant in fluid mechanics (the helicity of the velocity fi eld), potential vorticity, and Moffat's invariant in magnetohydrodynamics ( the cross helicity of the velocity field and the magnetic field) are the ki netic moment of the rigid body squared, the kinetic moment projection on th e direction of the gravity field in which the rigid body moves, and the kin etic moment projection on the direction of the electric current passing thr ough the electrically neutral medium of low density where an ideal rigid co nductor moves. A theorem is proved that there is no exact nonstationary sol ution to the equations of motion of an ideal homogeneous incompressible flu id in the form of a finite-dimensional dynamic system of the GRB type that would conserve a nonzero helicity. The validity of the Kelvin theorem for t he magnetic vector potential and the conservation of the magnetic field hel icity (Woltjer's invariant) follow from the fact that this field is immovab le with respect to the fluid, much as. for the Kirchhoff equations regarded as the equations of motion of an ideal rigid conductor with fixed point th rough a medium, the conservation of the density of electric current squared follows from its spatial constancy [15].