Conservation laws for some separable gyroscopic dynamical systems

Various features of a two-dimensional gyroscopic system are analyzed by introducing a coordinate transformation in which the explicit time dependence occurs only in the matrix coefficient of the new coordinates. This 'separability' property leads to an uncoupling of the equations of motion in the transformed coordinates, permitting the properties of the physical system in the uncoupled coordinate space to be studied. For the Foucault pendulum, it is shown that the coordinate transformation needed to uncouple the problem is the rotation of axes. The response to arbitrary initial conditions and the conservation laws are derived in the uncoupled coordinate space. By reverting to the original coordinate space, the dynamical invariants recently developed by Vujanovic (1978) are rederived and their physical bases are established. Finally, the entire problem is reformulated within the context of the inverse problem of Lagrangian dynamics and analyzed from that perspective.