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Lagrange's equations offer a systematic way to formulate the equations of motion of a mechanical system or a (flexible) structural system with multiple degrees Newton's equations are the fundamental laws of non-relativistic mechanics but their vector nature makes them simple to use only in Cartesian coordinates.The above procedure for finding the frequency of small oscillations can be summed up in three steps: (1) Find the equations of motion, (2) Find the equilibrium Derive the equations of motion for the two particles. Solution. It is desirable to use cylindrical coordinates for this problem. We have two degrees of freedom, Derive the equations of motion for a particle in a circular hoop spinning with angular velocity ω, as in these notes, but about a line that is a distance ϵ off.