First-order form, Lagrange’s form, and Gibbs–Appell’s form of Kane’s equations

Abstract

This paper proposes that Kane’s equations for a simple nonholonomic system are the first-order form of generalized speeds. When the first-order form of Kane’s equations is put in matrix form, the element of the mass matrix is identical to the inertia coefficient. Because the orthogonal set of partial velocities will decouple the first-order equations, one can use the orthogonal criterion to generate efficient equations of motion. With the presented first-order form, Kane’s equations are different from Maggi’s or Gibbs–Appell’s equations. Moreover, in order to clarify the relationship between Kane’s approach and classical approaches, we start from Kane’s equations and introduce kinetic energy or acceleration energy functions to derive Lagrange’s or Gibbs–Appell’s forms of Kane’s equations for the system. We found that the Lagrange’s forms of Kane’s equations can be used to solve nonholonomic systems without introducing Lagrangian multipliers. At last, the first-order form, Lagrange’s forms, and Gibbs–Appell’s forms of Kane’s equations are, respectively, used to depict the derivation of first-order equations of motion for a rolling coin.