Spatially distributed classical Lagrangian mechanics

It is well known that the existence of two nontrivial integrals of the motion makes it possible to parametrize the motion of a Lagrangian rigid body by two variables. On the basis of this fact it is shown that certain combinations of the quantities that characterize the trajectory of such a body satisfy well-known nonlinear equations: sine–Gordon, Korteweg–de Vries, Klein–Gordon, and nonlinear Schrödinger equation.