An example of Lagrangian for a non-holonomic system

Abstract

An adjustable two-mass-point Chaplygin Sleigh is used as an example of a non-holonomic system. Newtonian equations of motion based the assumption of zero virtual work done by constraints are calculated. A Lagrangian that reproduces these equations as its unmodified Euler-Lagrange equations is then explicitly given. The Lagrangian uses variables that are present in the Chaplygin Sleigh equations of motion, as well as some additional variables. Some of the Euler-Lagrange equations of that Lagrangian are non-differential. These non-differential equations automatically and completely reduce out all of these additional variables, so that only the variables that appear in the original equations of motion remain in the final dynamics of the system.