Nonholonomic Systems

mechanical systems that have imposed on them nonholonomic constraints (kinematic constraints that do not reduce to geometric constraints) in addition to purely geometric constraints. (SeeHOLONOMIC SYSTEMS.) A sphere rolling on a rough plane without slipping is an example of a nonholonomic system. In this case, the constraint imposed is a constraint not only on the position of the center of the sphere (geometric constraint) but also on the velocity of the point of contact between the sphere and the plane; this velocity must be zero at any moment of time, which is a kinematic constraint that does not reduce to a geometric constraint.

Mathematically, nonholonomic constraints are directly expressed by nonintegrable equations of the form

f (x_i, ẏ_i,z_i, ẋ_i, ẏ_i, ż_i, , t) = 0

where x_i, y_i, and z_i are the coordinates of the points of the mechanical system; and x_i, y_i, and z_i, are the components of the velocities of the points, equal to the derivatives of the coordinates with respect to time t.

The motion of nonholonomic systems is studied using special equations, such as the Chaplygin or Appell equations, or equations that can be obtained from differential variational principles of mechanics.