2.5.2.1 Holonomic constraints

Consider the special case when the constraint function $M$ does not depend on $y'$ , so that the constraints take the form

$$M(x,y(x))=0.$$ (2.53)

Alternatively, we might be able to integrate the constraints (2.52) to bring them to this form. We then say that the constraints are holonomic. The equation (2.54) gives us a constraint surface in the $(x,y)$ -space, and we have two options for studying our constrained optimization problem. The first one is to use the previous necessary condition involving a Lagrange multiplier function. The second option is to find fewer independent variables that parameterize the constraint surface, and reformulate the problem in terms of these variables. The problem then becomes an unconstrained one, and can be studied via the usual Euler-Lagrange equation. Sometimes this latter approach turns out to be more effective.

In control theory, one is often more interested in the opposite situation where the constraints are nonholonomic, i.e., cannot be integrated. In this case, two arbitrary points in the $(x,y)$ -space can be connected by a path satisfying the constraints, and there is no lower-dimensional constraint surface.