Nonholonomic Constraints

Nonholonomic Constraints in advanced mechanics. Theory, formulas, worked examples and applications.

Conceptual context

A coin rolling without slipping on a table can reach any position and orientation, and yet, at every instant, its velocity is not free: it is forced to point in the direction the coin is oriented. This is the hallmark of a nonholonomic constraint: it restricts the system's admissible velocities at each instant, without that restriction being integrable into a relation between the coordinates themselves.

This leaf covers constraints linear in the velocities, of the form Σ $A_{\alpha i}$·$\dot q_i$ + $a_\alpha$ = 0, and the Lagrange-multiplier method that allows incorporating them into the equations of motion without reducing the number of coordinates in the Lagrangian $L$.

🟢 Essential level

The essential idea is that there are two very different types of restrictions in mechanics: those that limit where the system can be (holonomic constraints, like a bead forced to move along a hoop) and those that limit how it can move instant by instant without limiting where it can end up (nonholonomic constraints, like rolling without slipping).

The practical difference is substantial: an independent holonomic constraint can be used to reduce the number of generalized coordinates. A genuinely nonholonomic constraint cannot be eliminated through a coordinate-only relation. For ideal constraints linear in the velocities, the Lagrange–d'Alembert approach used in this leaf incorporates them through multipliers $\lambda_\alpha$; other classes of constraint require checking which dynamical or variational principle applies.

A celebrated example illustrating the idea: a vertical disk rolling without slipping on a plane can reach any point of the plane with any orientation (there is no restriction on where it can be), but at every instant its translational velocity is completely determined by its rotational velocity and its current orientation.

🔵 Formal level

Basic formulas

This first formal layer presents the reference relation used to recognize the model and read its variables before entering more abstract derivations. Identify what each term represents, which assumptions make it valid, and how the chosen coordinates connect to the physical motion that the model is intended to describe.

The formal level rests on four relations:

Linear nonholonomic constraint on velocities

Virtual displacement compatible with the constraint

Lagrange equations with multipliers for nonholonomic constraints

Power delivered by the constraint force

Real student questions

"How do I know whether a constraint is holonomic or nonholonomic?" One checks whether the relation between velocities is the exact time derivative of some function of the coordinates; if no such function exists (which happens, for example, in the rolling of a disk that can rotate in two independent directions), the constraint is genuinely nonholonomic.

"Why can an 'ideal' constraint deliver power if it is supposed to do no work?" Because zero virtual work is defined at fixed time (with $\delta q_i$), while real power is measured along the system's actual motion, which includes the term $a_\alpha$ when the constraint itself moves in time.

Final synthesis

By the end of this leaf, the student should be able to identify whether a constraint is genuinely nonholonomic, write its linear form in the velocities, incorporate it into the equations of motion through a Lagrange multiplier, and compute the real power the constraint delivers to the system when it is rheonomic.

The relevant final check is one of consistency: verifying that the trajectory obtained by solving the complete system exactly satisfies, at every instant, the original nonholonomic constraint on the velocities.