Holonomic and nonholonomic constraints

A holonomic constraint can be written as a relation among configuration coordinates and time. Differentiating it produces a relation linear in velocities, but that differentiated relation has not become nonholonomic: it still integrates back to the original position restriction, with an integration constant fixed by initial data. Rolling along one permanently straight line gives a familiar example. If radius is a and spin angle is φ, xdot=a φdot integrates to x−aφ=constant on a chosen unwrapped angular chart.

A genuinely nonholonomic velocity restriction cannot locally be replaced by an equivalent collection of position constraints that describes all its admissible velocities. Its allowed directions change in a way that prevents such integration. Counting a linear velocity constraint as one fewer configuration coordinate is therefore generally wrong. The configuration may still require the original number of labels, although fewer independent velocity directions are available at each point. Some velocity restrictions become integrable after imposing additional restrictions, so specify the complete model before classifying it.

For an ideal upright disk on a horizontal plane, let x and y locate its center, θ specify its heading, and φ specify spin about its axle. No-slip rolling requires translational velocity to point along the heading and have magnitude a φdot. When heading is free to change, integrating the x and y relations depends on the whole steering history. Replacing them with x=aφ cos θ and y=aφ sin θ is incorrect: differentiating those proposed position equations introduces extra θdot terms absent from the actual rolling restrictions.

The no-sideways restriction alone can be illustrated by a knife edge: minus sin θ times xdot plus cos θ times ydot must vanish. At one heading it forbids one direction, but changing the heading changes which direction is forbidden. Short admissible motion sequences can produce net displacements unavailable in one instantaneous step. This history dependence is why steering maneuvers can position a vehicle without permitting direct lateral sliding. The ideal model ignores tire deformation, finite contact patches, and loss of grip.

For ideal constraints linear in velocities, Lagrange–d’Alembert dynamics uses reaction forces proportional to the constraint coefficients and restricts virtual displacements to the associated homogeneous relations. Solve the unconstrained Euler–Lagrange equations with these reaction terms together with the velocity constraints. The multipliers determine reactions. In contrast, placing velocity constraints into an action and varying with ordinary endpoint-fixed path variations generally defines a different, vakonomic problem. These procedures agree in special cases, but they are not interchangeable recipes for physical rolling.

The vector lab is a local geometry aid. Choose an angle and magnitude to represent the allowed forward velocity, predict its x and y components, and compare them with the rolling equations. Changing Angle changes the chosen instantaneous heading; it does not simulate a steering history, integrate the disk position, or enforce contact forces. Use separate written calculations to ask whether a proposed configuration relation differentiates to the actual velocity constraints. That differentiation test catches many attractive but invalid reductions before they contaminate a dynamics calculation.