[ \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}^j} \right) - \frac{\partial L}{\partial q^j} = \lambda_i a^i_j(q) ]
In nonholonomic systems, we cannot. The constraints are linear in velocities, so we can use Lagrange multipliers to enforce them. But here’s the deep part: (in the ideal case). That means D’Alembert’s principle still holds—but only for virtual displacements consistent with the constraints. dynamics of nonholonomic systems
This non-integrable velocity constraint is the hallmark of a nonholonomic system. The skateboard can access all possible $(x, y, \theta)$ configurations—no positional restriction—but it cannot move arbitrarily between them. Its velocity is constrained at every instant. In holonomic systems, we can reduce the problem: express velocities in terms of a smaller set of generalized coordinates and their derivatives. Lagrange’s equations then apply directly. Its velocity is constrained at every instant
[ \dot{x} \sin \theta - \dot{y} \cos \theta = 0 ] not its positions
In nonholonomic dynamics, the map is not the territory. The path is not reducible to positions. And the dance is, quite literally, in the derivatives. If you’d like to go further: look into the “Chaplygin sleigh,” “rolling penny,” or the “nonholonomic integrator” in geometric numerical integration. The rabbit hole is deep, and the wheels never slip.
But nonholonomic constraints are different. They restrict the velocities of a system, not its positions, in a way that cannot be integrated into a positional constraint. The classic example? A rolling wheel without slipping. Take a skateboard. Its position in the plane is given by $(x, y)$ and its orientation by $\theta$. That’s 3 degrees of freedom. Now impose the “no lateral slip” condition: the wheel’s velocity perpendicular to its orientation must be zero.