A disk rolls without slipping on a flat surface. Despite the no-slip condition v = ωr constraining velocity at every instant, which statement about accessible configurations is correct?
AThe disk can only reach configurations along straight lines, since rolling constrains it to forward motion
BThe disk can reach any position and heading on the plane, via sufficiently complex rolling paths
CThe disk's heading is permanently linked to its initial orientation and cannot change freely
DThe velocity constraint reduces accessible configurations just as a holonomic constraint would
This is the central insight about nonholonomic constraints: they restrict *instantaneous* allowable motions but do NOT restrict which configurations are accessible. The disk can reach any position and any heading on the plane — it just takes longer paths (like parallel parking a car). The constraint prevents sideways sliding at each instant, but does not remove any configuration from the reachable set. Option D reflects the classic misconception that velocity constraints reduce accessible configurations the same way position constraints do.
Question 2 Multiple Choice
A particle is constrained to remain on a sphere of radius R (x² + y² + z² = R²). How does this holonomic constraint change the system?
AIt adds a degree of freedom, enabling surface-specific motion that wasn't possible in free space
BIt reduces the particle's DOF from 3 to 2 and allows the constraint to be absorbed into the coordinate choice
CIt reduces the particle's DOF from 3 to 2, but the constraint must still be retained as a Lagrange multiplier
DIt has no effect on DOF since the particle still exists in 3D space
A holonomic constraint f(q, t) = 0 reduces DOF by exactly one. A free particle in 3D has 3 DOF; confined to a sphere, it has 2. Crucially, because the constraint is a position equation, it can be *absorbed* into the coordinate choice: you can work entirely in intrinsic surface coordinates (like latitude/longitude) and forget about the constraint equation entirely. This is the key practical advantage of holonomic constraints — they simplify analysis by reducing the problem's dimensionality. Nonholonomic constraints cannot be absorbed this way.
Question 3 True / False
A nonholonomic constraint limits which configurations a system can ever reach.
TTrue
FFalse
Answer: False
This is the central misconception about nonholonomic constraints. A nonholonomic constraint restricts *instantaneous allowable velocities* — at each moment, certain directions of motion are forbidden. But because the constraint doesn't fix positions, the system can still reach any configuration in its full configuration space by taking appropriate (possibly complex) paths. The rolling disk and the car are canonical examples: both can reach any position and heading on the plane despite their nonholonomic velocity constraints.
Question 4 True / False
A holonomic constraint can always be used to eliminate one generalized coordinate from the Lagrangian, simplifying the equations of motion.
TTrue
FFalse
Answer: True
This is the practical advantage of holonomic constraints: because they are algebraic equations relating coordinates, you can solve for one coordinate in terms of the others and substitute, reducing the number of independent variables by one. The constraint disappears into the coordinate choice. Nonholonomic constraints (velocity constraints that cannot be integrated to position equations) cannot be handled this way — they require Lagrange multipliers or specialized techniques because no coordinates can be eliminated.
Question 5 Short Answer
A car has a nonholonomic steering constraint — it cannot slide sideways at any instant. Does this mean there are parking spots it cannot reach? Explain why or why not, and what this reveals about nonholonomic constraints.
Think about your answer, then reveal below.
Model answer: No — a car can reach any parking spot despite its nonholonomic constraint. The constraint limits instantaneous motions (no sideways sliding) but not accessible configurations. Through sequences of allowed forward, reverse, and turning maneuvers (like parallel parking), any position and orientation is reachable. This reveals that nonholonomic constraints restrict the *paths* through configuration space, not the *set of reachable configurations* — a key distinction from holonomic constraints, which genuinely remove configurations from the accessible set.
This is what makes nonholonomic systems richer and more complex than holonomic ones. A ball on a sphere is genuinely restricted to the sphere — it cannot leave the surface. But a car on a flat plane can reach every point and every heading, despite its steering constraint. The constraint shapes the geometry of paths but doesn't shrink the configuration space. This is why nonholonomic systems require special treatment: you cannot simply reduce coordinates as with holonomic constraints, because no configurations have been removed.