What makes a constraint non-holonomic?

A non-holonomic constraint cannot be written as an algebraic relation among the coordinates alone; it typically involves velocities in a non-integrable way, as with a wheel rolling without slipping, and cannot be removed by a coordinate change.

Why are constraint forces convenient to eliminate?

Constraint forces are usually unknown and uninteresting, such as the normal force from a track. Because they do no work under virtual displacements consistent with the constraints, the Lagrangian method removes them from the equations of motion automatically.

Generalized Coordinates and Constraints

Generalized coordinates are any independent variables that specify a system's configuration, chosen to absorb constraints and reduce the number of degrees of freedom that must be tracked.

Definition

Generalized coordinates are a minimal set of independent parameters that uniquely specify the configuration of a mechanical system consistent with its constraints, reducing its description to its true number of degrees of freedom.

Scope

This topic covers the choice of generalized coordinates, the notion of configuration space and degrees of freedom, and the classification of constraints as holonomic or non-holonomic, scleronomic or rheonomic. It treats how holonomic constraints are eliminated by an appropriate coordinate choice and how the principle of virtual work and d'Alembert's principle handle constraint forces.

Core questions

How does choosing generalized coordinates reduce the number of variables in a problem?
What distinguishes holonomic from non-holonomic constraints?
How do d'Alembert's principle and virtual work eliminate unknown constraint forces?

Key concepts

Generalized coordinates
Degrees of freedom
Configuration space
Holonomic versus non-holonomic constraints
Virtual displacement and virtual work
Constraint forces

Key theories

Holonomic constraints and degrees of freedom: Holonomic constraints are expressible as equations among coordinates and time; each reduces the degrees of freedom by one and can be absorbed by choosing suitable generalized coordinates.
D'Alembert's principle and virtual work: By admitting only virtual displacements consistent with the constraints, the constraint forces, which do no virtual work, drop out, leaving equations of motion in terms of applied forces alone.

Clinical relevance

Choosing generalized coordinates that respect constraints is what makes the dynamics of linkages, robotic arms, gear trains, and articulated mechanisms tractable, and the holonomic/non-holonomic distinction is decisive for the control of rolling and wheeled systems.

History

D'Alembert's 1743 principle reduced dynamics to a problem of statics by combining inertial and applied forces, and Lagrange built on it to develop the method of generalized coordinates that eliminates constraint forces. The systematic classification of constraints, including the term non-holonomic, was sharpened in the late nineteenth century by Hertz and others.

Key figures

Jean le Rond d'Alembert
Joseph-Louis Lagrange
Heinrich Hertz

Seminal works

goldstein2002
lanczos1970

Frequently asked questions

What makes a constraint non-holonomic?: A non-holonomic constraint cannot be written as an algebraic relation among the coordinates alone; it typically involves velocities in a non-integrable way, as with a wheel rolling without slipping, and cannot be removed by a coordinate change.
Why are constraint forces convenient to eliminate?: Constraint forces are usually unknown and uninteresting, such as the normal force from a track. Because they do no work under virtual displacements consistent with the constraints, the Lagrangian method removes them from the equations of motion automatically.