Constraint (mechanics)

A condition that restricts the possible configurations or motions of a mechanical system. Constraints may be equations, inequalities, or rules that depend on time.

Configuration space: The set of all configurations available to a system, usually represented as a geometric space. A constraint selects which configurations in this space are allowed.
Constraint force: A force that enforces a mechanical restriction, such as a surface’s normal force preventing penetration. It redirects motion so the system remains within its allowed configurations.
Rolling without slipping: A rolling condition in which the contact point has zero velocity relative to the surface. It relates a wheel’s translation and rotation through a velocity constraint.
Friction: A force that resists relative motion or its tendency between contacting surfaces. Friction can impose velocity restrictions, but it can also dissipate energy.
Degree of freedom: An independent coordinate needed to specify a system’s configuration. Each independent constraint can reduce the number of independent coordinates.
Lagrange multiplier: An auxiliary variable used to impose an equation constraint in an optimization or mechanical formulation. In constrained mechanics, it often represents the strength of a constraint force.
Rigid body: An idealized body whose internal distances remain fixed during motion. Fixed distances between its points are holonomic constraints.
Constraint stabilization: Numerical methods that prevent accumulated error from violating constraints in simulated motion. It addresses drift from constraints during time integration.
Generalized coordinates: Coordinates chosen to describe a system’s configuration, such as angles or distances. They express constraints and motions without requiring Cartesian coordinates.
Lagrangian mechanics: A formulation of mechanics based on kinetic and potential energy and the principle of stationary action. Its equations incorporate holonomic constraints through generalized coordinates or multipliers.

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