classical mechanics Classical mechanics is a Theoretical physics, physical theory describing the motion of objects such as projectiles, parts of Machine (mechanical), machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics inv ...

, a constraint on a

system A system is a group of interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its open system (systems theory), environment, is described by its boundaries, str ...

is a

parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...

that the system must obey. For example, a box sliding down a slope must remain on the slope. There are two different types of constraints: holonomic and non-holonomic.

Types of constraint

First class constraint In physics, a first-class constraint is a dynamical quantity in a constrained Hamiltonian system whose Poisson bracket with all the other constraints vanishes on the constraint surface in phase space (the surface implicitly defined by the simultan ...

s and second class constraints * Primary constraints, secondary constraints, tertiary constraints, quaternary constraints *

Holonomic constraints In classical mechanics, holonomic constraints are relations between the position variables (and possibly time) that can be expressed in the following form: f(u_1, u_2, u_3,\ldots, u_n, t) = 0 where \ are generalized coordinates that describe t ...

, also called integrable constraints, (depending on time and the coordinates but not on the momenta) and

Nonholonomic system A nonholonomic system in physics and mathematics is a physical system whose state depends on the path taken in order to achieve it. Such a system is described by a set of parameters subject to differential constraints and non-linear constraints, s ...

Pfaffian constraint In dynamics, a Pfaffian constraint is a way to describe a dynamical system in the form: : \sum_^nA_du_s + A_rdt = 0;\; r = 1,\ldots, L where L is the number of equations in a system of constraints. Holonomic systems can always be written in Pfa ...

s * Scleronomic constraints (not depending on time) and rheonomic constraints (depending on time) *Ideal constraints: those for which the work done by the constraint forces under a

virtual displacement In analytical mechanics, a branch of applied mathematics and physics, a virtual displacement (or infinitesimal variation) \delta \gamma shows how the mechanical system's trajectory can ''hypothetically'' (hence the term ''virtual'') deviate very ...

vanishes.

References

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