classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...

, a constraint on a

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

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 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 simultaneous vanis ...

s and

second class constraints 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 simultaneous vanish ...

Primary constraint In Hamiltonian mechanics, a primary constraint is a relation between the coordinates and momenta that holds without using the equations of motion. A secondary constraint is one that is not primary—in other words it holds when the equations ...

secondary constraint In Hamiltonian mechanics, a primary constraint is a relation between the coordinates and momenta that holds without using the equations of motion. A secondary constraint is one that is not primary—in other words it holds when the equations ...

tertiary constraint In Hamiltonian mechanics, a primary constraint is a relation between the coordinates and momenta that holds without using the equations of motion. A secondary constraint is one that is not primary—in other words it holds when the equations ...

quaternary constraint The Quaternary ( ) is the current and most recent of the three periods of the Cenozoic Era in the geologic time scale of the International Commission on Stratigraphy (ICS). It follows the Neogene Period and spans from 2.58 million year ...

s. *

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 the ''n'' generalized coordinates that d ...

, 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 vanishes.

References

Classical mechanics fa: دستگاه‌های مقید {{classicalmechanics-stub