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 classi ...

, a constraint on a system 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 *

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 ...

s, 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 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, ...

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

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