Definition?

Any restriction on the freedom of movement of a system of particles in the form of an equation that relates a coordinate with other coordinates is called a constraint.

Examples:

• Rigid bodies, where the constraints on the motions of the particles keep the distances rij
  unchanged.
  The supporting wires on an abacus restrict the beads to one-dimensional mobility.
• The walls of a container restrict the movement of gas molecules within the container exclusively.
• The restriction that a particle on the surface of a solid sphere can only move on the surface or in the area outside the sphere applies to it.

Types:

There are two major types of constraints discussed below:

Holonomic Constraints:

If the conditions of constraint can be expressed as equations connecting the position coordinates and the time having the form f (r1, r2, r3, ..., t) = 0 then the constraints are called holonomic constraints.
Examples:

• The rigid body having a center of mass C is an example of holonomic constraints, where the constraints are expressed by equations of the form (ri - rj)  ²- cij  ² = 0
• An additional clear example of a holonomic constraint is a particle confined to move along any curve or on a specified surface; the equations describing the curve or surface serve as the equations of a constraint.

Scleronomic Type Holonomic Constraints:

If the constraints are explicitly independent of time, i.e., stationary constraint, then it is called scleronomic. Mathematically, the equation of constrain will be of the form

∂ f ( r, t) / ∂ t =0

The constraint in the case of a pendulum with fixed support is a scleronomic constraint.
A bead sliding on a rigid curved wire fixed in space is subject to a scleronomic constraint.

Rheonomic Type Holonomic Constraints:

If the constraints depend on time explicitly, i.e., non-stationary constraint, then it is called rheonomous. Mathematically, the equation of constrain will be of the form
∂ f ( r, t) / ∂ t =0
The particle can freely move only within the xy plane while for the z coordinate the constraint is
z(t) = vo (t – to) + zo
A simple pendulum whose length changes with time, its constraint expression is time dependent and this motion is rheonomic.

Non-holonomic Constraints:

The constraints which cannot be expressed as a function of position and time are called nonholonomic constraints. Systems having nonholonomic constraints involve constraints of both generalized coordinates and their derivatives. First-order nonholonomic constraints have the form
Examples:
g (r, t) >= 0

A non-holonomic limitation consists of a gas container's walls. In the example of a particle placed on a spherical surface, the constraint is likewise non-holonomic. The constraint can be expressed as an inequality
r ² - a ² >= 0
where a is the radius of the sphere, which is not of the form f ( vec r 1, vec r 2, dot r 3,...,t)=0. Thus, in a gravitational field, a particle placed on the top of the sphere will slide down the surface part of the way but will eventually fall off. 

Difficulties in Constraintsin:

• Difficulties in constraintsIn the solving of mechanical issues, constraints present two kinds of obstacles. Due to their connection through the constraint equations, the coordinates are not independent in the first place.
• Two types of forces are exerted on the gas particle or bead, respectively: the force of the wall or the wire. Imposing constraints on the system is simply another method of stating that there are forces present in the problem that cannot be specified directly but are known rather in terms of their effect on the motion of the system.

In the case of holonomic constraints, the first difficulty is solved by introducing generalized coordinates like Cartesian coordinates. A system of N particles in three dimensions, free from constraints, has 30 independent coordinates. If there exist k holonomic constraints, expressed in equations, then the system will have 3N – k independent coordinates.
This elimination of the dependent coordinates can be expressed by the introduction of new, 3N - k, independent variables q1, q2, ..., q3N - k in terms of which the old coordinates r1, r2,...,qN are expressed by equations of the form
vec r = vec r1 (q1, q2,...,q3 _{N - k},t)

vec rN = vec rN (q1, q2 ,...,q3 _{N - k} ,l)

containing the constraints implicitly. These are transformation equations from the Cartesian coordinates (ri) to generalized coordinates (qi).
In the case of a particle constrained to move on the surface of a sphere, the two angles expressing position on the sphere, say latitude and longitude, are possible generalized coordinates. For a double pendulum moving in a plane generalized coordinates are the two angles 0,,0,. Generalized coordinates, in the sense of coordinates other than Cartesian, are often useful in systems without constraints. Thus, in the problem of a particle moving in an external central force field (V=V(r)), there is no constraint involved, but it is clearly more convenient to use spherical polar coordinates than Cartesian coordinates.

It is not possible to delete the dependent coordinates using the equations describing the restriction if it is non-holonomic. The constraint of an object rolling across a rough surface without slipping is an example of a non-holonomic one. Typically, a set of coordinates representing the point of contact's location on the surface is combined with angular coordinates to identify the body's orientation when characterizing the system. These two coordinate sets are not independent; they are connected by the "rolling" constraint. The point of contact will always vary in orientation when its position changes. Yet we cannot reduce the number of coordinates, for the "rolling" condition is not expressible as an equation between the coordinates. It is more of a velocities-based requirement. That is to say, the point of touch remains fixed.

Degrees of Freedom:

• The minimum number of coordinates required to specify the configuration of a dynamical system which are compatible with the given constraints is called the degrees of freedom of the system.
  Applying this definition:
  For a single particle in a plane, two coordinates characterize its location so it has two degrees of freedom.
• A single particle in space requires three coordinates so it has three degrees of freedom. Two particles in space have a combined six degrees of flexibility.
• For a diatomic particle, the six arranges must fulfill a single limitation condition characterized by the remove formula. This decreases the degree of freedom of the system to five.