## Constraints

The limitations on the motion of a system are called constraints or the conditions imposed on the motion of a particle are called constraints. A constraint is a restriction or condition that limits the possible motion of a system of particles or objects in classical mechanics. These constraints can result from physical barriers, such as rigid connections or fixed boundaries, or from mathematical relationships between the coordinates describing the system. Constraints play a critical role in determining the system’s allowed configurations and motions, influencing its dynamics, and defining the set of valid solutions. The study of constraints is critical in analyzing and understanding the behaviour of mechanical systems in various branches of classical physics, ranging from simple pendulums to complex multi-body systems.

## Constrained Motion

A motion that can’t proceed in any arbitrary manner and obeys certain given conditions is known as *constrained motion*.

Examples of Constrained Motion:

(i) The motion of a rigid body is a constrained motion as the distance between any two particles remains constant.

(ii) The motion of a point mass of a simple pendulum is constrained because it is subjected to the following two conditions

- The point mass oscillates in a fixed plane.
- The point mass always remains at a constant distance from the point of suspension or the length of the string always remains constant.

## Types of Constraints

Constraints are classified as-

### Holonomic Constraints

Constraints are said to be *holonomic* if constraints imposed on the motion of the system can be expressed in the form of a mathematical equation as

Holonomic constraints are *independent of velocities*. If a constraint relation contains velocity and upon integration w.r.t. time it can be made free from velocity then those constraints are known as holonomic constraints.

e.g. The motion of a particle that is moving in a circular path in a plane i.e.

\overrightarrow{r_i}=k =>\overrightarrow{r_i}-k=0The motion of the rigid body as the distance between any two pairs of its particles remains constant.

|\overrightarrow{r_i}-\overrightarrow{r_j}|^2−|𝐶_{𝑖𝑗}|^2=0### Non-Holonomic Constraints

Constraints are said to be *non-holonomic* if constraints imposed on the motion of a system cannot be expressed in the form of mathematical equations i.e. if constraint relations are not holonomic then it is non-holonomic. Non-holonomic constraints are irreducible *functions of velocities*.

e.g. The motion of particles that is moving inside a hollow sphere. If gas molecules bound in a sphere of radius r_i , then the position of every molecule can be expressed as

r_i≤k### Scleronomic Constraints

Constraints that are independent of time are known as *scleronomic* constraints. Scleronomic constraints are also known as *stationary* constraints. General form of scleronomic constraint is f(\overrightarrow{r},\dot{\overrightarrow{r}})=0, f(\overrightarrow{r},\dot{\overrightarrow{r}})\geq0, \overrightarrow{r}, \dot{\overrightarrow{r}} denote respectively the position and velocity vectors at time *t*.

e.g. For a simple pendulum with fixed string length, the position vector of the bob \overrightarrow{r} with respect to the fixed fulcrum at any time *t* must satisfy the constraint relation |\overrightarrow{r}|^2=l^2, l being the constant length of the string of the pendulum. This relation is of the form f(r)=0. It is independent of time, so sceleronomic.

### Rheonomic Constraints

If the constraints relation depends explicitly on time then it is called *rheonomic* constraints. General form of rheonomic constraint is f(\overrightarrow{r},\dot{\overrightarrow{r}},t)=0, f(\overrightarrow{r},\dot{\overrightarrow{r}},t)\geq0, \overrightarrow{r}, \dot{\overrightarrow{r}} denote respectively the position and velocity vectors at time *t*.

e.g. The constraint on a bead on a uniformly rotating wire in a force-free space. The position of the particles depends on time.

### Conservative Constraints

If the *total mechanical energy *of the system remains constant during the constrained motion, then the constraint is known as the conservative constraint.

e.g. The constraints on a rigid body are conservative because the distance between any two arbitrary points remains constant.

### Dissipative Constraints

If the total mechanical energy of the system is *not conserved *during the constrained motion, then the constraint is known as the dissipative constraint.

e.g. An expanding or contracting spherical gas container. The motion each gas particle at any time t is constrained by the inequation given by |\overrightarrow{r}(t)|≤R(t), where R is the radius of the container.

### Bilateral Constraints

If at every point on the constraint surface both the forward and backward motions are possible, the constraint relation can be expressed in the form of an equation. In such cases, the constraint is known as the bilateral constraint. Constraint relations are in the form of *equations*. The general form of bilateral constraint is f(\overrightarrow{r},\dot{\overrightarrow{r}},t)=0.

e.g. Simple pendulum with fixed rigid support with fixed length. At any time *t* the constraint relation is |\overrightarrow{r}|^2=l^2, l being the constant length of the string of the pendulum.

### Unilateral Constraints

If the constraint relation *cannot be expressed *in the form of an equation, the constraint is called unilateral constraint. In such cases, forward motion is not possible at some point on the constraint surface. Constraint relations are in the form of inequalities. The general form of unilateral constraint is f(\overrightarrow{r},\dot{\overrightarrow{r}},t)\geq0.

e.g. The motion of a train or bicycle until and unless breaks are applied to it.

## Difference between Holonomic and Non-Holonomic Constraints

Holonomic | Non-Holonomic |

It can be expressed in algebraic equality | It obeys inequality relation |

It is velocity independent | It depends on velocity |

It is integrable | It depends on the velocity |

Some systems and their constraints

System | Type of constraint |

Rigid Body | Scleronomic, Holonomic, Bilateral, Conservative |

Deformable bodies | Rheonomic, Holonomic, Bilateral, Dissipative |

Simple pendulum with rigid support | Scleronomic, Holonomic, Bilateral, Conservative |

Pendulum with variable length | Rheonomic, Holonomic, Bilateral, Dissipative |

Gas filled with hollow sphere | Scleronomic, Holonomic, Unilateral, Conservative |

An expanding or contracting spherical container of gas | Rheonomic, Holonomic, Unilateral, Dissipative |

## Finite or Geometrical Constraints

The constraints which can be expressed as f( \overrightarrow{r_1}, \overrightarrow{r_2}, ... , \overrightarrow{r_N}, t)=0 are called *finite or geometrical constraints*. These constraints imposed restrictions on the *position* of the particle but do not impose restrictions on velocity.

At any instant of time, the particle cannot possess any arbitrary position.

## Differential or Kinematical Constraints

The constraints which can be expressed as f( \overrightarrow{r_1}, \overrightarrow{r_2}, ... , \overrightarrow{r_N}, \dot{\overrightarrow{r_1}}, \dot{\overrightarrow{r_2}}, ..., \dot{\overrightarrow{r_N}}, t)=0 are called *differential or kinematical constraints*. These constraints imposed restrictions on the *velocity* of the particle but do not impose restriction on the position of the particle.

At any instant of time, the particle can posses any arbitrary position.

## Conclusion

In classical mechanics, constraints are the invisible architects who shape the motion of objects from the macroscopic to the quantum level. They enrich mechanics by revealing a plethora of intricate phenomena and guiding our understanding of the physical world. Physicists continue to push the boundaries of classical mechanics by embracing and unravelling the secrets of constraints, revealing the beauty and complexity of nature’s dance. So, the next time you see an object in motion, consider the constraints that shape its path and the amazing interplay of forces that govern its every move.

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