lectures on classical mechanics-1

PBC Lecture Notes Series in Mechanics:

by Dr. Abhijit Kar Gupta, e-mail: [email protected]

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Classical Mechanics: Lecture-1 (degrees of freedom, constraints)

To describe the motion of a single particle or a system of particles we have to know the

coordinates. If N particles are moving freely in three dimensions, the motion is described

by N3 space coordinates. But all the coordinates are not independent when there are

constraints in the system. This means there are restrictions imposed on the coordinates;

change in one coordinate affects other coordinates. The ‘restrictions’ can be expressed in

terms of equations or appropriate conditions.

Examples:

Let us think of a simple pendulum oscillating in a two dimensional plane ( yx, ):

Next, we consider a particle moving on the surface of a sphere. The three

Cartesian coordinates satisfy: 2222 azyx . Thus in this case we have 2

independent coordinates. If we know x and y we can know 222 yxaz .

But if we say, the particle can be anywhere inside the sphere, we have the

following inequality condition: 2222 azyx .

In this case the constraint is not in the form of an equation as before.

Degrees of freedom:

Number of independent coordinates that are required to describe the motion of a system

is called degrees of freedom.

In a system of N -particles, if there are k -equations of constraints, we have

kNn 3 number of independent coordinates. n degrees of freedom.

Configuration space:

The motion of a system of N -particles having n -degrees of freedom can be imagined by

the motion of a point in the space of n -coordinates ( n -dimensional hypothetical space).

This space is called configuration space.

Classification of constraints: Constraints can be of broadly two types.

Holonomic

Non-Holonomic

X

Y

l

),( yx

The motion of the pendulum bob is

such that we have 222 lyx . This

equation is a relation between two

coordinates and thus is a constraint.

Therefore, the motion is described by a

single independent coordinate.



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Holonomic:

A Holonomic constraint is one that can be expressed in the form of an equation relating

the coordinates: 0)..,,,( 111 tzyxf

Example : simple pendulum, 222 lyx ; particle on a sphere, 2222 azyx .

A general example of holonomic constraint is a rigid body where the distance

between any two points is fixed: 22)( ijji crr .

Non-holonomic: The constraint which can not be expressed in the form of an equation relating the

coordinates is called non-holonomic.

Example : The motion of a particle inside anywhere a sphere, 2222 azyx .

The constraints are also classified into the following way:

Scleronomic: where the constraints are independent of time.

Rheonomic: where the constraints have explicit dependence on time.

More examples on the types of constraints:

#1. An object sliding down an inclined plane:

#2. An object sliding down an inclined plane where the inclination angle of the plane

varies with time: constraint, tx

ytan , here t

Type of constraint: Holonomic and Rheonomic.

X

Constraint: tanx

y

Type of constraint:

Holonomic and Scleronomic

Y



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#3. A disc rolling down an inclined plane (not slipping):

Note: Sometimes the constraints are written in terms of velocities and momentum etc. But they

can be integrated to find relations among coordinates.

#4. A disc rolling on the XY-plane:

#5. A sphere rolling over a larger sphere:

v

X

Y

Z

X

s

Y Constraint (No slipping condition):

dt

da

dt

ds

Integrating, as const., where

s is the distance measured on the

inclined plane.

Type of constraint:

Holonomic and Scleronomic

Condition of rolling:

av

Also we can write,

cosvx

; sinvy

Thus we have,

0.cos dadx ……(1) and

0.sin dady ……..(2).

The above two differential

equations for constraints are not

exact differential. Thus there can

not be any algebraic equation

relating the coordinates. Therefore,

the type of constrain is

Non-holonomic.

The constraint changes when the small

sphere is detached from the larger sphere

while rolling down. Thus we can not write

an equation relating the coordinates for the

entire process. The system is Non-

holonomic.

Since there is no explicit time dependence,

the system is scleronomic.



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NOTE:

Mind that for HOLONOMIC constraint, you have to have an equation relating the

coordinates (more precisely, an algebraic equation). In case you have a differential

equation and you know this can be integrated to convert into an algebraic one, you will

still consider this to be holonomic. One particularly interesting example is a cylinder (or a

disc or a sphere) rolling down an inclined plane. When you consider the relation between

the linear speed with the radius and angle that is created due to rotation, you have a

constraint of the kind as just said. Here it is assumed that the object is rolling down

straight over the inclined plane.

In case of a disc (or sphere) rolling on a horizontal plane, the object is rolling but its path

is not restricted to a straight line always, it is wandering around over the plane making a

curved path. So there is one additional coordinate (an angle) which is needed to describe

the motion but that is not known. So if you write the differential equations here, one

along x-axis and another along y-axis, that additional angle appears there. As the angles

are unknown, the differential equations can not be integrated to make an algebraic

equation in this case like before. So this is NON-HOLONOMIC.

lectures on classical mechanics-1

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