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Ashok Goyal

1. Introduction:

Classical mechanics deals with the description of a dynamical system which evolves in a three dimensional Euclidean space. The space and time are considered absolute and immutable entities. In the Newtonian world view, motion is described by Newton’s laws of motion which are valid in an inertial frame of reference. The dynamical system may be a point particle, a rigid body or a collection of particles. Classical mechanics was developed by Newton. An alternative and attractive formulation was developed by Lagrange, Euler, Hamilton, Poisoon, Jacobi and others. This formulation made the transition from classical mechanics to quantum mechanics easier and found important role in the theory of classical and quantum fields.

2. Constraints

In real physical situations the motion is often constrained to move in a way such that some of its coordinates or velocity components satisfy certain relations all through its motion. The relations can be expressed in the form of equations or inequalities. For example, the motion of a particle on a circle or on an ellipse in the X-Y plane satisfies

if it is moving on a circle of radius a or on an ellipse of semi-major axis a and semi-minor axis b. The coordinates of a particle confine to move within a sphere of radius a satisfies

The constrained motion results on account of certain forces called ‘constraining forces’ which arise when the particle is in contact with the surface or the curve on which it is constrained to move. Constraints can be classified in different classes depending on their nature ‘Holonomic’ constraints can be expressed in terms if algebraic equations in the form

and they can be made independent of velocities. If the constraints depend on velocities and cannot be expressed in ‘integrable equations’ they are termed ‘Non holonomic’. Non-holonomic constraints may involve frictional forces or expressed in terms of an inequality or in terms of non-integrable equations like . The constraints that do not depend on time explicitly are called ‘Scleronomic’ and Rheonomic’ if they depend explicitly on time. Further the constraints are said to be ‘conservative’ if the total mechanical energy during the motion is conserved and ‘dissipative’ when the constraint forces do work and the mechanical energy is not conserved.

Examples:

1. Constraint equation satisfied by a particle moving on the surface of a sphere of radius a is

2. Constraints satisfied by the motion of an atom of radius a moving in a rectangular cavity of size are

In view of the fact that the coordinates of a constrained system satisfy certain relations, the number of independent variables required to fix the position and configuration of a dynamical system is reduced in

A free particle moving in a three dimensional Euclidean space has three degrees of freedom ( , , ) in Cartesian coordinates each varying between −∞ to + ∞ . N free particles likewise have 3N degrees of freedom.

A diatomic molecule having two identical atoms joined together by electromagnetic forces such that the bond length remains fixed has 3 x 2 – 1=5 degrees of freedom. Each atom has 3 degrees of freedom with one constraint − =

A rigid body : A Rigid body is mathematically idealized as a system with large number of particles with fixed distances among themselves has 6 degrees of freedom. This is because any three points in a rigid body that are not collinear have 3 x 2 = 6 degrees of freedom and once any three arbitrary points in the rigid body are fixed, the rigid body is completely fixed.

How to choose the degrees of freedom?

There is a choice in choosing degrees of freedom in terms of

i) choice of the origin

ii) coordinate system (can be Cartesian, cylindrical, spherical etc.)

Simple Example of constrained motion

Example 1. A Block sliding without friction on an inclined plane

Coordinate system: X-axis is pointing along the surface of the plane, Y-axis normal to the plane.The  Block  is confined to slide  without  friction along the  plane,  therefore  the  constraint  is y = constant ne motion along Y-axis. The equations of motion are

Example 2. Atwoods’s Machine.

Two  masses  connected  by  a  mass  less  inextensible  string  of  fixed  length  passing  over  a frictionless pulley.Coordinate system: Since there is only vertical motion we use Y – coordinate in the down direction.

Equation of motion of  1 and  2 are given by

1  1 = − 1   +  ;  2  2 = − 2  +

Where T is the tension in the string. Since the length of the string is fixed

1 +   2 = −   . .  2 = −  −   1

So

Generalised cordinates:

We saw that for a system of N particles if we have k independent constraint equations, the number of degrees of freedom is 3N – k. The dynamical behavior of the system depends only on the coordinates corresponding to the degrees of freedom. Since these are fewer degrees of freedom then the position coordinates, required to specify the position of each particle in the system, can we eliminate the unnecessary coordinates? This will undoubtedly simplify the analysis of the motion particularly if we choose coordinates that take into account the constraints and are independent.

For holonomic constraints it is indeed possible to define a set of 3N-k independent coordinates called ‘Generalized Coordinates’  that specify the motion of the system subject to the given constraints and which are independent of each other. The Cartesian coordinates can then be expressed in terms of known function of generalized coordinates  i.e.

In terms of components

where . It may or may not be always possible to have an analytical expression for these functions. The generalized coordinates eliminate the constraints forces from the problem and they are independent. The generalized coordinates need not be Cartesian and can be chosen as suitable to the problem considered. For example, the motion of a particle moving on an ellipse

has only one degree of freedom which can be chosen to be the polar angle  and the particle coordinates X and Y are expressible in terms of the generalized coordinates  as

for which the constraint equation is automatically satisfied.

Just as the velocity components  are defined for poition coordinates, we can define generalized velocity

Velocities being defined as total time derivative of the said coordinate. For a function  the total time derivative is given by the chain rule:

The use of generalized coordinates in place of position coordinates of the system was to eliminate the nondynamical degrees of freedom from the system. In a similar manner, constraint forces can be eliminated leaving only the generalized forces.

Let  be the force acting on the ith particle of a N particle system.  The work done by the force for an arbitrary displacement   is

And the total work done on the system by all the forces is

Now RI ‘s are expressible in terms of generalized coordinates as

Then

Substituting in (1.17)

Define

as the Generalized force and the total work done on the system can now be written as

expressed only in terms of the generalized coordinates and forces.  The expression denotes the work done by all generalized forces which equal to the number of degrees of freedom of the system in the arbitrary displacement in generalized coordinates.  Let us illustrate this with the example of Atwood’s machine.

Choose the coordinates as shown  and  are the positions of masses  and  from a horizontal plane passing through the pulley. The y-axis points upwards since the length of the string is fixed and does not charge with time.

We can choose either  or  as generalized

Let us choose  as the generalized coordinates generalized velocity is given by

The Force  and acting on masses  and  are

Now

Problem 1: How many degrees of freedom does the double pendulum has? Write down the constraint equations.

Problem 2. Four masses 1, 2, 3 and 4 are hanging such that they can move only in the vertical direction. How many degrees of freedom does it have? Write down the constraint equations.

Summary :

• The constraints which can be expressed as algebraic or integrable differential equations are holonomic.
• If a halonomic constraint depends on time explicitly it is called Rhenomic, if there is no explicit time dependence, they are called Scleronomic.
• Non-holonomic constraints involve frictional forces or are expressed in the form of inequalities.
• Number of independent variables required for the description of a system is called the number of degrees of freedom.