Classical Mechanics Constraints and Degrees of freedom

1. Classical MechanicsConstraints and Degrees of freedom Dr.P.Suriakala Assistant Professor Department of Physics

2. What is Constraint • Restriction to the freedom of the body or a system of particles • Sometimes motion of a particle or system of particles is restricted by one or more conditions. The limitations on the motion of the system are called constraints. The number of coordinates needed to specify the dynamical system becomes smaller when constraints are present in the system. 

3. Types of constraint Holonomic - Constraints expressed in the form of equation f(X1,Y1, Z1,.........Xn,Yn,Zn,t)=0 Non holonomic - Constraints not expressed in this fashion Scleronomic - independent of time Rheonomic - Constraints containing time explicitly

4. Holonomic constraint The motion of a rigid body, the distance between any two particles of the body ramains fixed and do not change with time the it satisfies the holonomic set of constraints |ri − rj | − cij = 0 When a body moves on an inclined plan, it is constrained to move on the inclined plane surface. A bead on a circular wire

5. Holonomic constraint • Simple pendulum is a system composed of a weight and a string. The string is attached at the top end to a pivot and at the bottom end to a weight. Being inextensible, the string’s length is a constant. Therefore, this system is holonomic; it obeys the holonomic constraint

6. Consider a fluid inside a spherical vessel of radius. If the distance between the centre of the vessel and a fluid particle is then we have r ≤ a The motion of the particle placed on the surface of the sphere of radius a will be r-a ≥ 0 Non holonomic constraint

8. Degrees of freedom • The minimum number of independent variables coordinates required to specify the position of a dynamical system, consisting one or more particles called degrees of freedom • For example the motion of the particle moving freely in space, can be described by a set of coordinates (x,y,z) and hence the number of degrees of freedom is 3

9. Degrees of freedom • A system of two particles moving freely in space requires two sets of three coordinates i.e six coordinates to specify its position. The number of degrees of freedom is 6 • If a system consists of N particles moving freely in space, we need 3N independent coordinates to describe its position. Hence the The number of degrees of freedom is 3N

10. Determine the number of degrees of freedom • A particle moving on the circumference of the circle • Five particle moving freely in a plane • Two particle connected by rigid rod moving freely in a plane • A rigid body moving freely in space • A rigid body moving in space with one point fixed • The bob of simple pendulum oscillating in a plane • Dumbbell moving in space

11. Answer • 1 • 10 • 3 • 6 • 3 • 1 • 5

12. Thank you