2. Syllabus Note : Attendance is important because the theory and questions will be explained in the class. II ntroduction. LL agrange’s Equation. SS mall Oscillations. HH amilton’s equations of motion. CC anonical transformations and Generating functions. HH amilton-Jacobi Equation.

3. References 1. Dynamics by A S Ramsey 2. Motion of a particle Rigid body and a Particle S L Loni 3. Advanced Mechanics – Schaum Series 4.Any book related with Mechanics

4. System of Particles With respect to a system of 3-dimensional coordinates, we need 3n number of independent variables to describe the position of a system consisting of n number of particles. If there are k number of constraints (restrictions ), then we need only 3n-k number of independent variables. We define Degrees of freedom of the system as 3n-k. By looking at the system we can define these independent variables and they are called generalized coordinates and they are denoted by q 1, q 2,…, q n.

5. Constraints 3 constraints can be classified as 1) Holonomic constraints. 2) Non-Holonomic constraints. This is a non-Holonomic constraints. 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 E.g. (1) A Holonomic constraint can be expressed as a function of the space coordinates and time. A rigid body. Length between any two points is constant. So a Holonomic constraint. E.g. (2)

6. So Degrees of freedom of a system means the number of independent variables required to describe the system. Simple pendulum Since system has one particle, need 3 independent variables. Constraints : P article moves in a plane. L ength of the string is constant. Therefore need 3-2=1 independent variable. i.e. it has one degree of freedom. After observing the system, one can define the generalized coordinates. Define the generalized coordinate as the angle between downward vertical and the string.

7. Lagrange’s Equations Result I Suppose is the position vector of the i th particle and q j is the j th generalized coordinate.. Proof : Since Eq. 1

8. Therefore Hence.

9. Result II Proof : Since

10. Hence the result is proved. Result 1 Result 2 We have

11. Kinetic energy of the system is

12. Also Result 1By

13. Hence Result 3 Result 4

14. Suppose the system of particles changed slightly without changing the time t then Eq. 1 becomes Work done Here is the external force on the i th particle.

15. Here Also gives us is called generalized force associated with generalized coordinate.

16. Since are all independent above equation yields Eq. 2 Applying Newton’s 2 nd Law to the i th particle we have

17. ByResult 2

18. When and are applied in above equation, it reads as Result 3 Result 4 This is called Lagrange’s equation of motion.

19. Special case Suppose all the forces are conservative. i.e. there exists a scalar function called potential function. By the definition of the potential function Definition Lagrangian or Lagrange’s Function L of the system is defined as the difference of Kinetic energy and the Potential energy and denoted by L. i.e. L = T - V

20. Eq. 2 Here Hence by we get

21. Since This is the Lagrange’s equation for a conservative system

22. A particle of mass m moves in a conservative force field. Find the Lagrangian and equations of motions in cylindrical polar coordinates. E.g. ( Question No. 1 of Exercises ) P-particle X Y Z Q O Let OQ= r, P Then

23. Therefore Hence equations of motions are

24. E.g. ( Question No. 2 of Exercises ) Solution : The equations become Suppose that the particle, in the previous example moves in the Oxy plane and V=V(r) only. If at time t=0 the particle on the Ox axis of distance a and the velocity of the particle is v o in the direction of the positive Oy axis. Find the velocity of the particle.

25. So the equation becomes Further

26. Hence

27. Now we have velocity as a function of the distance from the origin. When the velocity potential function is given we can get the velocity using this equation..