Cyclic coordinates and conservative theorem present ation by haseeb

Cyclic coordinates and conservative theorem present ation by haseeb

1. CYCLIC COORDINATES AND CONSERVATIVE THEOREM CHAPTER NO 8 AND TOPIC 8.2

2. Cyclic Coordinates State : The coordinates that does not appear explicity in the lagrangian of a system are said to be cyclic or ignorable coordinates

3. PROVE As lagrangian L is the function of If q j are cyclic coordinates Then

4. Since generalized momentum So, So, momentum p j is a constant of motion.

5. Conservative Theorem State : The generalized conjugate momentum to the cyclic coordinates is conserved . or A coordinates that is cyclic will also be absent in hamiltonian .

6. PROVE We know that’s hamiltonian is the function of Taking derivative w.r.t “ t”

7. From Hamilton equation of motion By integration H is constant

8. The modified Hamiltonian is Since Where potential energy “V’ does not depend on velocity (depend on postion only)

9. Then Putting the value of p i in modified Hamiltonian Then

10. Then This shows that Hamiltonian is numerically equal to the total energy of the system.