Cyclic coordinates are those that do not appear explicitly in the Lagrangian of a system. The generalized momentum conjugate to cyclic coordinates is a constant of motion. The conservative theorem states that the generalized momentum conjugate to cyclic coordinates is conserved, and cyclic coordinates will be absent from the Hamiltonian. If a coordinate is cyclic, the Hamiltonian of a system will be numerically equal to the total energy and remain constant over time.

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 qj are cyclic coordinates
Then

4. Since generalized momentum
So,
So, momentum pj 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 pi in modified Hamiltonian
Then

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