I'm trying to calculate the proper time of a massive particle circulating Schwarzschild black hole, using EL equation of the following Lagrangian:
At first I get from the Euler-Lagrange equation of $r$:
Then, using energy conservation:
Now, I thought to integrate over $\tau$ (the proper time), as LHS does not depend on it, while at RHS it turns to integration over $\phi $ form $0$ to $2\pi$ .
Eventually, I end up with:
for the proper time for one circulation.
This result doesn't seem to make a lot of sense.
Where do I have wrong?