Integration help, Kepler's problem Lagrangian dynamics

1. Jan 17, 2013

black_hole

1. The problem statement, all variables and given/known data

Carry out the integration ψ = ∫[M(dr/r2)] / √(2m(E-U(r)) - (M2/r2))

E = energy, U = potential, M = angular momentum

using the substitution: u = 1/r for U = -α/r

2. Relevant equations

3. The attempt at a solution

This is as far as I've gotten: -∫ (Mdu) / √(2m(E + αu) - (M2u2))
I have no idea how to take this integral by hand which seems to be what the question is implying. Wolfram gives me something crazy looking.

My book gives the answer as ψ = arccos( (M/r - mα/M) / √(2mE + m2α2/M2) ) ???
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Jan 17, 2013

TSny

Try "completing the square" of the expression inside the square root in the denominator. Factor out the coefficient of u2 from the square root beforehand.

Last edited: Jan 17, 2013