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Integration help, Kepler's problem Lagrangian dynamics

  • Thread starter black_hole
  • Start date
  • #1
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Homework Statement



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

Homework Equations





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) ) ???

Homework Statement





Homework Equations





The Attempt at a Solution


Homework Statement





Homework Equations





The Attempt at a Solution


Homework Statement





Homework Equations





The Attempt at a Solution


Homework Statement





Homework Equations





The Attempt at a Solution


Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
TSny
Homework Helper
Gold Member
12,232
2,743
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:

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