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    Orbital mechanics solved by using substitution integrals

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    Show the integral below can be integrated to give Equation 1

    θ(r) = ∫ {[(L/r^2) dr] / [ 2μ (E+(k/r) - (L^2/ 2μr^2))]^.5} + Constant

    Using :

    U= L/r and r = minimum at θ = 0

    Equation 1----------cos (θ) =[ (L^2/(μkr)) - 1]/ [{1+((2EL^2)/(μk^2))}^.5]

    See attached file for full problem description.

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    https://brainmass.com/physics/classical-mechanics/orbital-mechanics-solved-substitution-integrals-103325

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    Solution Preview

    Referring back to Eqs. (7), (8) and (9) of the solution of the previous problem (with appropriate changes in notation), you see that:

    d theta/dt = L/(m r^2) (1)

    dr/dt = Plus/minus [(2/m) * (E - V(r)) ]^(1/2) (2)

    where V(r) = - k/r + L^2/(2m r^2) (3)

    If you divide (1) by (2) you get:

    d theta/dr = L/(m r^2) * [(2/m) * (E - V(r)) ]^(-1/2) =

    L/r^2 * [ 2m (E - V(r)) ]^(-1/2) =

    L/r^2 * [ 2m ...

    Solution Summary

    A detailed solution is given regarding orbital mechanics.

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