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Integration/Proving help! And check my work please!

  1. Oct 4, 2004 #1

    I am given the following equation.

    This is the total energy of a prticle moving in a central conservative field. m = mass, E = energy, L = angular momentum. The force the particle experiences is F = -Hmu^-3, where H is some constant, m is the mass of the particle, and u the distance. V(u), the potential, is just the negative integral of the force, and it is


    How can I show that the energy, E, if E < 0, then

    [tex]\alpha(t-t_{0})=\int_{R_{0}}^{R(\Theta)}\frac{du}{u\sqrt{a^2-u^2}} [/tex]

    , for real numbers alpha and a.

    And similarly, for E > 0, how can I show it's

    [tex]\beta(t-t_{0})=\int_{R_{0}}^{R(\Theta)}\frac{du}{u\sqrt{u^2+b}} [/tex]

    , for some real numbers beta and b?

    I really need help on this! :confused:

    I did show, for the E = 0 case, how it should be done. Please take the time check my work for this part.

    Since E = 0, the total energy equation simplifies to:


    then, plugging in V(u),


    [tex] Let s = m^2L^{-2}H[/tex]




    Since s is only a bunch of constants, we can factor it out.


    [tex] \ln {R_{\Theta}/R_{0}} = \sqrt{s-1)[/tex]

    then solve for [tex] R (\Theta) [/tex]
    [tex] R (\Theta) = R_{0}e^{(\Theta-\Theta_{0})\sqrt{s-1}}[/tex]
    Last edited: Oct 4, 2004
  2. jcsd
  3. Oct 4, 2004 #2


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    I don't know whether you did your integrations correctly but the problems as you stated them only involve factoring numbers out from the radicals and rearranging terms.
  4. Oct 4, 2004 #3
    Yes...I was asked to get the equation...carry out the integral and arrange the terms so that it resembles the ones above with the alpha and beta.
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