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Hetware
Posts:
148
Registered:
4/13/13


Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
Posted:
Apr 8, 2013 11:21 PM


On 4/8/2013 8:18 PM, Alfred Einstead wrote: > On Apr 7, 7:14 pm, Hetware <hatt...@speakyeasy.net> wrote: >> This is the geodesic equation under discussion: >> d^2(r)/dt^2 = r(dp/dt)^2 >> d^2(p)/dt^2 = (2/r)(dp/dt)(dr/dt). > > Where the problem comes from is not important, since all you're asking > about is how this is solved.
I don't fully agree with that view. The context of a problem often suggests a particular approach.
Clearly, what I posted as my effort to solve the problem suggested /an/ approach.
> Notice that it's independent of p and depends only on do/dt. So, > define v = dr/dt, f = dp/dt and write the system as > dv/dt = rf^2, df/dt = 2fv/r (along with dr/dt = v, dp/dt = f). > > The second equation can be rewritten as > 0 = 1/f df/dt + 2/r dr/dt = 1/(f r^2) d(f r^2)/dt > Therefore, f = K/r^2, for some constant K. > > The first equation then becomes, dv/dt = K^2/r^3. The standard trick > is to turn this into a conservation of energy integral: > v dv/dt = K^2/r^3 v = K^2/r^3 dr/dt, > from which it follows that > d/dt (v^2/2) = d/dt (K^2/2 1/r^2). > The solution is > v^2 = v_0^2  K^2/r^2 > for some constant v_0. > > You can take it on from here; solving for r as a function of t, and > then putting this into the equation for f (i.e. dp/dt) to get p as a > function of t.
I've typed all of the above into Mathematica for typesetting. I have a hard time following math in ASCII. I appreciate your help. It did get me further along than I was. Differential equations is not a strong suit of mine. Unfortunately, I can't spend any more time on it tonight.
> If you go back to the original problem, the following properties hold > true.
The exercise is titled "A sheet of paper in polar coordinates."


Date

Subject

Author

4/7/13


Guest

4/7/13


Lord Androcles, Zeroth Earl of Medway

4/8/13


Lord Androcles, Zeroth Earl of Medway

4/8/13


Hetware

4/8/13


Lord Androcles, Zeroth Earl of Medway

4/8/13


Hetware

4/9/13


Lord Androcles, Zeroth Earl of Medway

4/9/13


Hetware

4/8/13


Hetware

4/8/13


Lord Androcles, Zeroth Earl of Medway

4/9/13


Hetware

4/9/13


Lord Androcles, Zeroth Earl of Medway

4/9/13


Hetware

4/9/13


Lord Androcles, Zeroth Earl of Medway

4/9/13


Hetware

4/8/13


Dirk Van de moortel

4/8/13


Lord Androcles, Zeroth Earl of Medway

4/8/13


rotchm@gmail.com

4/9/13


Dirk Van de moortel

4/13/13


Dono

4/13/13


rotchm@gmail.com

4/13/13


Dono

4/13/13


Dono

4/13/13


Dono

4/13/13


rotchm@gmail.com

4/13/13


Dono

4/13/13


rotchm@gmail.com

4/13/13


Dono

4/13/13


Dono

4/13/13


rotchm@gmail.com

4/13/13


Dono

4/13/13


rotchm@gmail.com

4/13/13


Dono

4/9/13


Guest

4/9/13


Dirk Van de moortel

4/9/13


Lord Androcles, Zeroth Earl of Medway

4/8/13


Rock Brentwood

4/8/13


Hetware

4/8/13


Lord Androcles, Zeroth Earl of Medway


