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Hetware
Posts:
148
Registered:
4/13/13


Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
Posted:
Apr 9, 2013 12:23 AM


On 4/8/2013 10:03 PM, Lord Androcles, Zeroth Earl of Medway wrote: > "Hetware" wrote in message > news:CeSdneS1Wf0B_7MnZ2dnUVZ_vGdnZ2d@megapath.net... > > On 4/7/2013 11:34 PM, Lord Androcles, Zeroth Earl of Medway wrote: >> "Hetware" wrote in message >> news:AuGdnVu7TbVslv_MnZ2dnUVZ_g6dnZ2d@megapath.net... >> >> 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). >> >> r is radius in polar coordinates, p is the angle, and t is a path >> parameter. >> >> The authors ask me to "[S]olve the geodesic equation for r(t) and p(t), >> and show that the solution is a uniformly parametrized straight >> line(x===r cos(p) = at+p for some a and b; y===r sin(p) = jt+k for some >> j and k). >> >> I tried the following: >> >> (d^2(p)/dt^2)/(dp/dt) = (2/r)(dr/dt) >> >> f=dp/dt >> >> (df/dt)/f = (2/r)(dr/dt) >> >> 1/2 ln(f) + k = ln(r) >> >> a(f^(1/2)) = r >> >> a(dp/dt)^(1/2) = r >> >> And substitute for r in: >> >> d^2(r)/dt^2 = r(dp/dt)^2 >> >> to get >> >> d^2(r)/dt^2 = a(dp/dt)^(3/2) >> >> But there I'm stuck. >> >> How should the problem be handled? >> ============================================= >> What you have is a second order differential equation. >> Unlike the solution to the general polynomial equation, >> ax +bx^2 + cx^3 + ... + kx^n = 0, where you seek a value >> for x given values for a,b,c etc., the solution for a >> differential equation is a FUNCTION. >> In other words you cannot obtain a numerical or algebraic >> value (you don't have enough information and that is not >> the idea anyway) but you can find functions r(t) and p(t) . >> The authors have already told you the solution is a straight >> line, which is of course a function. >> http://search.snap.do/?q=solving+differential+equations&category=Web >> HTH, because we don't do homework for you. > > That was the kick in the head that I needed. > > I was getting close. I remembered what Wheeler told me: "Anytime you > are struggling to understand or explain something, draw a picture." > > http://www.speakeasy.org/~hattons/chapter08.nb > > I worked the problem in pencil today while away from my computer. Just > use p=arctan(t/r_0) where r_0 is a radial vector perpendicular to the > "curve". r=(r_0^2+t^2)=r_0/cos(p). The rest is just a matter of > plugging these into the geodesic equation. > ========================================== > Good. Note that what you are doing is strictly NOT anything > to do with relativity or gravitation or even physics, it is just > math. > Pity Einstein didn't work the examples or he'd have made fewer > blunders like this one: > http://www.androcles01.pwp.blueyonder.co.uk/DominoEffect.GIF > >  This message is brought to you from the keyboard of > Lord Androcles, Zeroth Earl of Medway. > When the fools chicken farmer Wilson and Van de faggot present an > argument I cannot laugh at I'll retire from usenet. > > > > > >
I don't have time to follow up on this, but this is from a guy who doesn't spend his time trying to prove how smart he is. He just shows you what he things in the clearest manner available to him. The mark of a true genius.
http://people.oregonstate.edu/~drayt/Courses/MTH437/2007/hw/geodesic.pdf


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


