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Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
Posted:
Apr 8, 2013 9:27 AM


Hetware <hattons@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). > > 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).
Normally we'd write dotted variables, but with quotes it's easier. So write r' = dr/dt r'' = d^2(r)/dt^2 p = dp/dt p'' = d^2(p)/dt^2 then you have r'' = r (p')^2 [1] p'' = 2/r p' r' [2]
Deriving [1] gives r''' = r' (p')^2 + 2 r p' p'' which with [2] gives r''' = 3 r' (p')^2 which again with [1] gives r r''' + 3 r'' r' = 0
So if you find some r(t) that has r''(t) = r'''(t) = 0 for all t, you're done. Easy. Take r(t) = A t + B [3] then r'(t) = A r''(t) = 0 r''"(t) = 0 so indeed r r''' + 3 r'' r' = 0
Now, from [1] and [3] you get 0 = ( A t + B ) (p')^2 so p' = 0 so p(t) = C [4]
So you get r(t) = A t + B p(t) = C Check it out with [1] and [2]. Trivial.
So x(t) = (A t + B ) cos(C) y(t) = (A t + B ) sin(C)
There's probably another solution, but seems to be the one they're after.
Dirk Vdm


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


