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Misner, Thorne and Wheeler, Exercise 8.5 (c)
Posted:
Apr 7, 2013 8:14 PM


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?


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


