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MathematicalPhysicist
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I have another question, if I am given a metric and I want to find the constants of motion of that system, then how do I do it?
Thanks!
Thanks!
Then, for the example I gave I get that the constants of motion are:In your expression, t is certainly not cyclic, since the integrand depends on it. You do not get ##\partial L/\partial\dot t = const.##
The general way of finding the constants of motion is to look for the Killing vectors and Killing tensors of your manifold, ie, its symmetries. However, there is no guarantee that this is easy to do and no general recipe.
Also, you cannot guarantee to find all constants of motion by looking at what coordinates are cyclic. Your coordinates may be chosen in a way that obscures this.
Also, you have yet to complete the original task in this thread so I am not sure why you put another question in here (that should probably go in its own thread).
Done, posts broken out of the previous thread into this new thread.should probably go in its own thread
How to find ##U^\mu = dx^\mu/d\tau## and ##t(\tau)##?
Then, for the example I gave I get that the constants of motion are:
$$p_x=2a^2(t)\dot{x}+2f(t)\dot{t}$$
$$p_y=2a^2(t)\dot{y}+2h(t)\dot{t}$$
$$p_z= 2a^2(t)\dot{z}+2g(t)\dot{t}$$
How to find ##U^\mu = dx^\mu/d\tau## and ##t(\tau)##?
##L## is also a constant (it has value 1).
So you get an equation involving ##\dot{t}##. You can solve that for ##t## (presumably) and use the equation for ##\dot{x}## to get a differential equation for ##x##. Similarly, you can solve for ##y## and ##z##.
Actually it's ##-1## with the signature convention the OP is using.
This won't give a unique solution since there will be constants of integration involved.
Also, this method assumes that the object in question is traveling on a geodesic. It won't work for objects on non-geodesic worldlines.
I thought he was asking for how to find the geodesics.