Proving Orbit Derivation in External Gravitational Field

[bemigh]

Hey everyone,
I need to prove something explictly, problem is, i don't know where to start.
In orbits, we can show that the motion of 2 bodies interacting with each other only by central forces can be reduced to a an equivalent one-body problem. This is by using a Lagrangian, and by introducing an 'equivalent' mass.
The question is to show that this is possible even if these bodies are in an external uniform gravitational field.
I don't know where to start. I can't see how a gravitational field will affect the Lagrangian, because a graviatational field will just affect the potential energy term. This isn't really solving it explicity though...
any ideas where to start?
Cheers

 

[topsquark]

Hey everyone,
I need to prove something explictly, problem is, i don't know where to start.
In orbits, we can show that the motion of 2 bodies interacting with each other only by central forces can be reduced to a an equivalent one-body problem. This is by using a Lagrangian, and by introducing an 'equivalent' mass.
The question is to show that this is possible even if these bodies are in an external uniform gravitational field.
I don't know where to start. I can't see how a gravitational field will affect the Lagrangian, because a graviatational field will just affect the potential energy term. This isn't really solving it explicity though...
any ideas where to start?
Cheers

If I'm following your argument correctly, you've made the correct connection. The process by which the coordinates are modified to produce an equivalent one body problem do not depend on a specific form of a potential, so the process will still go through. All you should have to do to prove the process is possible is to input the change of variables into the Lagrangian explicitly.

-Dan