Which coordinate is cyclic in this case

[aaaa202]

Consider a simple two particle system with two point masses of mass m at x1 and x2 with a potential energy relative to each other which depends on the difference in their coordinates V = V(x1-x2)

The lagrangian is:

L = ½m(x₁')² + ½m(x₂')² + V(x₁-x₂)

Obviously their total momentum is conserved d/dt(mx₁' + mx₂') = 0, which can be verified by plugging into the lagrangian. But there is no cyclic coordinates in the lagrangian. Is it possible to put it in a form where this hidden cyclic coordinate is shown?

 

[mfb]

Hint: Try to express your lagrangian in terms of
y₁=x₁+x₂
y₂=x₁-x₂

 

[aaaa202]

ahh nice.
So you get:

1/4my1² + 1/4my2² - V(y2) = 0

Is it possible to transform to a situation with all coordinates cyclic?

 

[raopeng]

It seems to be the motion in an inertial frame of reference. Well in that case x1 - x2 must be constant I think, so potential energy can be omitted as a constant. What is clear is that potential energy is a function of generalised coordinates, so it has to be Cartesian coordinates as well.

 

[vanhees71]

First of all the Lagrangian in the new coordinates
$$L=\frac{\mu}{2}(\dot{y}_1^2+\dot{y}_2^2)-V(y_2)$$
with $\mu=m/2$.

To answer the question, whether you can find a set of coordinates, which all are cyclic, you should read about the Hamilton-Jacobi partial differential equation and action-angle variables.

In your case there is for sure another conserved quantity! Think which that might be!

 

[aaaa202]

well that's the energy but that has nothing to do with cyclic coordinates.
Also I did read Hamilton-Jacobi theory but that takes its basis in the hamiltonian formulation. So I guess it's not really possible to transform to a frame with all coordinates cyclic UNLESS you use the hamiltonian formulation with more freedom to vary your conjugate variables?

 

[mfb]

To get a second cyclic variable, you would need some parameter which describes the time-evolution of your (y2-)system with fixed energy. If V is quadratic (or at least gives oscillations in some way), this would be the phase of the oscillation, for example.
I doubt that you can do this transformation in an explicit way with a general V.