Which coordinate is cyclic in this case

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Discussion Overview

The discussion revolves around identifying cyclic coordinates in a two-particle system described by a Lagrangian that includes a potential energy dependent on the difference in their coordinates. Participants explore the possibility of transforming the system into a form where all coordinates are cyclic.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant describes a two-particle system with a potential energy function V = V(x1 - x2) and notes that while total momentum is conserved, no cyclic coordinates are apparent in the Lagrangian.
  • Another participant suggests expressing the Lagrangian in terms of new coordinates y1 = x1 + x2 and y2 = x1 - x2 to potentially reveal cyclic coordinates.
  • A subsequent post confirms the transformation and presents a new Lagrangian but questions whether it is possible to achieve a situation with all coordinates cyclic.
  • One participant proposes that in an inertial frame, the difference x1 - x2 could be constant, implying that potential energy might be omitted as a constant, but emphasizes that potential energy is a function of generalized coordinates.
  • Another participant reformulates the Lagrangian in the new coordinates and mentions the Hamilton-Jacobi equation and action-angle variables as relevant concepts for finding cyclic coordinates.
  • A later reply states that while energy is conserved, it does not relate to cyclic coordinates and suggests that achieving all cyclic coordinates may only be possible using the Hamiltonian formulation with more flexibility in conjugate variables.
  • One participant posits that a second cyclic variable could be obtained if a parameter describing the time evolution of the system is introduced, particularly if the potential is quadratic, but expresses doubt about the feasibility of such a transformation with a general potential V.

Areas of Agreement / Disagreement

Participants express differing views on the possibility of transforming the system to have all cyclic coordinates, with some suggesting it may be feasible under specific conditions while others remain skeptical, indicating an unresolved discussion.

Contextual Notes

The discussion involves assumptions about the nature of the potential energy function and the conditions under which cyclic coordinates can be identified, which remain unspecified and unresolved.

aaaa202
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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?
 
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Hint: Try to express your lagrangian in terms of
y1=x1+x2
y2=x1-x2
 
ahh nice.
So you get:

1/4my12 + 1/4my22 - V(y2) = 0

Is it possible to transform to a situation with all coordinates cyclic?
 
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.
 
First of all the Lagrangian in the new coordinates
[tex]L=\frac{\mu}{2}(\dot{y}_1^2+\dot{y}_2^2)-V(y_2)[/tex]
with [itex]\mu=m/2[/itex].

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!
 
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?
 
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.
 

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