Double Pendulum (One pendulum hanging from the other)

[voko]

I think you made a reasonable assumption in #29. Yet the result is clearly incompatible with the requirements of the problem. This may be because your x1 and x2 are not exactly what the problem requires. It may be that both x1 and x2 should be measured from the single fixed vertical line (the equilibrium line). Then I think all coupling in kinetic energy will vanish (by an argument similar to #29), so you get a diagonal matrix just like the problem requires.

 

[CAF123]

I think you made a reasonable assumption in #29. Yet the result is clearly incompatible with the requirements of the problem. This may be because your x1 and x2 are not exactly what the problem requires. It may be that both x1 and x2 should be measured from the single fixed vertical line (the equilibrium line). Then I think all coupling in kinetic energy will vanish (by an argument similar to #29), so you get a diagonal matrix just like the problem requires.

This is exactly the case. Defining x1 and x2 both from the equilibrium line, I get the required form with the potential no longer decoupled as was the case earlier. Is there a physical reasoning why simply defining my x2 differently yields a decoupled potential!

It seems to be that this question can only be done via the 'right' method. I would have thought given this the question actually state where x1 and x2 are defined from because I did spend a lot of time yesterday trying to figure out why I couldn't proceed.

 

[voko]

This is exactly the case. Defining x1 and x2 both from the equilibrium line, I get the required form with the potential no longer decoupled as was the case earlier. Is there a physical reasoning why simply defining my x2 differently yields a decoupled potential!

I am not exactly sure what you mean here. What "earlier" potential are you referring to?

It seems to be that this question can only be done via the 'right' method. I would have thought given this the question actually state where x1 and x2 are defined from because I did spend a lot of time yesterday trying to figure out why I couldn't proceed.

Frankly, I think this problem should have been done using angles rather than displacements.

 

[CAF123]

I am not exactly sure what you mean here. What "earlier" potential are you referring to?

The potential written in post #12. The potential redefining x2 from the vertical line will have the first term the same but the second will be $\sqrt{a^2 - (x_2 -x_1)^2}$ and hence coupled. The potential in #12 is not at all coupled. So depending on my definitions, I decouple the potential term. Does this seem to make physical sense?

Frankly, I think this problem should have been done using angles rather than displacements.

This is exactly how we did it in lectures and we are told to do it in terms of x1 and x2 for this question.

 

[voko]

When the coordinates of the two masses are defined in such a way that that the motion of one mass can be described by one set of coordinates, and another mass by another, disjoint, set of coordinates, kinetic energy becomes decoupled. But it has to be somewhere, so it appears in potential energy.

 

[CAF123]

This makes sense, the energy has to go somewhere so it appears in the potential term. But in both defintions of x1 and x2, the kinetic energy is decoupled (or at least decoupled after neglection of small elements) while the potential is only decoupled in one definition. Is there a reason for this?

 

[voko]

I think you are mistaken. If x2 is defined as it was originally, the horizontal displacement from the upper mass, then kinetic energy is not decoupled, you have a product of x1 and x2's derivatives, and the kinetic energy matrix has non-zero non-diagonal elements.

 

[CAF123]

I think you are mistaken. If x2 is defined as it was originally, the horizontal displacement from the upper mass, then kinetic energy is not decoupled, you have a product of x1 and x2's derivatives, and the kinetic energy matrix has non-zero non-diagonal elements.

Indeed I was mistaken. Thanks voko, you have been a big help!