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I am asked to find the conserved quantities for the following Lagrangian for a three-particle system in three dimensions

$$L = \left[\sum_{i=1}^{3} \frac{1}{2} m_i \left(|\dot{\bf{r}}|^2 - \omega^2 x_i^2 \right) \right] - U_{12} (y_1 - y_2) \cos [\lambda (m_1 - m_2) t] - U_{13} (y_1 - y_3) - V(z_1 - z_2),$$

where $U$, $V$, $\omega$ and $\lambda$ are constants. How do I interpret this potential? Is $z_3$ the only cyclic coordinate? Additionally, while I understand cosine is an even function, my gut is telling me that time symmetry only exists for equal masses $m_1$ and $m_2$ – is this correct? Lastly, I was given a hint that there is a conserved quasi-angular momentum when the masses are all equal, but I can’t spot it – how do I show this? Thank you!

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closed as off-topic by Kyle Kanos, sammy gerbil, Emilio Pisanty, heather, ZeroTheHero Jul 8 '18 at 2:48

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Since this looks like a "homework-like" problem, but it does have some conceptual confusions that need to be cleared up, I'll answer in broad strokes.

Is $z_3$ the only cyclic coordinates?

It appears so, at least in the form you've written it. However, just because none of the remaining coordinates are cyclic doesn't mean that there can't be some other combination of the remaining coordinates that's cyclic. Take a closer look at the other two pairs of $y$ and $z$ coordinates. Or, as a simpler example, take a look at the Lagrangian $\mathcal{L} = \frac{1}{2}(m_1 \dot{x}_1^2 + m_2 \dot{x}_2^2) + U(x_1 - x_2)$ and see if you can figure out a change of coordinates for it which results in one coordinate being cyclic.

Additionally, while I understand cosine is an even function, my gut is telling me that time symmetry only exists for equal masses $m_1$ and $m_2$—is this correct?

As far as conserved quantities go, there is a conserved quantity associated with time translation symmetry. This requires that if you change $t \to t + \alpha$, you have the same Lagrangian for all values of $\alpha$. Your comment about "an even function" makes me think that you're thinking about time-reversal symmetry, $t \to - t$; but since this is a discrete symmetry, there's no conserved quantity associated with it.

Lastly, I was given a hint that there is a conserved quasi-angular momentum when the masses are all equal, but I can’t spot it – how do I show this?

For "real" angular momentum, you need a rotational symmetry between combinations of $x$, $y$, and $z$. This means that $x$, $y$, and $z$ must appear identically in the the Lagrangian (roughly speaking). In this case, you can't rotate between $x$, $y$, and $z$; instead, you need to "rotate" between three other coordinates that appear identically in the Lagrangian.

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  • $\begingroup$ Hi there, I’ll have to ask my questions in parts since there’s a lot to take in. Firstly, how do you notice that combinations of coordinates are cyclic? Does this require a complete redefinition with my generalised coordinates? Aren’t the q_i linearly independent? $\endgroup$ – user107224 Jun 23 '18 at 20:24
  • $\begingroup$ @user107224: Honestly, noticing that various combinations are cyclic is something that comes from experience. Your Lagrangian has a property that pops up frequently in physics that it's good to be able to recognize. See my edit to the first part. $\endgroup$ – Michael Seifert Jun 23 '18 at 20:39

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