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gionole

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- TL;DR Summary
- Question about symmetry coordinate transformation

Let's say we got one particle in x direction only and we got some motion x(t) which we figured it out through Lagrangian.

In Noether's theorem for coordinate transformation symmetry, we start with the following:

x(t)' = x(t) +εf(t) (ε - some number) - I denoted new path with x(t)'

I'm focusing now on εf(t). In Noether's theorem, εf(t) can't really be any arbitrary function(note that for 2 particles, it must be the same, but I'm not asking that). The reason it can't be arbitrary is if it is kind of wiggling function, then adding it to x(t) will result in a x(t)' where every coordinate of the old path didn't move by the same distance to result in the new path.

Could you say/explain where I'm making wrong statements ? what would you feel like I'm missing ? The best way would be to follow the single particle example first.

In Noether's theorem for coordinate transformation symmetry, we start with the following:

x(t)' = x(t) +εf(t) (ε - some number) - I denoted new path with x(t)'

I'm focusing now on εf(t). In Noether's theorem, εf(t) can't really be any arbitrary function(note that for 2 particles, it must be the same, but I'm not asking that). The reason it can't be arbitrary is if it is kind of wiggling function, then adding it to x(t) will result in a x(t)' where every coordinate of the old path didn't move by the same distance to result in the new path.

**Q1**: Is the εf(t) really arbitrary and can be anything ?(I know it's small, but that doesn't mean it can't be such a function where middle of it is much bigger than left one, which could cause the new path trajectory definitely not symmetrically moved from old one). Namely, I thought what we mean by coordinate transformation is moving every bit of point on the current trajectory by the same amount. I'm asking as I haven't seen such restrictions in textbooks and if it can be anything, then trajectory has not moved symmetrically -**while in Lagrangian, it can be any wiggling function.**Could you say/explain where I'm making wrong statements ? what would you feel like I'm missing ? The best way would be to follow the single particle example first.

**Q2:**If every point on the trajectory didn't move by the same distance because of εf(t), then we wouldn't have δS = 0 right ? How do I prove that**ONLY**the symmetric movement of the whole trajectory would cause δS = 0 ? If you assume that δS = 0, then you can get the momentum conservation, but I'm more interested in the proof of δS = 0 in terms of εf(t) and Noether's theorem.
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