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Homework Statement:

Use Noether's theorem to find the constants of motion of the transformation:
## q_1' = e^{\epsilon} q_2 ##
## q_2' = e^{\epsilon} q_1 ##
Relevant Equations:

Lagrangian:
## L = \dot q_1 \dot q_2  \omega q_1 q_2 ##
Transformation:
## q_1' = e^{\epsilon} q_2 ##
## q_2' = e^{\epsilon} q_1 ##
Noether's theorem:
## K = \frac {\partial L} {\partial \dot q_i} \delta q_i + (L  \frac {\partial L} {\partial \dot q_i} \dot q_1) \delta t ##
Hi!
I am given the lagrangian:
## L = \dot q_1 \dot q_2  \omega q_1 q_2 ##
(Which corresponds to a 2D harmonic oscillator) And I am given two transformations and I am asked to say if there is a constant of motion associated to each transformation and to find it (if that's the case).
I am not sure which is the literature my course is following, but I understand I have to:
So, first transformation is:
## q_1' = e^{\epsilon} q_1 ' ##
## q_2' = e^{\epsilon} q_2 ' ##
Which is easy to verify as a symmetry of the action and, as ##\epsilon## is really small, I can do ##e^{\epsilon} \approx 1 + \epsilon ## to identify ## \delta q_1 = \epsilon## and ## \delta q_2 = \epsilon## to plug them into Noether's theoren and calculate the constant of motion.
Case 2
Second transformation is where I am stuck:
## q_1' = e^{\epsilon} q_2 ##
## q_2' = e^{\epsilon} q_1 ##
If I plug that transformation into ##S'[q_1', q_2']##, it is easy to check it is a symmetry of the action which suggests there exists a constant of motion for that transformation, but the derivation of Noether's theorem we did in my course (and the one I have always found on the internet) is for transformations of the shape:
## q_1' = q_1 + \delta q_1 ##
Which my professor labeled as ##finite## in the sense that if I make the variation ##\delta q_1## really small, ## q_1'## always maps back to ##q_1##. But notice that my transformation, after doing ##e^{\epsilon} = 1 + \epsilon ## is:
## q_1' = q_2 + \epsilon q_2 = q_2 + \delta q_1 ##
Where ## \delta q_1 = \epsilon q_2 ##. My transformation is mapping ## q_1'## to ##q_2## and viceversa, which sugggests I can't use Noether's theorem as is. At least, not as I wrote it above.
How should I proceed here?
I was thinking of doing Noether's theorem derivation again but for transformations of the kind ## q_i' = q_j + \delta q_i ## to get the constant of motion ## K ## associated to that kind of transformations, but I am not sure if this would be a good approach or if there is something easier I could do or where I can read about these type of transformations.
I am given the lagrangian:
## L = \dot q_1 \dot q_2  \omega q_1 q_2 ##
(Which corresponds to a 2D harmonic oscillator) And I am given two transformations and I am asked to say if there is a constant of motion associated to each transformation and to find it (if that's the case).
I am not sure which is the literature my course is following, but I understand I have to:
 Check that the transformation is a symmetry of the action. That is, if:
## S[q_1, q_2] = \int L(q_1, q_2, t) dt ##
## S'[q_1', q_2'] = \int L'(q_1'(t'), q_2'(t'), t') dt' ##
Then when ## S = S' ##, it means there is a constant of motion associated to that transformation.  Use Noether's theorem to calculate the constant of motion ##K##:
## K = \frac {\partial L} {\partial \dot q_i} \delta q_i + (L  \frac {\partial L} {\partial \dot q_i} \dot q_1) \delta t ##
So, first transformation is:
## q_1' = e^{\epsilon} q_1 ' ##
## q_2' = e^{\epsilon} q_2 ' ##
Which is easy to verify as a symmetry of the action and, as ##\epsilon## is really small, I can do ##e^{\epsilon} \approx 1 + \epsilon ## to identify ## \delta q_1 = \epsilon## and ## \delta q_2 = \epsilon## to plug them into Noether's theoren and calculate the constant of motion.
Case 2
Second transformation is where I am stuck:
## q_1' = e^{\epsilon} q_2 ##
## q_2' = e^{\epsilon} q_1 ##
If I plug that transformation into ##S'[q_1', q_2']##, it is easy to check it is a symmetry of the action which suggests there exists a constant of motion for that transformation, but the derivation of Noether's theorem we did in my course (and the one I have always found on the internet) is for transformations of the shape:
## q_1' = q_1 + \delta q_1 ##
Which my professor labeled as ##finite## in the sense that if I make the variation ##\delta q_1## really small, ## q_1'## always maps back to ##q_1##. But notice that my transformation, after doing ##e^{\epsilon} = 1 + \epsilon ## is:
## q_1' = q_2 + \epsilon q_2 = q_2 + \delta q_1 ##
Where ## \delta q_1 = \epsilon q_2 ##. My transformation is mapping ## q_1'## to ##q_2## and viceversa, which sugggests I can't use Noether's theorem as is. At least, not as I wrote it above.
How should I proceed here?
I was thinking of doing Noether's theorem derivation again but for transformations of the kind ## q_i' = q_j + \delta q_i ## to get the constant of motion ## K ## associated to that kind of transformations, but I am not sure if this would be a good approach or if there is something easier I could do or where I can read about these type of transformations.
Last edited: