Classical Mechanics, Lecture 7: Noether's Theorem. Two Body Problem.

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Lecture 7 of my Classical Mechanics course at McGill University, Winter 2010. Noether's Theorem. Two Body Problem.

The course webpage, including links to other lectures and problem sets, is available at

The written notes for this lecture are available at

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The focus on symmetries of the EOM is confusing. Noether's theorem applies to symmetries of the lagrangian (more specifically of the action) and not the EOM. For example the EOM for a particle falling under gravity is ẍ=-g. This is clearly invariant under a continuous spacial translation x→x+s. Thus, the EOM is invarant under this continuous spacial translation but momentum is definitely not conserved. The lack of conservation of momentum is seen by the fact that the lagrangian (and hence the action) is not invariant under x→x+s. Specifically the potential energy goes trom mgx to mgx+mgs.

brad

these lectures are extremely useful for me thankyou

mattotonton

You've helped clear up something Honerkamp didn't specify in his Theoretical Physics. Thank you!!

kristenchou

I really wished I had a cool teacher like you. I sincerely wish I studied in your uni and in your class. Thornton marion hasn't been enough for me and neither is my professor.

admiralhyperspace

15:46 better to use partial derivative wrt s here, I think.
You need to take this derivative at s=0, as ∂L/∂Q needs to be taken at s=0 so that Q ---> q and Euler - Lagrange can be applied.

17:50 You don't explain why d/dt (∂Q/ ∂ s) = ∂ / ∂s (dQ/dt). This is not trivial.
You need to write dQ/dt = ∂Q/∂s . ds/dt + ∂Q/∂t = ∂Q/∂t as ds/dt = 0 since s is meant to be time independent.
Similarly, d/dt (∂Q/ ∂ s) = ∂/∂t (∂Q/ ∂ s) and the result follows using interchange of partial derivatives.
This proof, although only a few lines long, is clearly subtle. Just because it may be relatively easy to memorize the small number of manipulations in writing it down in an exam does not make it "kid's stuff".
However, despite my annoyance at having to fill so many omissions, overall I am grateful for your videos and accompanying notes.

eamon_concannon

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