How do you prove conservation of angular momentum via Hamiltonian Mechanics?
I managed to figure out linear momentum with some velocity. I can't imagine angular momentum is much different, but I'm not quite sure how to show a rotational variance.
Hamiltonian Mechanics
Tyler Miller
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Alexander Fisher
The proof should be virtually identical with the slight caveat that when working with angular momentum you have a cross product but just recognizing that the time derivative of position and the tangential momentum are parallel so that term is zero, what you're left with is the positions crossed with the time derivative of the tangential velocity. you should be able to finish it from here with the same arguments as for the linear momentum proof.
Samuel Jenkins
From my work last week, I can't seem to remember why dH/dq, or rather the sum of those, is 0. From there you can obviously relate to the canonical equations and get the result, but why is dH/dq when q --> q + dq equal to 0?
Pic related.
Isaiah Mitchell
Bump
Jace Collins
Don't study physics if you don't know basic analysis.
Ryder Morris
Remember your hamiltonian should be invariant under the infinitesimal translation q--> q +dq so the variation should be zero (that's the del H part). When taking the limit as before this turns the sum of variations of the hamiltionian with respect to the position coordinates to a sum of derivatives of the hamiltonian with respect to the position coordinates which again must sum up to zero. (Apologies if this is poorly worded, this type of thing is easier to explain irl)
Isaac Moore
Ohhhh! They must sum up to 0 because dH=0. So Σ(dH/dq)=0 because the total change is 0, invariant. God I feel so stupid.
And it's only invariant under that infinitesimal translation because we are making that assumption for this argument? Or is there a deeper reason?
Carson Jackson
>And it's only invariant under that infinitesimal translation because we are making that assumption for this argument? Or is there a deeper reason?
Yeah, basically Noether's theorem states that for ever invariant property there is conservation law, this is especially true for canonically conjugate variables like position and momentum, or time and energy (i.e. in-variance under infinitesimal translation of position leads to momentum conservation and in-variance under infinitesimal translation of time leads to energy conservation)
Aiden Perez
Thank you. I've never had a formal introduction to differentials and what they mean (dH??) so this stuff has been quite hard. Any short resources you recommend just to get the gist of them and how to interpret and work with them?
Jaxon Roberts
im taking hamiltonian mechanics this upcoming semester. AS A SOPHOMORE (fuck yeah im a badass don't fuk wit me)