So that's why my professor called Hamilton mechanics "very elegant" , I mean, this is absolutely beautiful. For me the beauty lies within the smooth transition between pure mathematics and theoretical physics that the Hamiltonian formalism provides. It's so obvious and easy to see, yet so sophisticated when digging in deeper. Lie-Theory, differential geometry, topology you name it... Thank you for this video!
Thanks for the comment! Yes, I completely agree. And the power of the formalism is proved by how it generalizes to other areas of physics, even/especially Quantum Mechanics!
Fantastic, eloquent and simply put, it really made me interested from start to end, this lecture was just beautiful. Masha Allah!!! May Allah reward you and give hidayah Ameen.
Thank you so much for the good work. At some point Gauge invariance is addressed: I am comfortable with the visualization of the Gauge choice as a term added to the Lagrangian which gives you an equivalent Lagrangian, but I don't see how this translates in the language of Hamiltonian mechanics. Any further explaination or video about it? Thanks again
A gauge transformation is like a kind of symmetry transformation in that it does not affect any measurable or observable physical quantity. There is a natural redundancy in the Lagrangian description, and the Lagrangian is not unique: there are various different Lagrangians you can write down that give you the same equation of motion and physics. It's exactly the same for the Hamiltonian. This must be true because the Hamiltonian can always be obtained from the Lagrangian by a Legendre transformation.
So we could say that a canonical transformation is by definition one that transforms one set of canonical coordinates into another set of canonical coordinates?
Yes that's exactly right. And a given set of coordinates is canonical if they satisfy the fundamental poisson bracket relations, which is easy to check. So therefore we can also think of a canonical transformation as one that preserves these poisson brackets.
I cannot thank you enough for the clarity you have provided me through this video!
totally life saver! I stared at my prof's note and book and googling for almost 3 hours. Can't get my mind straight until I saw your lecture
So that's why my professor called Hamilton mechanics "very elegant" , I mean, this is absolutely beautiful. For me the beauty lies within the smooth transition between pure mathematics and theoretical physics that the Hamiltonian formalism provides. It's so obvious and easy to see, yet so sophisticated when digging in deeper. Lie-Theory, differential geometry, topology you name it... Thank you for this video!
Thanks for the comment! Yes, I completely agree. And the power of the formalism is proved by how it generalizes to other areas of physics, even/especially Quantum Mechanics!
Great video! Very well explained thanks Dr Mitchell!
Fantastic, eloquent and simply put, it really made me interested from start to end, this lecture was just beautiful. Masha Allah!!! May Allah reward you and give hidayah Ameen.
Thank you so much for the good work. At some point Gauge invariance is addressed: I am comfortable with the visualization of the Gauge choice as a term added to the Lagrangian which gives you an equivalent Lagrangian, but I don't see how this translates in the language of Hamiltonian mechanics. Any further explaination or video about it? Thanks again
A gauge transformation is like a kind of symmetry transformation in that it does not affect any measurable or observable physical quantity. There is a natural redundancy in the Lagrangian description, and the Lagrangian is not unique: there are various different Lagrangians you can write down that give you the same equation of motion and physics. It's exactly the same for the Hamiltonian. This must be true because the Hamiltonian can always be obtained from the Lagrangian by a Legendre transformation.
Hello there. I was wondering if you have an example of showing if a transformation is canonical usin PB but in a system with 2 dof?
Very nive video! i understand everythong ;-; tysm.
Do you about where a can I find more examples with systems of 2 or 3 dof?
So we could say that a canonical transformation is by definition one that transforms one set of canonical coordinates into another set of canonical coordinates?
Yes that's exactly right. And a given set of coordinates is canonical if they satisfy the fundamental poisson bracket relations, which is easy to check. So therefore we can also think of a canonical transformation as one that preserves these poisson brackets.