Poisson Brackets and Canonical Transformations

Classical Mechanics and Relativity: Lecture 13
Theoretical physicist Dr Andrew Mitchell presents an undergraduate lecture course on Classical Mechanics and Relativity at University College Dublin. This is a complete, self-contained, and standalone course in which everything is derived from scratch.
In this lecture I discuss the time-dependence of dynamical variables in classical mechanics, and how they can be obtained from so-called Poisson Brackets. We will see how Hamilton's equations of motion can themselves be cast in terms of Poisson Brackets, and we will study their general properties. Canonical transformations constitute a more general class of coordinate transformation that involve all canonical coordinates (positions and momenta). A canonical transformation is one that preserves the fundamental Poisson Bracket relations. Poisson Brackets are canonical invariants, as are Hamilton's equation's.
Full lecture course playlist: • Classical Mechanics an...
Course textbooks:
"Classical Mechanics" by Goldstein, Safko, and Poole
"Classical Mechanics" by Morin
"Relativity" by Rindler

[@Sherlock14-d6x]

I cannot thank you enough for the clarity you have provided me through this video!

[@yulupeng3952]

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

[@MoguinYT]

Great video! Very well explained thanks Dr Mitchell!

[@DBg429]

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!

[@drmitchellsphysicschannel2955]

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!

[@riccardofasano1040]

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

[@drmitchellsphysicschannel2955]

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.

[@mariaporras1864]

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?

[@joelcurtis562]

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?

[@drmitchellsphysicschannel2955]

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.

[@juksaykaisey8772]

Very nive video! i understand everythong ;-; tysm.
Do you about where a can I find more examples with systems of 2 or 3 dof?

[@AbdAllah_54564]

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.