Can you recommend any good books? I downloaded this free book on classical field theory written by some guy from the university of Utah. It wasn't bad until he got to the chapter on Hamilton mechanics. That's why I'm here.

Thanks!

@Geoffry Gifari The divergence does not change if the H contour lines are not closed, it will be zero as well. The difference is that open lines represent "non-bound states". If you imagine you have a potential well, the "bound states" will keep circling inside the well, while the others will escape the well and fly off to infinity. That will correspond to closed and open contour lines. Does that help?

@@gcarcassi oh i got the picture now. Area enclosed is something that's defined for bound states only

Hi Hory! The literature is vast... what type level/particular issue are you interested in?

I wasn't familiar with the Cauchy Riemann equations... but, at a first glance, I think there is a link: they both seem to preserve a symplectic form. In the case of Hamilton's equations, it's the symplectic form that corresponds to the areas/volumes in phase space. In the case of Cauchy Riemann equations it is the preservation of the complex structure. See en.wikipedia.org/wiki/Linear_complex_structure and en.wikipedia.org/wiki/Hamiltonian_system. In both articles you'll essentially find the same block diagonal matrix, the preservation of which is what switches the order and adds the minus sign. I am not sure, though, whether the similarity ends here or there is something more...

Thank you very much sir =)

Thanks and welcome to the channel! :-)

Noted. Though that would unfortunately open a gigantic can of worms... Conjugate momentum is technically non-physical: it is not gauge invariant, and anything that we measure must be gauge invariant. What we actually measure is kinetic momentum, which is a different thing. Conjugate momentum is basically "the other variable we use to keep track of states". The Hamiltonian has similar status. My take is that going through these issues would only confuse things...

he clearly says he does in part 2, ya have to finish the video,its at the end.

I think he says that in this video he will stick to geometric perspective only.

@@gcarcassi right thats what we want, we want to open the whole can of worms and know what each one tastes like, so math can be a USEFUL TOOL and not a secret code. If the viewer of the math doesnt have and definitions for the symbols than the symbols are arbitrary and i believe they are not arbitrary but are the constitution of the terms in the equations

@@andrewfetterolf7042 :-) Part of my research is indeed going back to the foundations of mathematics and trace all the mathematical definitions from the very beginning, so that we have a clear physical motivation for all of them, and so that we only have mathematical objects with physical motivations (see assumptionsofphysics.org/book/ ). It's a long journey... What is a real number? What is a manifold? What does a differentiable structure represent? All of these things need to be answered to do it properly. But it can be done and gives so much insight in both math and physics. The problem is having the time and resources to create other videos... :-/

Thanks!

I think the problem is the coma. The square brackets are used for all linear algebra elements, including vectors and matrices. Unfortunately, there is no way to fix it at this point. FYI, there is no consistent notation that spans all the different areas touched by this video series. There are plenty of other cases where one would object. Remaining in the realm of differential geometry, IMHO, the use of grad/curl (instead of the exterior derivative) would probably be a bigger offender...

Gabriele Carcassi Yes I agree on that, having a notation that fits everything is always a problem. Nevertheless, when it comes to Hamiltonian / symplectic math and physics the notation is already quite standard across books. Also, grad is standard in differential geometry, it is the Riemannian musicality of df. And one could define curl via Hodge duality.

Unfortunately the coordinate free notation used in differential geometry is good for the math, but not good at highlighting the physics. So that's a no go.

Glad they helped!!! As for the sign of the angle, it's 90 degrees clockwise. If you follow the right-hand rule then it's -90. See en.wikipedia.org/wiki/Right-hand_rule . If you follow the left-hand rule then it's +90. Makes sense?

Not exactly. The gradient rotated 90 degrees kind of looks like a curl in 2D, but it does not generalize correctly... If you imagine each d.o.f. represented as q + i p, then multiplying by i is like rotating 90 degrees... but I think that complicates things in the end.

@@gcarcassi Yes, thank you. Then it is a rotation matrix, or covector. I have never seen that notation before. Thanks for your brilliant idea and analysis.

Well, this is the first video of a series, and is titled "the math"... then you have other videos: "measurements", "thermodynamics", ...

@@gcarcassi Thanks. I wasn't complaining, just identifying that the focus on the maths without the physical context lost me. Nothing to hang my hat on.

Both x and q are used in the literature, with different frequency in different context. For example, in classical mathematical physics it usually is q, while in quantum mechanics and intro classical physics is usually x. The greek letter xi is also used in symplectic geometry for both q and p. The intended audience of the videos is intro level physics, so I went for what I believe is most common in that case.

Victor Ovidiu Slupic Because q refers to any generalized coordinate system that applies to keep track of the state of the object of study. The implication behind using x is that those coordinates are Cartesian, but this is not necessarily the case, so using x would be misleading. It is for the same reason r is used instead of x to denote position in multiple dimensions.