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Symplectic Geometry in 2D – Points, Lines, Circles

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Re: Symplectic Geometry in 2D – Points, Lines, Circles

#11

Post by Koshkin »

blablabla123 wrote: post #8
Koshkin wrote: post #2

A very nice article. On the other hand, I have a feeling that symplectic geometry (in 3D) is being pushed by its proponents onto the unsuspecting public as the best framework for understanding Hamiltonian mechanics, similar to how geometric algebra people claim that theirs is the best mathematical framework for physics. Personally, I find both largely unintuitive and, at deeper levels, too complicated to be useful.

Not sure, vector calculus isn't very intuitive either if you start with it, with Pauli/gamma matrices it's even worse. Having studied Physics myself, I haven't encountered one lecture where they were able to give a reasonable geometric explanation. (Symplectic Geometry and GA provide it) IMHO if the the same amount of effort was used to force vector calculus into people's heads, it should be doable with these tools a…

> vector calculus isn't very intuitive either

That's, by the way, why we have the calculus of differential forms which, unlike vectors with all their flavors (free; polar; axial/pseudo), have a clear geometric meaning, and with which many statements about fields acquire an especially simple form. There are many excellent guides; for the motivation, see, for example, https://www.jpier.org/PIER/pier148/09.14063009.pdf

Re: Symplectic Geometry in 2D – Points, Lines, Circles

#12

Post by ogogmad »

I've read a bit of this textbook on projective geometry: https://www.amazon.co.uk/Perspectives-Projective-Geometry-Th...

To some extent, the book justifies Arthur Cayley (the inventor of matrix algebra)'s adage that "Projective geometry is all geometry". Towards the end of the book, models of non-Euclidean geometries are built within CP^2. I've written up an overview in this Wikipedia sandbox: https://en.wikipedia.org/wiki/User:Svennik/sandbox

Re: Symplectic Geometry in 2D – Points, Lines, Circles

#13

Post by jacobolus »

Qem wrote: post #10
jacobolus wrote: post #7

Earlier quoted context omitted.

Gyrovectors are a generally poor representation for rotations compared to quaternions (or the like). In a spherical context, a “spherical gyrovector” can represent any rotation of the sphere whose axis is on the equator, with the representation being the point where the north pole gets sent. This gets you 2 out of 3 degrees of freedom for spherical rotations. Then you can represent an arbitrary rotation of the sphere…

Can you recommend good geometric algebra books starting at a elementary level, say, not assuming starting knowledge much beyond high school mathematics? I did some preliminary research, but I had the impression the intended audience for most books in this area is people that already master the conventional approach but are open to see the subject under a new light, so a lot of previous knowledge is assumed.

What I have repeatedly found with GA is that I can solve some problem I have using some other brute-forceish tools with a few pages of tricky error-prone scratch-work that balloons out to a complicated mess before simplifying back down at the end, and then afterward think about it a bit and come up with 2–6 lines of simple GA identities showing the same thing in a much higher-level coordinate-free way, and with most of the steps geometrically interpretable, rather than just opaque calculation. But coming up with the simple version at the beginning is hard.

The tricky part about it is that there are a lot of useful identities that can be written down, and properly learning a decent number of them and figuring out which ones to apply in which situation takes probably years practice, ideally with some guidance/support from someone who knows more than you. (I do not feel like I have mastered the subject.) The same thing happens using whatever other formalism, with the difference that many identities that are pretty short to write down in GA are much more complicated to write down, so people don’t even try to use them.

I’m not sure if there’s really a good beginner source, but I haven’t ever really sat down and tried to go comprehensively through the exercises in any books pitched at a relatively elementary level. You could try Alan MacDonald’s book Linear and Geometric Algebra which is designed as an intro undergraduate textbook. If you want to also learn some mechanics, you could try Hestenes’s book New Foundations for Classical Mechanics.

Re: Symplectic Geometry in 2D – Points, Lines, Circles

#14

Post by [deleted] »

Koshkin wrote: post #11
blablabla123 wrote: post #8

Earlier quoted context omitted.

Not sure, vector calculus isn't very intuitive either if you start with it, with Pauli/gamma matrices it's even worse. Having studied Physics myself, I haven't encountered one lecture where they were able to give a reasonable geometric explanation. (Symplectic Geometry and GA provide it) IMHO if the the same amount of effort was used to force vector calculus into people's heads, it should be doable with these tools a…

> vector calculus isn't very intuitive either That's, by the way, why we have the calculus of differential forms which, unlike vectors with all their flavors (free; polar; axial/pseudo), have a clear geometric meaning, and with which many statements about fields acquire an especially simple form. There are many excellent guides; for the motivation, see, for example, https://www.jpier.org/PIER/pier148/09.14063009.pdf

[deleted]

Re: Symplectic Geometry in 2D – Points, Lines, Circles

#16

Post by fango »

ogogmad wrote: post #12

I've read a bit of this textbook on projective geometry: https://www.amazon.co.uk/Perspectives-Projective-Geometry-Th... To some extent, the book justifies Arthur Cayley (the inventor of matrix algebra)'s adage that "Projective geometry is all geometry" . Towards the end of the book, models of non-Euclidean geometries are built within CP^2. I've written up an overview in this Wikipedia sandbox: https://en.wikipedia.o…

I've read 'Geometriekalküle': https://www.amazon.de/Geometriekalk%C3%BCle-Springer-Lehrbuc... from the same author ... thanks for your Wikipedia link.

Re: Symplectic Geometry in 2D – Points, Lines, Circles

#17

Post by fango »

Koshkin wrote: post #11
blablabla123 wrote: post #8

Earlier quoted context omitted.

Not sure, vector calculus isn't very intuitive either if you start with it, with Pauli/gamma matrices it's even worse. Having studied Physics myself, I haven't encountered one lecture where they were able to give a reasonable geometric explanation. (Symplectic Geometry and GA provide it) IMHO if the the same amount of effort was used to force vector calculus into people's heads, it should be doable with these tools a…

> vector calculus isn't very intuitive either That's, by the way, why we have the calculus of differential forms which, unlike vectors with all their flavors (free; polar; axial/pseudo), have a clear geometric meaning, and with which many statements about fields acquire an especially simple form. There are many excellent guides; for the motivation, see, for example, https://www.jpier.org/PIER/pier148/09.14063009.pdf

yes ... V.I.Arnold - important symplectic geometry author tells us: "Hamilton mechanics cannot be understood without differential forms ...". Thanks for your link.

Re: Symplectic Geometry in 2D – Points, Lines, Circles

#18

Post by onedognight »

Koshkin wrote: post #11
blablabla123 wrote: post #8

Earlier quoted context omitted.

Not sure, vector calculus isn't very intuitive either if you start with it, with Pauli/gamma matrices it's even worse. Having studied Physics myself, I haven't encountered one lecture where they were able to give a reasonable geometric explanation. (Symplectic Geometry and GA provide it) IMHO if the the same amount of effort was used to force vector calculus into people's heads, it should be doable with these tools a…

> vector calculus isn't very intuitive either That's, by the way, why we have the calculus of differential forms which, unlike vectors with all their flavors (free; polar; axial/pseudo), have a clear geometric meaning, and with which many statements about fields acquire an especially simple form. There are many excellent guides; for the motivation, see, for example, https://www.jpier.org/PIER/pier148/09.14063009.pdf

Differential forms are a sub-algebra of the geometric algebra, so you don’t give up any of the beautiful things you mentioned.
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