Symplectic Geometry in 2D – Points, Lines, Circles
researchgate.netI'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
I've read 'Geometriekalküle': https://www.amazon.de/Geometriekalk%C3%BCle-Springer-Lehrbuc... from the same author ... thanks for your Wikipedia link.
This is also SL_2-invariant geometry, in that most of these results admit a ternary operator generalization to 3D.
Symplectic geometry feels different once area and volume diverge.
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 as well. Unfortunately there is already a lack of books about the topics
> 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.
yes ... V.I.Arnold - important symplectic geometry author tells us: "Hamilton mechanics cannot be understood without differential forms ...". Thanks for your link.
... and don't forget that gyrovector spaces are the best mathematical framework for relativistic mechanics - much better than Lorentz transformations:
https://en.wikipedia.org/wiki/Gyrovector_space
Just mentioning ... ;-)
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 as the composition of a “gyrovector” and a rotation about the north pole. But the details here are tricky and unintuitive and a lot of the symmetries of spherical rotation are not reflected in the representation.
The deficits of this system are a bit less obvious in a context (hyperbolic space) that students are less familiar with. But if you represent the hyperbolic plane as a paraboloid in pseudo-Euclidean space (akin to representing a sphere as a surface embedded in Euclidean space), a tool similar to unit quaternions is also a more convenient and natural representation for hyperbolic rotation.
* * *
Geometric algebra as a language makes it easy and natural to understand and describe the meaning and relationships between various rotation representations, and is much better for this purpose than e.g. matrices.
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.
Interesting definition of 'Euclidian geometry'. I am used to call 'something' Euclidean, if the parallel postulate of Euclid holds (that is, there exists exactly one parallel line to a given line through a point (which does not lie on the given line)).
To be fair, the term they are using is "Euclidean structure," which, strictly speaking, is more algebraic than (purely) geometric. The notion of the "angle" (and from there that of parallelism) logically follows from it.
I guess he isnusing it in the metric sense (Euclidean metric).
> In symplectic geometry an area is the fundamental quantity, whereas Euclidean geometry measures lengths and angles.
> yield the results in a coordinate, matrix and trigonometry-free manner
Some related ideas, for simplifying and generalising geometry:
Euclidean geometry is characterised by inner-product/symmetric-bilinear-form, shown in Section 2.1:
𝐚𝐛 = a₁×b₁ + a₂×b₂
𝐚𝐛 = Σaₙbₙ
|𝐚| = √(𝐚𝐚)
Remarkably, we can do a lot of geometry without using length at all, hence not requiring square roots, and therefore generalising our results to many more fields. Instead, we just work with quantities like 𝐚𝐚 directly, which can be interpreted as the area of a square with side-length |𝐚| (AKA a "quadrance"). An obvious example is Pythagoras' theorem, which relates the quadrances of a right-triangle's sides.
This use of area is probably connected to symplectic geometry, but I haven't looked into that yet.
The approach described above is called Rational Trigonometry; which also avoids transcendental functions like cos/sin, by replacing angles with "spreads" (equivalent to the sin^2 of an angle), which range from 0 = parallel to 1 = perpendicular.
Looking again at the inner-product 𝐚𝐛, there's another degree of freedom lurking in there if we interpret it as matrix multiplication 𝐚𝐛ᵀ (the rules of matrix multiplication require us to transpose the 1×n row-vector 𝐛 into the n×1 column-vector 𝐛ᵀ).
By default, this matrix formulation doesn't alter the inner product: it's still Σaₙbₙ. However, it gives us the flexibility to introduce an n×n matrix 𝐌 in-between the vectors: 𝐚𝐌𝐛ᵀ
If 𝐌 is the identity matrix [[1, 0], [0, 1]] (denoted 𝐈 in the article), then we again keep the original behaviour. In this sense, Euclidean geometry is characterised by 𝐈 (encoding its symmetric bilinear form).
If we use other n×n matrices we get different geometries. In particular, the matrix [[1, 0], [0, -1]] gives us the "red" inner-product a₁×b₁ - a₂×b₂; and [[0, 1], [1, 0]] gives us the "green" inner-product a₁×b₂ + a₂×b₁. These are closely related to each other (one is a rotation of the other; both are 2D analogues of special-relativity), and to the "blue" Euclidean geometry. This colour-coding come from Chromogeometry, which studies their relations.
These are explained more in An Introduction to Rational Trigonometry and Chromogeometry (which I just submitted at https://news.ycombinator.com/item?id=30418194 )