What is Symplectic Geometry? (2016) [pdf]
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Re: What is Symplectic Geometry? (2016) [pdf]
#2For those into physics, I wholeheartedly recommend Marsden and Ratiu's book, "Introduction to Mechanics and Symmetry", which deals mainly with the different formulations of physics applied to symplectic and associated geometries.
Re: What is Symplectic Geometry? (2016) [pdf]
#3This was quite a treat !
Re: What is Symplectic Geometry? (2016) [pdf]
#4This really doesn't make it clear what symplectic geometry... is, or why I should care about it. I have eventually figured out an answer that was satisfactory to me, after much frustration: it is math on a manifold that has a concept of paired-off coordinates, like (x,v) in mechanics. Typically this is interesting because it is an alternate characterization of the mathematics of a space where the relevant quantities are a variable and its derivative.
(In classic mechanics, particularly, there is an unusual symmetry to position and velocity, such that the laws of mechanics look roughly a rotation x -> v, v -> -x, which is why this works so well.)
Re: What is Symplectic Geometry? (2016) [pdf]
#5Importantly, the Hamiltonian formulation of classical mechanics has symplectic form, with the conjugate variables (position, momentum) making up the dimensions.
Re: What is Symplectic Geometry? (2016) [pdf]
#6This really doesn't make it clear what symplectic geometry... is, or why I should care about it. I have eventually figured out an answer that was satisfactory to me, after much frustration: it is math on a manifold that has a concept of paired-off coordinates, like (x,v) in mechanics. Typically this is interesting because it is an alternate characterization of the mathematics of a space where the relevant quantities…
Correct, and it also appears in Electromagnetism and Quantum Physics (in several ways), and in Lie theory that is useful for rotations and differential applications.
Re: What is Symplectic Geometry? (2016) [pdf]
#7This really doesn't make it clear what symplectic geometry... is, or why I should care about it. I have eventually figured out an answer that was satisfactory to me, after much frustration: it is math on a manifold that has a concept of paired-off coordinates, like (x,v) in mechanics. Typically this is interesting because it is an alternate characterization of the mathematics of a space where the relevant quantities…
> (In classic mechanics, particularly, there is an unusual symmetry to position and velocity, such that the laws of mechanics look roughly a rotation x -> v, v -> -x, which is why this works so well.)
In fact, every symplectic manifold locally looks like this (Darboux's theorem).
Re: What is Symplectic Geometry? (2016) [pdf]
#8This really doesn't make it clear what symplectic geometry... is, or why I should care about it. I have eventually figured out an answer that was satisfactory to me, after much frustration: it is math on a manifold that has a concept of paired-off coordinates, like (x,v) in mechanics. Typically this is interesting because it is an alternate characterization of the mathematics of a space where the relevant quantities…
> (In classic mechanics, particularly, there is an unusual symmetry to position and velocity, such that the laws of mechanics look roughly a rotation x -> v, v -> -x, which is why this works so well.) In fact, every symplectic manifold locally looks like this (Darboux's theorem).
Yeah, but I'm interested in understanding things the other direction: if there is not some major class of problems that are best described by symplectic manifolds, then why care about Darboux's theorem at all? If there are, why isn't that front-and-center? This article conspicuously avoids motivating symplectic geometry _at all_, which is so frustrating. It mentions connections to subjects, but it doesn't mention why symplectic geometry is _necessary_, rather than sufficient, for these connections.
Re: What is Symplectic Geometry? (2016) [pdf]
#9This really doesn't make it clear what symplectic geometry... is, or why I should care about it. I have eventually figured out an answer that was satisfactory to me, after much frustration: it is math on a manifold that has a concept of paired-off coordinates, like (x,v) in mechanics. Typically this is interesting because it is an alternate characterization of the mathematics of a space where the relevant quantities…
What does "paired-off coordinates, like (x,v) in mechanics" mean? What is x and v here?
Thanks!
Re: What is Symplectic Geometry? (2016) [pdf]
#10This really doesn't make it clear what symplectic geometry... is, or why I should care about it. I have eventually figured out an answer that was satisfactory to me, after much frustration: it is math on a manifold that has a concept of paired-off coordinates, like (x,v) in mechanics. Typically this is interesting because it is an alternate characterization of the mathematics of a space where the relevant quantities…
Symplectic geometry is a collection of facts having to do with symplectic manifolds. Just like euclidean geometry is a collection of facts having to do euclidean manifolds.
That's the way in which terms like "____ geometry" are defined and understood.