Symplectic geometry

the idea is to generalise phase space mechanics to abstract (not necessarily eucledian spaces)

we want a method to turn hamiltonian function H to a vector field V, then dynamics is the flow across integral curves of this field

requirements:

1. depend only on dH (global shift doesn't matter)
   
2. linear dependence on dH
   

he claims that tensor field, a section of Hom(T*M, TM), or Hom(TM, T*M) = (TM -> T*M) = T*M tensor T*M does that

NOTE: vector bundle – vector space, depending on parameter (point)
NOTE: tensor field by definition is some section on tensor bundle. mmm.
NOTE: vector fields on manifold – a section of tangent bundle. Okay, sort of makes sense. although; you could have said that it's a mapping F: (x: X) -> T(x)
NOTE: huh, so f: M -> R, df: {m: M} -> T(M) -> R. TODO wonder if it's interesting that number of arguments is increasing?
in general: f: M -> N, df: TM -> TN, meaning that df: {exists m: M} (T(m) -> T(f(m)))
TODO shit, I need agda here?…

jesus, they just don't have nice notation

tensor field is F: (x: X) -> T(x) ; T(x) is the space of all tensors at x. and that's it!

note from wikipedia: if f: M -> N, then df: TM -> TN

ok, so if H: M -> R; then dH: TM -> R = T*M
NOTE: when we think of dH, we consider it as a section, sort of with implicit {m: M} argument.

hmm, what is the tangent space of point on R. still R right? Yes, because it's basically space of 'velocities'
so dH is a covector field, ok https://ru.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D1%84%D0%BE%D1%80%D0%BC%D0%B0#%D0%9F%D1%80%D0%B8%D0%BC%D0%B5%D1%80%D1%8B

so we want to turn dH: T(m) -> R into V: (m: M) -> T(m)

to do that, for every point m, we need a way to map dH_m into T(m). dH_m: (T(m) -> R)

U* x V = Hom(U, V) = U -> V

process : (T(m) -> R) -> T(m) =

good point on first page: a more obvious set of equations is gradient flow:
dp_i/dt = -df/dp_i
dq_i/dt = -df/dq_i

it's a flow along a vector field ∇_f, which comes from: taking -df (1-form); then using inner product on R²ⁿ to dualise and get a vector field from 1-form.
that is:

f -> ∇_f: <∇_f, x> = -df. Ok, makes sense. We can just substitute vector fields in forms to get forms of lower rank.

Hamilton's equations are similar, but instead the form is symplectic, not an inner product.

Sometimes X_H is called symplectic gradient.

Flow along the gradient of f changes f as fast as possible
Flow along the symplectic gradient of f keeps f constant

since dH(X_H) = -w(X_H, X_H) = 0

NOTE: I guess that's natural, we want to keep energy constant along the phase space movement.

TODO blah blah something about hamiltonian vector fields and poisson brackets

NOTE! Right, so contangent bundle is actually just an example (!) of a symplectic manifold, with some canonical structure. Another example is Kahler manifold

M = T* N
Canonical one-form theta an point (a, b) ∈ T*n by theta_(a,b)(v) = b(Proj_a v)
Then the symplectic form w = d theta

TODO interesting that we throw away most of v's information. I guess that has to do with degeneracy??

hmm, https://en.wikipedia.org/wiki/Tautological_one-form
apparently this theta is called canonical one-form

ok, so his stuff is pretty similar.
positions q in conf space M
momentum p: in cotangent bundle T_q* M
phase space is the cotangent bundle T* M

note that velocity lies in tangent bundle T M, but momentum is defined as dL/dq', so it's in a different space

TODO so most of time, they are kind of same things… but not always. I guess I need a better sense of what momentum actually is.
I guess they are same if Lagrangian depends on v'²/2. What would be some interesting and physical examples of Lagrangians where it's not the case?

ok, he say same thing that Hamilton's equation is analogue to gradient flow for H on the manifold T*M, but w.r.t. the symplectic form.

NOTE gradient flow is a curve, such that: x'(t) = - \Nabla F(x(t)).
Okay, kinda makes sense. Time evolution in the direction of steepest descent.

TODO something about definition via observables

maybe visualise a hamiltonian on n-torus?

https://mathoverflow.net/a/16462/29889

https://mathoverflow.net/a/16537/29889

Following [2].

K = 1/2 Sum m_a (v^a)²
F_a = -d_a V(r)

TODO err, they define cartesian coordinates via generalised coordinates; then generalised velocities; and then rewrite K as 1/2 Sum q'ⁱ g_ik q^k'.

shit, I don't really follow this :(

lagrangian is L: T Q -> R, so defined on tangent bundle.

Next, define the generalised momenta as p_i = dL/dq'ⁱ.
Ans, f_i = dL/dqⁱ is generalised force.

In generalised coordinates: L(q, q') = 1/2 Sum q'ⁱ g_ij(q, m) q'^j - V(q)

Components of generalised momentum: p_i = dL/dq'ⁱ = Sum_j g_ij q'^j.

ok, that's sort of interesting
so, if Q is Riemannian manifold (got metric), then there is a diffeomorphism T Q -> T^* Q from tangent space to cotangent space.

metric tensor is 2-form; which means that when it acts on a vector field, we get 1-form (so momentum is 1-form)

I don't understand why they started using kinetic and potential energy staright away.

Ok, good point on page 23:

• in Lagrangian formalism, dynamics takes place on T (T Q)
  
• in Hamiltonian formalise, dynamics takes place on T (T^* Q)
  

on page 24:

x'(t) = X_Hx(t) = J dH (x); where J is the symplectic matrix; dH is gradient of hamiltonian function. Hamiltonian vector field.

pull-backs:

consider O: M -> T* M – 1-form on phase space – ok, as a member of T* M.
consider a: Q -> T* Q – 1-form on conf space.

right, so a is a linear map from Q to M; and O is a 1-form on M. We can pull back O to Q to get the 1-form a* O. | err so what??

canonical poincare 1-form Theta = \Sum_i p_i dqⁱ, which satisfies a^* Theta = a for all a in ….

so they are actually what's called covectors??

Suppose \(f\) is observable. It can only depend on position, momentum and possibly time:

\(f(p_i, q^i)\): \(\dv{f}{t} = \sum_i \pdv{f}{q^i} \dot{q}^i + \pdv{f}{p_i} {\dot p}_i = \sum_i \pdv{f}{q^i} \pdv{H}{p_i} - \pdv{f}{p_i} \pdv{H}{q_i} := \{f, H\}\)

\(f(p_i, q^i, t)\) – if time dependent , then \(\dv{f}{t} = \{f, H\} + \pdv{f}{t}\) – makes sense! kinda like flow derivative from Segal

any 2n-dimensional symplectic manifold looks like R²ⁿ, w₀

https://github.com/BartoszMilewski/gravity-sim
haskell or rust?… not sure

try L = 1/2x'² + x' - 1/2 x²

TODO what if we make linear dependency on position? That's pretty much coordinate transformation right?
soo, if it's a linear dependency on velocity… it's kind of time coordinate transformation!!

  historically, we empirically noticed/observed that principle of least action works (Lagrangian formulation), and there is an alternative and completely equivalent formulation (Hamiltonian).
When we study symplectic geometry, we ask: what are the minimum requirements to *define* familiar basic physical concepts (energy/time), and so that we still have the usual properties (e.g. Hamiltonian flow).

As a bonus, we drop the requirement for configuration space, and considering its cotangent bundle -- it can be *any* symplectic space.

https://mathoverflow.net/questions/19932/what-is-a-symplectic-form-intuitively/19935#19935

soo, all we can do is correlate pairs of coordinate between each other? otherwise we can't tell which are position, which are momentum?

Suppose that (M, ω) is a symplectic manifold. Since the symplectic form ω is nondegenerate, it sets up a fiberwise-linear isomorphism

https://twitter.com/johncarlosbaez/status/1065727111120310272

I thought about it longer and realized what was going on.
You get equations like Hamilton's whenever a system *extremizes something subject to constraints*.   A moving particle minimizes action; a box of gas maximizes entropy.
Read how it works:

Legendre transformation, named after Adrien-Marie Legendre, is an involutive transformation on the real-valued convex functions

https://math.stackexchange.com/questions/2447269/what-moment-map-is-as-a-physical-concept-in-sympletic-geometry

https://physics.stackexchange.com/questions/203653/intuition-about-momentum-maps

https://mathoverflow.net/questions/16460/how-to-see-the-phase-space-of-a-physical-system-as-the-cotangent-bundle/16462#16462

https://en.wikipedia.org/wiki/Legendre_transformation#Further_properties

For a strictly convex function, the Legendre transformation can be interpreted as a mapping between the graph of the function and the family of tangents of the graph.

https://en.wikipedia.org/wiki/Symplectic_integrator

https://physics.stackexchange.com/questions/33897/is-there-an-analogue-of-configuration-space-in-quantum-mechanics

http://www.cplusplus.com/forum/general/103744/

https://www.reddit.com/r/explainlikeimfive/comments/47lwch/eli5_what_is_symplectic_geometry/d0duju7/

It's geometry done on a curved surface that carries enough structure to measure area. In the same way that in Calc III you could do double integral over f dA to measure a sort of 'weighted area', on a symplectic manifold you can (in a suitable sense) take a double integral over f d(omega).

https://www.quora.com/What-is-an-intuitive-way-of-explaining-the-Poisson-bracket
{fg, h} = f {g, h} + g {f, h}
looks like leibnitz rule!
poisson bracket is derivation of the first arg w.r.t. to second
symplectic manifold: equipped with a symplectic form, that maps any function to a vector on manifold
so basially {f, g} is applying the form to g, and differentiating f along the integral curves of that vector field
for hamiltonian,
df/dt = {f, H} – makes sense, motion along integral curve is motion in time

The nice thing about the Hamiltonian formulation is that you can use lots of different coordinates. They will all work, as long as their Poisson brackets satisfy the equations above.

If you've studied quantum mechanics, this should look familiar as the algebra of commutators, and Poisson brackets are a nice way to see the connection between classical and quantum mechanics.

algebra with poisson bracket (lie bracket) is lie algebra

Thus, the time evolution of a function f on a symplectic manifold can be given as a one-parameter family of symplectomorphisms (i.e., canonical transformations, area-preserving diffeomorphisms), with the time t being the parameter

phase space: R³ x R³ – it's a base space for manifold
Hamiltonaian is defined on manifold, not on its tangent bundle!

hmm, but lagrangian is defined on the tangent bundle of configuration space?? https://math.stackexchange.com/questions/1210047/why-is-the-lagrangian-a-function-on-the-tangent-bundle confusing…

To summarize so far: a path in the tangent space "corresponds" (in the sense demonstrated in the example of h and H) to a path in the base space only if, in physicist notation, the function t↦q˙(t) turns out to be the time-derivative of the function t↦q(t). THAT is what that cryptic thing in your text is trying to say.

but this time, really really do exercises!

https://learn-anything.xyz

https://en.wikipedia.org/wiki/Energy_drift

Energy drift - usually damping - is substantial for numerical integration schemes that are not symplectic, such as the Runge-Kutta family.

https://github.com/chakravala/Grassmann.jl

The Grassmann.jl package provides tools for doing computations based on multi-linear algebra, differential geometry, and spin groups using the extended tensor algebra known as Leibniz-Grassmann-Clifford-Hestenes geometric algebra.

One of the main roles of the tangent bundle is to provide a domain and range for the derivative of a smooth function. Namely, if f : M → N is a smooth function, with M and N smooth manifolds, its derivative is a smooth function Df : TM → TN.

Quite interestingly, Symplectic Geometry is currently under review/investigation for some of the foundational papers in the field having serious gaps and outright errors after closer inspection. These concerns were always spoken of in hush hush tones and only in recent times have people stated their concerns publically. Some of the original authors refuse to retract their papers despite being assured their academic positions (which realistically, came through the reputation built up by these papers) are secure. Here's a quanta article about this fiasco: https://www.quantamagazine.org/the-fight-to-fix-symplectic-g...

Symplectic geometry is like the evil twin of Euclidean geometry: instead of a dot product with v⋅w = w⋅v, we have one with v⋅w = -w⋅v.   But it's not really evil.   Check out Jonathan Lorand's talk!

https://twitter.com/johncarlosbaez/status/1087416965012942849