Table of Contents

[20181110]
ok, trying to break down this thing http://math.mit.edu/~cohn/Thoughts/symplectic.html 
[20181116]
Topics in Representation Theory: Hamiltonian Mechanics and Symplectic Geometry [20181117]
Terence Tao: Phase Space
[[momentum vs velocity]]

[[https://physics.stackexchange.com/questions/213991/whyspecifythestateofaparticleintermsofpositionandmomentumnotveloci]]
 [[tl;dr: In classical mechanics, specifying a particle's state in terms of momentum is equivalent to specifying it in terms of velocity, but the specification in terms of momenta often has computational advantages.]]
 [[Quantum mechanics does not have welldefined trajectories q(t), so the notion of a velocity does not make sense. On the contrary, the momentum operator can still be defined as relating to the position operator in the same way as in Hamiltonian mechanics, by replacing the classical Poisson bracket by the quantum commutator of operators.]]
 [[hmm, so velocities (q') are just additional 'data' which happens to be related via q = q'(t). initially, you don't have to treat it as derivative.]]

[[https://physics.stackexchange.com/questions/213991/whyspecifythestateofaparticleintermsofpositionandmomentumnotveloci]]
 [[that's pretty interesting https://physics.stackexchange.com/questions/123725/whatkindofmanifoldcanbethephasespaceofahamiltoniansystem]]
 [[equations of motion are dx/dt = {x, H} and dp/dt = {p, H} β hmm, that's interestingβ¦]]
 [[https://www.quora.com/Whatisthesignificanceofasymplecticmanifold]]
 [something interesting about the fact that not all symplectic forms can be exact (if the space is not T*Q for some Q)](#smthngntrstngbtthfctthtntrmscnbxctfthspcsnttqfrsmq TIDDLYLINK)
 [[where to put it?]]
 [some graphical intuition about covectorsβ¦ http://www.physicsinsights.org/pbp_one_forms.html](#smgrphclnttnbtcvctrswwwphyscsnsghtsrgpbpnfrmshtml TIDDLYLINK)
[20181110]
digression: [Goldberg] A Little Tase of Symplectic Geometry.pdf β very cool!!! [something interesting about basis of tangent vectors?β¦ https://math.stackexchange.com/a/454663/15108 v β T_{p} M = v^{i} Ξ΄_{i}?](#smthngntrstngbtbssftngntvssmthstckxchngcmvntpmvdlt TIDDLYLINK)
 [hmm, why are alternating forms important? https://en.wikipedia.org/wiki/Multilinear_form#Alternating_multilinear_forms](#hmmwhyrltrntngfrmsmprtntsltlnrfrmltrntngmltlnrfrms TIDDLYLINK)
 '[Powell] Aspects of Symplectic Geometry in Physics.pdf'
 [[something about symplectomorphismsβ¦]]
[20181118]
ok, I should implement some simple phase space portrait plotting first
[20181121]
understood A LOT while waiting for Metric concert!
[https://en.wikipedia.org/wiki/Hamiltonian_mechanics#Deriving_Hamilton's_equations](#snwkpdrgwkhmltnnmchncsdrvnghmltnsqtns TIDDLYLINK)
 [[mnemonic : p then q (dp/dt = dH/dq)]]
 [[In fact, as is shown below, the Hamiltonian is the Legendre transform of the Lagrangian when holding q and t fixed and defining p as the dual variable, and thus both approaches give the same equations for the same generalized momentum. The main motivation to use Hamiltonian mechanics instead of Lagrangian mechanics comes from the symplectic structure of Hamiltonian systems.]]
 [[Hamilton's equation easily derived by looking at the total differential of Lagrangian on time]]
 [[generalised coordinates: just any coordinates that (injectively??) map onto system configuration]]
 [[Since this calculation was done offshell, one can associate corresponding terms from both sides of this equation to yield:]]
 [[If the transformation equations defining the generalized coordinates are independent of t, and the Lagrangian is a sum of products of functions (in the generalized coordinates) which are homogeneous of order 0, 1 or 2, then it can be shown that H is equal to the total energy E = T + V.]]
 [[The solutions to the HamiltonβJacobi equations for this Hamiltonian are then the same as the geodesics on the manifold. In particular, the Hamiltonian flow in this case is the same thing as the geodesic flow. The existence of such solutions, and the completeness of the set of solutions, are discussed in detail in the article on geodesics. See also Geodesics as Hamiltonian flows.]]

[https://ru.wikipedia.org/wiki/%D0%93%D0%B0%D0%BC%D0%B8%D0%BB%D1%8C%D1%82%D0%BE%D0%BD%D0%BE%D0%B2%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0](#srwkpdrgwkddbdbcdbdbbdcddbddbdbdbdbcdbddbdbddbdbdb TIDDLYLINK)
 [[Π ΠΏΠΎΠ»ΡΡΠ½ΡΡ ΠΊΠΎΠΎΡΠ΄ΠΈΠ½Π°ΡΠ°Ρ ΠΎΠ±ΠΎΠ±ΡΡΠ½Π½ΡΠΉ ΠΈΠΌΠΏΡΠ»ΡΡ, ΡΠΎΠΎΡΠ²Π΅ΡΡΡΠ²ΡΡΡΠΈΠΉ ΡΠ³Π»ΠΎΠ²ΠΎΠΉ ΡΠΊΠΎΡΠΎΡΡΠΈ, β ΡΠΈΠ·ΠΈΡΠ΅ΡΠΊΠΈΠΉ ΡΠ³Π»ΠΎΠ²ΠΎΠΉ ΠΌΠΎΠΌΠ΅Π½Ρ. ΠΠ»Ρ ΠΏΡΠΎΠΈΠ·Π²ΠΎΠ»ΡΠ½ΠΎΠ³ΠΎ Π²ΡΠ±ΠΎΡΠ° ΠΎΠ±ΠΎΠ±ΡΡΠ½Π½ΡΡ ΠΊΠΎΠΎΡΠ΄ΠΈΠ½Π°Ρ ΡΡΡΠ΄Π½ΠΎ ΠΏΠΎΠ»ΡΡΠΈΡΡ ΠΈΠ½ΡΡΠΈΡΠΈΠ²Π½ΡΡ ΠΈΠ½ΡΠ΅ΡΠΏΡΠ΅ΡΠ°ΡΠΈΡ ΡΠΎΠΏΡΡΠΆΡΠ½Π½ΡΡ ΡΡΠΈΠΌ ΠΊΠΎΠΎΡΠ΄ΠΈΠ½Π°ΡΠ°ΠΌ ΠΈΠΌΠΏΡΠ»ΡΡΠΎΠ² ΠΈΠ»ΠΈ ΡΠ³Π°Π΄Π°ΡΡ ΠΈΡ Π²ΡΡΠ°ΠΆΠ΅Π½ΠΈΠ΅, Π½Π΅ ΠΈΡΠΏΠΎΠ»ΡΠ·ΡΡ ΠΏΡΡΠΌΠΎ ΠΏΡΠΈΠ²Π΅Π΄ΡΠ½Π½ΡΡ Π²ΡΡΠ΅ ΡΠΎΡΠΌΡΠ»Ρ.]]
 [[ΠΡΡΡΠ΄Π°, Π² ΡΠ°ΡΡΠ½ΠΎΡΡΠΈ, ΡΠ»Π΅Π΄ΡΠ΅Ρ, ΡΡΠΎ Π΅ΡΠ»ΠΈ ΠΊΠ°ΠΊΠ°ΡΡΠΎ ΠΊΠΎΠΎΡΠ΄ΠΈΠ½Π°ΡΠ° ΠΎΠΊΠ°Π·Π°Π»Π°ΡΡ ΡΠΈΠΊΠ»ΠΈΡΠ΅ΡΠΊΠΎΠΉ, ΡΠΎ Π΅ΡΡΡ Π΅ΡΠ»ΠΈ ΡΡΠ½ΠΊΡΠΈΡ ΠΠ°Π³ΡΠ°Π½ΠΆΠ° ΠΎΡ Π½Π΅Ρ Π½Π΅ Π·Π°Π²ΠΈΡΠΈΡ, Π° Π·Π°Π²ΠΈΡΠΈΡ ΡΠΎΠ»ΡΠΊΠΎ ΠΎΡ Π΅Ρ ΠΏΡΠΎΠΈΠ·Π²ΠΎΠ΄Π½ΠΎΠΉ ΠΏΠΎ Π²ΡΠ΅ΠΌΠ΅Π½ΠΈ, ΡΠΎ Π΄Π»Ρ ΡΠΎΠΏΡΡΠΆΡΠ½Π½ΠΎΠ³ΠΎ Π΅ΠΉ ΠΈΠΌΠΏΡΠ»ΡΡΠ° {\displaystyle {\dot {p}}=0} {\dot {p}}=0, ΡΠΎ Π΅ΡΡΡ ΠΎΠ½ ΡΠ²Π»ΡΠ΅ΡΡΡ ΠΈΠ½ΡΠ΅Π³ΡΠ°Π»ΠΎΠΌ Π΄Π²ΠΈΠΆΠ΅Π½ΠΈΡ (ΡΠΎΡ ΡΠ°Π½ΡΠ΅ΡΡΡ Π²ΠΎ Π²ΡΠ΅ΠΌΠ΅Π½ΠΈ), ΡΡΠΎ Π½Π΅ΡΠΊΠΎΠ»ΡΠΊΠΎ ΠΏΡΠΎΡΡΠ½ΡΠ΅Ρ ΡΠΌΡΡΠ» ΠΎΠ±ΠΎΠ±ΡΡΠ½Π½ΡΡ ΠΈΠΌΠΏΡΠ»ΡΡΠΎΠ².]]
 [[ΠΡΠ±Π°Ρ Π³Π»Π°Π΄ΠΊΠ°Ρ ΡΡΠ½ΠΊΡΠΈΡ {\displaystyle H: Mβ \mathbb {R} } H: Mβ \mathbb{R} Π½Π° ΡΠΈΠΌΠΏΠ»Π΅ΠΊΡΠΈΡΠ΅ΡΠΊΠΎΠΌ ΠΌΠ½ΠΎΠ³ΠΎΠΎΠ±ΡΠ°Π·ΠΈΠΈ {\displaystyle M} M ΠΌΠΎΠΆΠ΅Ρ ΠΈΡΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°ΡΡΡΡ, ΡΡΠΎΠ±Ρ ΠΎΠΏΡΠ΅Π΄Π΅Π»ΠΈΡΡ Π³Π°ΠΌΠΈΠ»ΡΡΠΎΠ½ΠΎΠ²Ρ ΡΠΈΡΡΠ΅ΠΌΡ. Π€ΡΠ½ΠΊΡΠΈΡ {\displaystyle H} H ΠΈΠ·Π²Π΅ΡΡΠ½Π° ΠΊΠ°ΠΊ Π³Π°ΠΌΠΈΠ»ΡΡΠΎΠ½ΠΈΠ°Π½ ΠΈΠ»ΠΈ ΡΠ½Π΅ΡΠ³Π΅ΡΠΈΡΠ΅ΡΠΊΠ°Ρ ΡΡΠ½ΠΊΡΠΈΡ. Π‘ΠΈΠΌΠΏΠ»Π΅ΠΊΡΠΈΡΠ΅ΡΠΊΠΎΠ΅ ΠΌΠ½ΠΎΠ³ΠΎΠΎΠ±ΡΠ°Π·ΠΈΠ΅ Π½Π°Π·ΡΠ²Π°ΡΡ ΡΠ°Π·ΠΎΠ²ΡΠΌ ΠΏΡΠΎΡΡΡΠ°Π½ΡΡΠ²ΠΎΠΌ. ΠΠ°ΠΌΠΈΠ»ΡΡΠΎΠ½ΠΈΠ°Π½ ΠΏΠΎΡΠΎΠΆΠ΄Π°Π΅Ρ ΡΠΏΠ΅ΡΠΈΠ°Π»ΡΠ½ΠΎΠ΅ Π²Π΅ΠΊΡΠΎΡΠ½ΠΎΠ΅ ΠΏΠΎΠ»Π΅ Π½Π° ΡΠΈΠΌΠΏΠ»Π΅ΠΊΡΠΈΡΠ΅ΡΠΊΠΎΠΌ ΠΌΠ½ΠΎΠ³ΠΎΠΎΠ±ΡΠ°Π·ΠΈΠΈ, ΠΈΠ·Π²Π΅ΡΡΠ½ΠΎΠΌ ΠΊΠ°ΠΊ ΡΠΈΠΌΠΏΠ»Π΅ΠΊΡΠΈΡΠ΅ΡΠΊΠΎΠ΅ Π²Π΅ΠΊΡΠΎΡΠ½ΠΎΠ΅ ΠΏΠΎΠ»Π΅.]]
 [[Π‘ΠΈΠΌΠΏΠ»Π΅ΠΊΡΠΈΡΠ΅ΡΠΊΠΎΠ΅ Π²Π΅ΠΊΡΠΎΡΠ½ΠΎΠ΅ ΠΏΠΎΠ»Π΅ (ΡΠ°ΠΊΠΆΠ΅ Π½Π°Π·ΡΠ²Π°Π΅ΡΡΡ Π³Π°ΠΌΠΈΠ»ΡΡΠΎΠ½ΠΎΠ²ΡΠΌ Π²Π΅ΠΊΡΠΎΡΠ½ΡΠΌ ΠΏΠΎΠ»Π΅ΠΌ) ΠΏΠΎΡΠΎΠΆΠ΄Π°Π΅Ρ Π³Π°ΠΌΠΈΠ»ΡΡΠΎΠ½ΠΎΠ² ΠΏΠΎΡΠΎΠΊ Π½Π° ΠΌΠ½ΠΎΠ³ΠΎΠΎΠ±ΡΠ°Π·ΠΈΠΈ. ΠΠ½ΡΠ΅Π³ΡΠ°Π»ΡΠ½ΡΠ΅ ΠΊΡΠΈΠ²ΡΠ΅ Π²Π΅ΠΊΡΠΎΡΠ½ΠΎΠ³ΠΎ ΠΏΠΎΠ»Ρ ΡΠ²Π»ΡΡΡΡΡ ΠΎΠ΄Π½ΠΎΠΏΠ°ΡΠ°ΠΌΠ΅ΡΡΠΈΡΠ΅ΡΠΊΠΈΠΌ ΡΠ΅ΠΌΠ΅ΠΉΡΡΠ²ΠΎΠΌ ΠΏΡΠ΅ΠΎΠ±ΡΠ°Π·ΠΎΠ²Π°Π½ΠΈΠΉ ΠΌΠ½ΠΎΠ³ΠΎΠΎΠ±ΡΠ°Π·ΠΈΡ Ρ ΠΏΠ°ΡΠ°ΠΌΠ΅ΡΡΠΎΠΌ, Π½Π°Π·ΡΠ²Π°Π΅ΠΌΡΠΌ Π²ΡΠ΅ΠΌΡ. ΠΠ²ΠΎΠ»ΡΡΠΈΡ Π²ΠΎ Π²ΡΠ΅ΠΌΠ΅Π½ΠΈ Π·Π°Π΄Π°ΡΡΡΡ ΡΠΈΠΌΠΏΠ»Π΅ΠΊΡΠΎΠΌΠΎΡΡΠΈΠ·ΠΌΠ°ΠΌΠΈ. ΠΠ· ΡΠ΅ΠΎΡΠ΅ΠΌΡ ΠΠΈΡΠ²ΠΈΠ»Π»Ρ ΡΠ»Π΅Π΄ΡΠ΅Ρ, ΡΡΠΎ ΠΊΠ°ΠΆΠ΄ΡΠΉ ΡΠΈΠΌΠΏΠ»Π΅ΠΊΡΠΎΠΌΠΎΡΡΠΈΠ·ΠΌ ΡΠΎΡ ΡΠ°Π½ΡΠ΅Ρ ΡΠΎΡΠΌΡ ΠΎΠ±ΡΡΠΌΠ° Π² ΡΠ°Π·ΠΎΠ²ΠΎΠΌ ΠΏΡΠΎΡΡΡΠ°Π½ΡΡΠ²Π΅. ΠΠ½ΠΎΠΆΠ΅ΡΡΠ²ΠΎ ΡΠΈΠΌΠΏΠ»Π΅ΠΊΡΠΎΠΌΠΎΡΡΠΈΠ·ΠΌΠΎΠ², ΠΏΠΎΡΠΎΠΆΠ΄Π°Π΅ΠΌΡΡ Π³Π°ΠΌΠΈΠ»ΡΡΠΎΠ½ΠΎΠ²ΡΠΌ ΠΏΠΎΡΠΎΠΊΠΎΠΌ, ΠΎΠ±ΡΡΠ½ΠΎ Π½Π°Π·ΡΠ²Π°ΡΡ Π³Π°ΠΌΠΈΠ»ΡΡΠΎΠ½ΠΎΠ²ΠΎΠΉ ΠΌΠ΅Ρ Π°Π½ΠΈΠΊΠΎΠΉ Π³Π°ΠΌΠΈΠ»ΡΡΠΎΠ½ΠΎΠ²ΠΎΠΉ ΡΠΈΡΡΠ΅ΠΌΡ.]]
 [[ΠΠ°ΠΌΠΈΠ»ΡΡΠΎΠ½ΠΎΠ²ΠΎ Π²Π΅ΠΊΡΠΎΡΠ½ΠΎΠ΅ ΠΏΠΎΠ»Π΅ ΡΠ°ΠΊΠΆΠ΅ ΠΏΠΎΡΠΎΠΆΠ΄Π°Π΅Ρ ΡΠΏΠ΅ΡΠΈΠ°Π»ΡΠ½ΡΡ ΠΎΠΏΠ΅ΡΠ°ΡΠΈΡ β ΡΠΊΠΎΠ±ΠΊΠ° ΠΡΠ°ΡΡΠΎΠ½Π°. Π‘ΠΊΠΎΠ±ΠΊΠ° ΠΡΠ°ΡΡΠΎΠ½Π° Π΄Π΅ΠΉΡΡΠ²ΡΠ΅Ρ Π½Π° ΡΡΠ½ΠΊΡΠΈΠΈ Π½Π° ΡΠΈΠΌΠΏΠ»Π΅ΠΊΡΠΈΡΠ΅ΡΠΊΠΎΠΌ ΠΌΠ½ΠΎΠ³ΠΎΠΎΠ±ΡΠ°Π·ΠΈΠΈ, ΡΠ°ΠΊΠΈΠΌ ΠΎΠ±ΡΠ°Π·ΠΎΠΌ ΠΏΡΠΈΠ΄Π°Π²Π°Ρ ΠΏΡΠΎΡΡΡΠ°Π½ΡΡΠ²Ρ ΡΡΠ½ΠΊΡΠΈΠΉ Π½Π° ΠΌΠ½ΠΎΠ³ΠΎΠΎΠ±ΡΠ°Π·ΠΈΠΈ ΡΡΡΡΠΊΡΡΡΡ Π°Π»Π³Π΅Π±ΡΡ ΠΠΈ.]]
 [[phase: from any point on phase space, evolution is unique. it's kinda like initial data for a differential equation]]

[https://en.wikipedia.org/wiki/Hamiltonian_mechanics#Deriving_Hamilton's_equations](#snwkpdrgwkhmltnnmchncsdrvnghmltnsqtns TIDDLYLINK)

[20181120]
why symplectic spaces? [20181110]
intuition  What is a symplectic form intuitively?  MathOverflow [[Hamiltonian vector field  Wikipedia]]
 [[Tweet from John Carlos Baez (@johncarlosbaez), at Nov 22, 22:01]]
 [[Legendre transformation  Wikipedia]]
[20181110]
What moment map is (as a physical concept) in sympletic geometry  Mathematics Stack Exchange[20181110]
classical mechanics  Intuition about Momentum Maps  Physics Stack Exchange[20181114]
sg.symplectic geometry  How to see the Phase Space of a Physical System as the Cotangent Bundle  MathOverflow[20181119]
Legendre transformation  Wikipedia[20181118]
Symplectic integrator  Wikipedia[20181119]
mathematical physics  Is there an analogue of configuration space in quantum mechanics?  Physics Stack Exchange[20181118]
Applying RungeKutta method to circular  C++ Forum[20180913]
Chel_{of}_{the}_{sea} comments on ELI5: What is symplectic geometry?[20180813]
poisson brackets are related to symplectic geometry (explains phase space)
[20180813]
poisson brackets explained 
[20181115]
 [[Spivak's Physics For Mathematicians,]]
 [configuration space: M (a Manifold). Phase space: cotangent bundle over M (T_{*} M). Also a manifold?? Yeah, ok T_{*} M is also a manifold with dimension twise as what M got.](#cnfgrtnspcmmnfldphsspcctnlsmnfldwthdmnsntwsswhtmgt TIDDLYLINK)
 [[right, and symplectic form is defined on the contagent bundle of configuration space! (or on a phase space?)]]
 [[learn basic QED and QFT]] [[study]] [[qed]]
[20181215]
symplectic geometry  Learn Anything[20190618]
Energy drift  Wikipedia [[symplectic]][20200118]
chakravala/Grassmann.jl: β¨LeibnizGrassmannCliffordβ© differential geometric algebra / multivector simplicial complex [[symplectic]][20181110]
Tangent bundle  Wikipedia https://en.wikipedia.org/wiki/Tangent_bundle [[symplectic]] [[drill]] [[Symplectic geometry foundations]]
 [[Tweet from John Carlos Baez (@johncarlosbaez), at Jan 21, 18:29]] [[symplectic]] [[towatch]]
[20181115]
ok, so trying to consolidate everything
 https://sbseminar.wordpress.com/2012/01/09/whatisasymplecticmanifoldreally/
TODO how to link in org mode?? e.g. citations?
here they try to deduce phase space from some reasonable mathematical assumptions (time evolution, energy conservation, flows)
 The Geometry of Hamiltonian Mechanics
So, we have a configuration space N, which is a manifold. Typically, it's the set of positions, more generally, it's the set of all possible 'snapshots' of the system at a certain time.
(TODO how does that correspond to wavefunctions? how is position special? e.g. momentum is not any worse right?).
Then take (M = T^* N) (cotangent bundle) β it's the phase space, and itself a manifold.
TODO twice the dimensions?
NOTE: ugh, looks like they are confusing p and q in [1], typically q is position/state and p is monentum. So I'm using the more common notation.
Some coordinates in M are position coordinates, some are momentum coordinates. (x_{q}, x_{p}). x_{q} corresponds to N, x_{p} corresponds to T* N at x_{q}?
Ok, consider time derivative.
Time derivative of x_{q} is a vector on N [1]. Note: ok, sort of makes sense.. I guess by vector they mean a point from tangent space? E.g. consider 2sphere, time derivative is indeed a vector on that sphere.
Time derivative of x_{p} is a covector on N [1]. TODO: this still makes sense I guess? Not really.. I guess my confusion has to do with not understanding what's a thing from T* N?
digression:
consider 1sphere S (radius 1). Its configurations X are angles phi from 0 to 2 pi. If you consider tangent space though, it's gonna have all possible velocities from inf to inf.
ok. but what about cotangent space??  TODO what does it have to do with 1forms
is it just the space of all {mul by v  v in (inf, inf)}. And then what??
so time derivative of position is a vector on N. Agree. I guess we're using the fact that N is a manifold, thus locally it's a vector space.
time derivative of momentum is a covector on N. Well, that's a bit more subtle. I mean, it kinda makes sense, but it's a different space than T*_{x} N. Right?
suppose we have a functional F(t): N > R, and we want to compute its derivative. By definition, F'(t) = (F(t+dt)  F(t)) / dt. But F(x) = <f, x>. Then, F'(t) = (<f(t + dt), x>  <f(t), x>) / dt = <(f(t + dt)  f(t))/dt, x>. So, it's dual to f', which is a time derivative of a vector, thus a covector. Ugh, ok.
Again, following [1]. Consider a function E: M > R.
Its differential w.r.t. space coordinates is a covector, and w.r.t momentum coordinates is a vector. well, ok
dE/d(q,p) = (dE/dq1.. dE/dqn, dE/dp1β¦de/dpn)
Ok, I suppose you could call dE/dq a covector. why though?β¦ what does that mean? I guess that if we plug
TODO shit. don't think I understand that bit really intuitivelyβ¦ but whatever
Ok. so we established that
dx_{q}/dt and dE/dx_{p} are both vectors
dx_{p}/dt and dE/dx_{q} and both covectors.
so that?? why the minus sign?
[20181110]
ok, trying to break down this thing http://math.mit.edu/~cohn/Thoughts/symplectic.html
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:
 depend only on dH (global shift doesn't matter)
 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 β vector space, depending on parameter (point)
NOTE: tensor field by definition is some section on tensor bundle. mmm.
NOTE: vector fields on manifold β 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?β¦
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) =
[20181118]
ok, use conservation of energy
that means that Hamiltonian is constant along the flow. for any H: dH(V_{H}) = 0, hence w(V_{H}, V_{H}) = 0. For every V, we can find its H, s.t. V_{H} = V, so w(V, V) = 0 for all V? so the form is alternating
TODO nondegeneracy?
fuck, I don't really understand the closedness thing and it might actually be crutial. maybe, laterβ¦
[20181116]
Topics in Representation Theory: Hamiltonian Mechanics and Symplectic Geometry
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 β_{f}, which comes from: taking df (1form); then using inner product on R^{2n} to dualise and get a vector field from 1form.
that is:
f > β_{f}: <β_{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 oneform theta an point (a, b) β 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_oneform
apparently this theta is called canonical oneform
print it!β¦
looks like really good paperβ¦ read more from it (or references?)
[1] Bryant, R., An Introduction to Lie Groups and Symplectic Geometry, in
Geometry and Quantum Field Theory, Freed, D., and Uhlenbeck, K., eds.,
American Mathematical Society, 1995.
[2] Guillemin, V. and Sternberg, S., Symplectic Techniques in Physics, Cam
bridge University Press, 1984.
[20181117]
Terence Tao: Phase Space
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 thingsβ¦ 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
momentum vs velocity
https://physics.stackexchange.com/questions/213991/whyspecifythestateofaparticleintermsofpositionandmomentumnotveloci
tl;dr: In classical mechanics, specifying a particle's state in terms of momentum is equivalent to specifying it in terms of velocity, but the specification in terms of momenta often has computational advantages.
Quantum mechanics does not have welldefined trajectories q(t), so the notion of a velocity does not make sense. On the contrary, the momentum operator can still be defined as relating to the position operator in the same way as in Hamiltonian mechanics, by replacing the classical Poisson bracket by the quantum commutator of operators.
hmm, so velocities (q') are just additional 'data' which happens to be related via q = q'(t). initially, you don't have to treat it as derivative.
that's pretty interesting https://physics.stackexchange.com/questions/123725/whatkindofmanifoldcanbethephasespaceofahamiltoniansystem
maybe visualise a hamiltonian on ntorus?
equations of motion are dx/dt = {x, H} and dp/dt = {p, H} β hmm, that's interestingβ¦
https://mathoverflow.net/a/16462/29889
https://www.quora.com/Whatisthesignificanceofasymplecticmanifold
I believe the significance for physics boils down to the following: it turns out that a twoform is precisely what is required to translate an energy functional on phase space (a Hamiltonian) into a flow (a vector field).
mmmβ¦
So in some sense, "conservation of symplectic form" is the second most basic conservation law. (The most basic is conservation of energy, which is essentially the definition o
something interesting about the fact that not all symplectic forms can be exact (if the space is not T*Q for some Q)
https://mathoverflow.net/a/16537/29889
where to put it?
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/dq^{i} 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 2form; which means that when it acts on a vector field, we get 1form (so momentum is 1form)
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_{H}_{x}(t) = J dH (x); where J is the symplectic matrix; dH is gradient of hamiltonian function. Hamiltonian vector field.
pullbacks:
consider O: M > T* M β 1form on phase space β ok, as a member of T* M.
consider a: Q > T* Q β 1form on conf space.
right, so a is a linear map from Q to M; and O is a 1form on M. We can pull back O to Q to get the 1form a* O.  err so what??
canonical poincare 1form Theta = \Sum_{i} p_{i} dq^{i}, which satisfies a^{*} Theta = a for all a in β¦.
some graphical intuition about covectorsβ¦ http://www.physicsinsights.org/pbp_one_forms.html
[20181110]
digression: [Goldberg] A Little Tase of Symplectic Geometry.pdf β very cool!!!
something interesting about basis of tangent vectors?β¦ https://math.stackexchange.com/a/454663/15108 v β T_{p} M = v^{i} Ξ΄_{i}?
hmm, why are alternating forms important? https://en.wikipedia.org/wiki/Multilinear_form#Alternating_multilinear_forms
so they are actually what's called covectors??
'[Powell] Aspects of Symplectic Geometry in Physics.pdf'
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)) β if time dependent , then (\dv{f}{t} = {f, H} + \pdv{f}{t}) β makes sense! kinda like flow derivative from Segal
something about symplectomorphismsβ¦
any 2ndimensional symplectic manifold looks like R^{2n}, w_{0}
[20181118]
ok, I should implement some simple phase space portrait plotting first
https://github.com/BartoszMilewski/gravitysim
haskell or rust?β¦ not sure
try L = 1/2x'^{2} + x'  1/2 x^{2}
TODO what if we make linear dependency on position? That's pretty much coordinate transformation right?
soo, if it's a linear dependency on velocityβ¦ it's kind of time coordinate transformation!!
[20181121]
understood A LOT while waiting for Metric concert!
https://en.wikipedia.org/wiki/Hamiltonian_mechanics#Deriving_Hamilton's_equations
mnemonic : p then q (dp/dt = dH/dq)
In fact, as is shown below, the Hamiltonian is the Legendre transform of the Lagrangian when holding q and t fixed and defining p as the dual variable, and thus both approaches give the same equations for the same generalized momentum. The main motivation to use Hamiltonian mechanics instead of Lagrangian mechanics comes from the symplectic structure of Hamiltonian systems.
Hamilton's equation easily derived by looking at the total differential of Lagrangian on time
generalised coordinates: just any coordinates that (injectively??) map onto system configuration
Since this calculation was done offshell, one can associate corresponding terms from both sides of this equation to yield:
hmm, wonder if that's kinda like dimensionality argument
In Cartesian coordinates, the generalized momenta are precisely the physical linear momenta. In circular polar coordinates, the generalized momentum corresponding to the angular velocity is the physical angular momentum. For an arbitrary choice of generalized coordinates, it may not be possible to obtain an intuitive interpretation of the conjugate momenta.
right, that's interesting that it's not possible to obtain intuitive sense
One thing which is not too obvious in this coordinate dependent formulation is that different generalized coordinates are really nothing more than different coordinate patches on the same symplectic manifold (see Mathematical formalism, below).
TODO should read more on thatβ¦ what do they call coordinate patches?
If the transformation equations defining the generalized coordinates are independent of t, and the Lagrangian is a sum of products of functions (in the generalized coordinates) which are homogeneous of order 0, 1 or 2, then it can be shown that H is equal to the total energy E = T + V.
not sure, that might be interesting
The solutions to the HamiltonβJacobi equations for this Hamiltonian are then the same as the geodesics on the manifold. In particular, the Hamiltonian flow in this case is the same thing as the geodesic flow. The existence of such solutions, and the completeness of the set of solutions, are discussed in detail in the article on geodesics. See also Geodesics as Hamiltonian flows.
https://en.wikipedia.org/wiki/Hamiltonian_mechanics#Riemannian_manifolds
https://ru.wikipedia.org/wiki/%D0%93%D0%B0%D0%BC%D0%B8%D0%BB%D1%8C%D1%82%D0%BE%D0%BD%D0%BE%D0%B2%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0
Π ΠΏΠΎΠ»ΡΡΠ½ΡΡ ΠΊΠΎΠΎΡΠ΄ΠΈΠ½Π°ΡΠ°Ρ ΠΎΠ±ΠΎΠ±ΡΡΠ½Π½ΡΠΉ ΠΈΠΌΠΏΡΠ»ΡΡ, ΡΠΎΠΎΡΠ²Π΅ΡΡΡΠ²ΡΡΡΠΈΠΉ ΡΠ³Π»ΠΎΠ²ΠΎΠΉ ΡΠΊΠΎΡΠΎΡΡΠΈ, β ΡΠΈΠ·ΠΈΡΠ΅ΡΠΊΠΈΠΉ ΡΠ³Π»ΠΎΠ²ΠΎΠΉ ΠΌΠΎΠΌΠ΅Π½Ρ. ΠΠ»Ρ ΠΏΡΠΎΠΈΠ·Π²ΠΎΠ»ΡΠ½ΠΎΠ³ΠΎ Π²ΡΠ±ΠΎΡΠ° ΠΎΠ±ΠΎΠ±ΡΡΠ½Π½ΡΡ ΠΊΠΎΠΎΡΠ΄ΠΈΠ½Π°Ρ ΡΡΡΠ΄Π½ΠΎ ΠΏΠΎΠ»ΡΡΠΈΡΡ ΠΈΠ½ΡΡΠΈΡΠΈΠ²Π½ΡΡ ΠΈΠ½ΡΠ΅ΡΠΏΡΠ΅ΡΠ°ΡΠΈΡ ΡΠΎΠΏΡΡΠΆΡΠ½Π½ΡΡ ΡΡΠΈΠΌ ΠΊΠΎΠΎΡΠ΄ΠΈΠ½Π°ΡΠ°ΠΌ ΠΈΠΌΠΏΡΠ»ΡΡΠΎΠ² ΠΈΠ»ΠΈ ΡΠ³Π°Π΄Π°ΡΡ ΠΈΡ Π²ΡΡΠ°ΠΆΠ΅Π½ΠΈΠ΅, Π½Π΅ ΠΈΡΠΏΠΎΠ»ΡΠ·ΡΡ ΠΏΡΡΠΌΠΎ ΠΏΡΠΈΠ²Π΅Π΄ΡΠ½Π½ΡΡ Π²ΡΡΠ΅ ΡΠΎΡΠΌΡΠ»Ρ.
sadβ¦
ΠΡΡΡΠ΄Π°, Π² ΡΠ°ΡΡΠ½ΠΎΡΡΠΈ, ΡΠ»Π΅Π΄ΡΠ΅Ρ, ΡΡΠΎ Π΅ΡΠ»ΠΈ ΠΊΠ°ΠΊΠ°ΡΡΠΎ ΠΊΠΎΠΎΡΠ΄ΠΈΠ½Π°ΡΠ° ΠΎΠΊΠ°Π·Π°Π»Π°ΡΡ ΡΠΈΠΊΠ»ΠΈΡΠ΅ΡΠΊΠΎΠΉ, ΡΠΎ Π΅ΡΡΡ Π΅ΡΠ»ΠΈ ΡΡΠ½ΠΊΡΠΈΡ ΠΠ°Π³ΡΠ°Π½ΠΆΠ° ΠΎΡ Π½Π΅Ρ Π½Π΅ Π·Π°Π²ΠΈΡΠΈΡ, Π° Π·Π°Π²ΠΈΡΠΈΡ ΡΠΎΠ»ΡΠΊΠΎ ΠΎΡ Π΅Ρ ΠΏΡΠΎΠΈΠ·Π²ΠΎΠ΄Π½ΠΎΠΉ ΠΏΠΎ Π²ΡΠ΅ΠΌΠ΅Π½ΠΈ, ΡΠΎ Π΄Π»Ρ ΡΠΎΠΏΡΡΠΆΡΠ½Π½ΠΎΠ³ΠΎ Π΅ΠΉ ΠΈΠΌΠΏΡΠ»ΡΡΠ° {\displaystyle {\dot {p}}=0} {\dot {p}}=0, ΡΠΎ Π΅ΡΡΡ ΠΎΠ½ ΡΠ²Π»ΡΠ΅ΡΡΡ ΠΈΠ½ΡΠ΅Π³ΡΠ°Π»ΠΎΠΌ Π΄Π²ΠΈΠΆΠ΅Π½ΠΈΡ (ΡΠΎΡ ΡΠ°Π½ΡΠ΅ΡΡΡ Π²ΠΎ Π²ΡΠ΅ΠΌΠ΅Π½ΠΈ), ΡΡΠΎ Π½Π΅ΡΠΊΠΎΠ»ΡΠΊΠΎ ΠΏΡΠΎΡΡΠ½ΡΠ΅Ρ ΡΠΌΡΡΠ» ΠΎΠ±ΠΎΠ±ΡΡΠ½Π½ΡΡ ΠΈΠΌΠΏΡΠ»ΡΡΠΎΠ².
Π ΡΡΠΎΠΉ ΡΠΎΡΠΌΡΠ»ΠΈΡΠΎΠ²ΠΊΠ΅, Π·Π°Π²ΠΈΡΡΡΠ΅ΠΉ ΠΎΡ Π²ΡΠ±ΠΎΡΠ° ΡΠΈΡΡΠ΅ΠΌΡ ΠΊΠΎΠΎΡΠ΄ΠΈΠ½Π°Ρ, Π½Π΅ ΡΠ»ΠΈΡΠΊΠΎΠΌ ΠΎΡΠ΅Π²ΠΈΠ΄Π΅Π½ ΡΠΎΡ ΡΠ°ΠΊΡ, ΡΡΠΎ ΡΠ°Π·Π»ΠΈΡΠ½ΡΠ΅ ΠΎΠ±ΠΎΠ±ΡΡΠ½Π½ΡΠ΅ ΠΊΠΎΠΎΡΠ΄ΠΈΠ½Π°ΡΡ ΡΠ²Π»ΡΡΡΡΡ Π² Π΄Π΅ΠΉΡΡΠ²ΠΈΡΠ΅Π»ΡΠ½ΠΎΡΡΠΈ Π½Π΅ ΡΠ΅ΠΌ ΠΈΠ½ΡΠΌ, ΠΊΠ°ΠΊ ΡΠ°Π·Π»ΠΈΡΠ½ΡΠΌΠΈ ΠΊΠΎΠΎΡΠ΄ΠΈΠ½Π°ΡΠΈΠ·Π°ΡΠΈΡΠΌΠΈ ΠΎΠ΄Π½ΠΎΠ³ΠΎ ΠΈ ΡΠΎΠ³ΠΎ ΠΆΠ΅ ΡΠΈΠΌΠΏΠ»Π΅ΠΊΡΠΈΡΠ΅ΡΠΊΠΎΠ³ΠΎ ΠΌΠ½ΠΎΠ³ΠΎΠΎΠ±ΡΠ°Π·ΠΈΡ.
ΠΡΠ±Π°Ρ Π³Π»Π°Π΄ΠΊΠ°Ρ ΡΡΠ½ΠΊΡΠΈΡ {\displaystyle H: Mβ \mathbb {R} } H: Mβ \mathbb{R} Π½Π° ΡΠΈΠΌΠΏΠ»Π΅ΠΊΡΠΈΡΠ΅ΡΠΊΠΎΠΌ ΠΌΠ½ΠΎΠ³ΠΎΠΎΠ±ΡΠ°Π·ΠΈΠΈ {\displaystyle M} M ΠΌΠΎΠΆΠ΅Ρ ΠΈΡΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°ΡΡΡΡ, ΡΡΠΎΠ±Ρ ΠΎΠΏΡΠ΅Π΄Π΅Π»ΠΈΡΡ Π³Π°ΠΌΠΈΠ»ΡΡΠΎΠ½ΠΎΠ²Ρ ΡΠΈΡΡΠ΅ΠΌΡ. Π€ΡΠ½ΠΊΡΠΈΡ {\displaystyle H} H ΠΈΠ·Π²Π΅ΡΡΠ½Π° ΠΊΠ°ΠΊ Π³Π°ΠΌΠΈΠ»ΡΡΠΎΠ½ΠΈΠ°Π½ ΠΈΠ»ΠΈ ΡΠ½Π΅ΡΠ³Π΅ΡΠΈΡΠ΅ΡΠΊΠ°Ρ ΡΡΠ½ΠΊΡΠΈΡ. Π‘ΠΈΠΌΠΏΠ»Π΅ΠΊΡΠΈΡΠ΅ΡΠΊΠΎΠ΅ ΠΌΠ½ΠΎΠ³ΠΎΠΎΠ±ΡΠ°Π·ΠΈΠ΅ Π½Π°Π·ΡΠ²Π°ΡΡ ΡΠ°Π·ΠΎΠ²ΡΠΌ ΠΏΡΠΎΡΡΡΠ°Π½ΡΡΠ²ΠΎΠΌ. ΠΠ°ΠΌΠΈΠ»ΡΡΠΎΠ½ΠΈΠ°Π½ ΠΏΠΎΡΠΎΠΆΠ΄Π°Π΅Ρ ΡΠΏΠ΅ΡΠΈΠ°Π»ΡΠ½ΠΎΠ΅ Π²Π΅ΠΊΡΠΎΡΠ½ΠΎΠ΅ ΠΏΠΎΠ»Π΅ Π½Π° ΡΠΈΠΌΠΏΠ»Π΅ΠΊΡΠΈΡΠ΅ΡΠΊΠΎΠΌ ΠΌΠ½ΠΎΠ³ΠΎΠΎΠ±ΡΠ°Π·ΠΈΠΈ, ΠΈΠ·Π²Π΅ΡΡΠ½ΠΎΠΌ ΠΊΠ°ΠΊ ΡΠΈΠΌΠΏΠ»Π΅ΠΊΡΠΈΡΠ΅ΡΠΊΠΎΠ΅ Π²Π΅ΠΊΡΠΎΡΠ½ΠΎΠ΅ ΠΏΠΎΠ»Π΅.
Π‘ΠΈΠΌΠΏΠ»Π΅ΠΊΡΠΈΡΠ΅ΡΠΊΠΎΠ΅ Π²Π΅ΠΊΡΠΎΡΠ½ΠΎΠ΅ ΠΏΠΎΠ»Π΅ (ΡΠ°ΠΊΠΆΠ΅ Π½Π°Π·ΡΠ²Π°Π΅ΡΡΡ Π³Π°ΠΌΠΈΠ»ΡΡΠΎΠ½ΠΎΠ²ΡΠΌ Π²Π΅ΠΊΡΠΎΡΠ½ΡΠΌ ΠΏΠΎΠ»Π΅ΠΌ) ΠΏΠΎΡΠΎΠΆΠ΄Π°Π΅Ρ Π³Π°ΠΌΠΈΠ»ΡΡΠΎΠ½ΠΎΠ² ΠΏΠΎΡΠΎΠΊ Π½Π° ΠΌΠ½ΠΎΠ³ΠΎΠΎΠ±ΡΠ°Π·ΠΈΠΈ. ΠΠ½ΡΠ΅Π³ΡΠ°Π»ΡΠ½ΡΠ΅ ΠΊΡΠΈΠ²ΡΠ΅ Π²Π΅ΠΊΡΠΎΡΠ½ΠΎΠ³ΠΎ ΠΏΠΎΠ»Ρ ΡΠ²Π»ΡΡΡΡΡ ΠΎΠ΄Π½ΠΎΠΏΠ°ΡΠ°ΠΌΠ΅ΡΡΠΈΡΠ΅ΡΠΊΠΈΠΌ ΡΠ΅ΠΌΠ΅ΠΉΡΡΠ²ΠΎΠΌ ΠΏΡΠ΅ΠΎΠ±ΡΠ°Π·ΠΎΠ²Π°Π½ΠΈΠΉ ΠΌΠ½ΠΎΠ³ΠΎΠΎΠ±ΡΠ°Π·ΠΈΡ Ρ ΠΏΠ°ΡΠ°ΠΌΠ΅ΡΡΠΎΠΌ, Π½Π°Π·ΡΠ²Π°Π΅ΠΌΡΠΌ Π²ΡΠ΅ΠΌΡ. ΠΠ²ΠΎΠ»ΡΡΠΈΡ Π²ΠΎ Π²ΡΠ΅ΠΌΠ΅Π½ΠΈ Π·Π°Π΄Π°ΡΡΡΡ ΡΠΈΠΌΠΏΠ»Π΅ΠΊΡΠΎΠΌΠΎΡΡΠΈΠ·ΠΌΠ°ΠΌΠΈ. ΠΠ· ΡΠ΅ΠΎΡΠ΅ΠΌΡ ΠΠΈΡΠ²ΠΈΠ»Π»Ρ ΡΠ»Π΅Π΄ΡΠ΅Ρ, ΡΡΠΎ ΠΊΠ°ΠΆΠ΄ΡΠΉ ΡΠΈΠΌΠΏΠ»Π΅ΠΊΡΠΎΠΌΠΎΡΡΠΈΠ·ΠΌ ΡΠΎΡ ΡΠ°Π½ΡΠ΅Ρ ΡΠΎΡΠΌΡ ΠΎΠ±ΡΡΠΌΠ° Π² ΡΠ°Π·ΠΎΠ²ΠΎΠΌ ΠΏΡΠΎΡΡΡΠ°Π½ΡΡΠ²Π΅. ΠΠ½ΠΎΠΆΠ΅ΡΡΠ²ΠΎ ΡΠΈΠΌΠΏΠ»Π΅ΠΊΡΠΎΠΌΠΎΡΡΠΈΠ·ΠΌΠΎΠ², ΠΏΠΎΡΠΎΠΆΠ΄Π°Π΅ΠΌΡΡ Π³Π°ΠΌΠΈΠ»ΡΡΠΎΠ½ΠΎΠ²ΡΠΌ ΠΏΠΎΡΠΎΠΊΠΎΠΌ, ΠΎΠ±ΡΡΠ½ΠΎ Π½Π°Π·ΡΠ²Π°ΡΡ Π³Π°ΠΌΠΈΠ»ΡΡΠΎΠ½ΠΎΠ²ΠΎΠΉ ΠΌΠ΅Ρ Π°Π½ΠΈΠΊΠΎΠΉ Π³Π°ΠΌΠΈΠ»ΡΡΠΎΠ½ΠΎΠ²ΠΎΠΉ ΡΠΈΡΡΠ΅ΠΌΡ.
ΠΠ°ΠΌΠΈΠ»ΡΡΠΎΠ½ΠΎΠ²ΠΎ Π²Π΅ΠΊΡΠΎΡΠ½ΠΎΠ΅ ΠΏΠΎΠ»Π΅ ΡΠ°ΠΊΠΆΠ΅ ΠΏΠΎΡΠΎΠΆΠ΄Π°Π΅Ρ ΡΠΏΠ΅ΡΠΈΠ°Π»ΡΠ½ΡΡ ΠΎΠΏΠ΅ΡΠ°ΡΠΈΡ β ΡΠΊΠΎΠ±ΠΊΠ° ΠΡΠ°ΡΡΠΎΠ½Π°. Π‘ΠΊΠΎΠ±ΠΊΠ° ΠΡΠ°ΡΡΠΎΠ½Π° Π΄Π΅ΠΉΡΡΠ²ΡΠ΅Ρ Π½Π° ΡΡΠ½ΠΊΡΠΈΠΈ Π½Π° ΡΠΈΠΌΠΏΠ»Π΅ΠΊΡΠΈΡΠ΅ΡΠΊΠΎΠΌ ΠΌΠ½ΠΎΠ³ΠΎΠΎΠ±ΡΠ°Π·ΠΈΠΈ, ΡΠ°ΠΊΠΈΠΌ ΠΎΠ±ΡΠ°Π·ΠΎΠΌ ΠΏΡΠΈΠ΄Π°Π²Π°Ρ ΠΏΡΠΎΡΡΡΠ°Π½ΡΡΠ²Ρ ΡΡΠ½ΠΊΡΠΈΠΉ Π½Π° ΠΌΠ½ΠΎΠ³ΠΎΠΎΠ±ΡΠ°Π·ΠΈΠΈ ΡΡΡΡΠΊΡΡΡΡ Π°Π»Π³Π΅Π±ΡΡ ΠΠΈ.
phase: from any point on phase space, evolution is unique. it's kinda like initial data for a differential equation
[20181120]
why symplectic spaces?
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.
now figure out why is that interesting. lol
[20181110]
intuition  What is a symplectic form intuitively?  MathOverflow
https://mathoverflow.net/questions/19932/whatisasymplecticformintuitively/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?
Hamiltonian vector field  Wikipedia
Suppose that (M, Ο) is a symplectic manifold. Since the symplectic form Ο is nondegenerate, it sets up a fiberwiselinear isomorphism
Tweet from John Carlos Baez (@johncarlosbaez), at Nov 22, 22:01
<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  Wikipedia
Legendre transformation, named after AdrienMarie Legendre, is an involutive transformation on the realvalued convex functions
[20181110]
What moment map is (as a physical concept) in sympletic geometry  Mathematics Stack Exchange
[20181110]
classical mechanics  Intuition about Momentum Maps  Physics Stack Exchange
https://physics.stackexchange.com/questions/203653/intuitionaboutmomentummaps
[20181114]
sg.symplectic geometry  How to see the Phase Space of a Physical System as the Cotangent Bundle  MathOverflow
[20181119]
Legendre transformation  Wikipedia
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.
[20181118]
Symplectic integrator  Wikipedia
https://en.wikipedia.org/wiki/Symplectic_integrator
[20181119]
mathematical physics  Is there an analogue of configuration space in quantum mechanics?  Physics Stack Exchange
[20181118]
Applying RungeKutta method to circular  C++ Forum
http://www.cplusplus.com/forum/general/103744/
[20180913]
Chel_{of}_{the}_{sea} comments on ELI5: What is symplectic geometry?
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).
[20180813]
poisson brackets are related to symplectic geometry (explains phase space)
[20180813]
poisson brackets explained
https://www.quora.com/WhatisanintuitivewayofexplainingthePoissonbracket
{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} β 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 oneparameter family of symplectomorphisms (i.e., canonical transformations, areapreserving diffeomorphisms), with the time t being the parameter
[20190612]
need to add this to drill..
[20181115]
phase space: R^{3} x R^{3} β 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/whyisthelagrangianafunctiononthetangentbundle confusingβ¦
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 tβ¦qΛ(t) turns out to be the timederivative of the function tβ¦q(t). THAT is what that cryptic thing in your text is trying to say.
Spivak's Physics For Mathematicians,
configuration space: M (a Manifold). Phase space: cotangent bundle over M (T_{*} M). Also a manifold?? Yeah, ok T_{*} M is also a manifold with dimension twise as what M got.
right, and symplectic form is defined on the contagent bundle of configuration space! (or on a phase space?)
learn basic QED and QFT [[study]] [[qed]]
but this time, really really do exercises!
Feynman QED strange theory of light
takeaways
feynman diagrams are way of computing quantum amplitude. sum over all diagrams
to compute individual jump probabilities: use propagators solutions for free electron (Dirac) and photon (KleinGordon).
renormalization: meh, but only theory parameters depend on it, measurable stuff is not impacted
Zee, Feynman. QED: Strange Theory of Light and Matter
(long time ago) Tong, some notes in study/qft
[20181215]
symplectic geometry  Learn Anything
[20190618]
Energy drift  Wikipedia [[symplectic]]
https://en.wikipedia.org/wiki/Energy_drift
Energy drift  usually damping  is substantial for numerical integration schemes that are not symplectic, such as the RungeKutta family.
[20200118]
chakravala/Grassmann.jl: β¨LeibnizGrassmannCliffordβ© differential geometric algebra / multivector simplicial complex [[symplectic]]
https://github.com/chakravala/Grassmann.jl
The Grassmann.jl package provides tools for doing computations based on multilinear algebra, differential geometry, and spin groups using the extended tensor algebra known as LeibnizGrassmannCliffordHestenes geometric algebra.
[20181110]
Tangent bundle  Wikipedia https://en.wikipedia.org/wiki/Tangent_bundle [[symplectic]] [[drill]]
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 fΒ : M β N is a smooth function, with M and N smooth manifolds, its derivative is a smooth function DfΒ : TM β TN.
Symplectic geometry foundations
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/thefighttofixsymplecticg...
Intro to symplectic geom
For 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.
Tweet from John Carlos Baez (@johncarlosbaez), at Jan 21, 18:29 [[symplectic]] [[towatch]]
Symplectic geometry is like the evil twin of Euclidean geometry: instead of a dot product with vβ
w = wβ
v, we have one with vβ
w = wβ
v. But it's not really evil. Check out Jonathan Lorand's talk!
<https://twitter.com/johncarlosbaez/status/1087416965012942849 >
 public document at doc.anagora.org/symplectic
 video call at meet.jit.si/symplectic