On the computational of Poisson Geometry

on the computational aspect of Poisson geometry, this presentation delves into the introductory aspects as well as application sides.

1. On Computational Poisson Geometry
Pablo Suarez-Serrato,
Geometric Intelligence Lab, UC Santa Barbara
Instituto de Matematicas UNAM
October 9th, 2024

2. Simeon Denis Poisson, 1809

3. Newton's Second Law

4. Harmonic Oscillator
Movement Equation:
x=c x;c>0
IResting StateIPerturbations

Fk=kx, Hooke

&

Fk=mx, Newton

5. Harmonic Oscillator
Movement Equation:
x=x;(c= 1)
?
y
System of Equations:
_q=p
_p=q

q=x,p= _x

Phase Space:

6. Harmonic Oscillator, it's a Hamiltonian System
System of Equations:
_q=p
_p=q
IH(q;p) =
1
2
q
2
+
1
2
p
2
_q=
@H
@p
_p=
@H
@q

7. Harmonic Oscillator, it's a Hamiltonian System
IH(q;p) =
1
2
q
2
+
1
2
p
2
_q=
@H
@p
_p=
@H
@q
Observe that, ifX= (q;p)
>
, then
_
X=JrH
ith
J=

0 1
1 0

and rH=

@H
@q
@H
@p
!

8. Symplectic structure for the Harmonic Oscillator
_
X=JrH;J=

0 1
1 0

Given two vector eldsu= (u1;u2)
>
yv= (v1;v2)
>
,ui;vi2C
1
R
2,
!(u;v) =u
>
Jv=u1v2u2v1
Jinduces a diferential 2-form.

9. Poisson bracket for the Harmonic Oscillator
!(u;v) =u
>
Jv=u1v2u2v1
IDenef;g:C
1
R
2C
1
R
2!C
1
R
2
ff;gg!:=!(rf;rg) =
@f
@p
@g
@q

@f
@q
@g
@p
IThen,
_q=fH;qg!; _p=fH;pg!
IIn general, for an 'observable' :
df
dt
=fH;fg!;f=f(q;p)

10. Poisson Bracket onR
2
f;g:C
1
R
2C
1
R
2!C
1
R
2
IR-linear.
IAntisimetric:ff;gg=fg;fg
IJacobiIdentity:
ff;fg;hgg=fff;gg;hg+fg;ff;hgg
ILeibnizRule:
ff;ghg=g ff;hg+h ff;gg
Note:C
1
R
2:=ff:M!R

fsmoothg

11. Hamiltonian Systems inR
2n
=f(q1;:::;qn;p1;:::;pn)g
GivenH2C
1
R
2n:
_qi=
@H
@pi
; _pi=
@H
@qi
IDenef;g:C
1
R
2nC
1
R
2n!C
1
R
2npor
ff;gg:=
n
X
i=1
@f
@pi
@g
@qi

@f
@qi
@g
@pi
IThen,
_qi=fH;qig; _pi=fH;pig

12. Poisson Bracketf;g:C
1
M
C
1
M
!C
1
M
IR-lineal.
IAntisymmetric:ff;gg=fg;hg
IJacobi Identity:
ff;fg;hgg=fff;gg;hg+fg;ff;hgg
ILeibniz Rule:
ff;ghg=gff;hg+hff;gg

13. Example
InR
3
x, given
:R
3
!R
3
; (x) =

1(x); 2(x); 3(x)

>
IDene
ff;gg =

;rf rg

IJacobi Identity:

;rot

= 0
Note:ff;gg =rf
>
;rg, where
=
0
@
0 3 2
30 1
2 10
1
A

14. Poisson Tensors
^
2
TM3 such that [[ ;]]SN= 0:
Jacobi identity inR
n
=fx= (x
1
; : : : ;x
n
)g:

is
@
jk
@x
s
+
js
@
ki
@x
s
+
ks
@
ij
@x
s
= 0,
=
1
2

ab@
@x
a^
@
@x
b.

15. Brackets$Tensors
Poissonf;g ! s.t. [[;]]SN= 0:
ff;gg= (df;dg)

16. Foliations Induced by Vector Fields
IExistence and UniquenessIFoliation by
Integral Curves

17. Vector Fields vs Distributions
IVector FieldIDistribution

18. Foliations by Singular Distributions
IStefan-SussmanIFoliation by Integral
Manifolds
R
n
3p7! DpTpR
n
,
Dpsubspace of tangent vectors atp

19. Poisson Structure$Symplectic Foliation
Poisson tensor:
D

:=fXf

f2C
1
M
g,
withXf(g) =ff;gg.
Then:
D

is integrable.
#
we obtain a symplecticfoliation.

20. Example :so(3)
I 1=x1, 2=x2, 3=x3
I =
0
@
0 x3x2
x30 x1
x2x10
1
A;ff;gg =

;rf rg

21. Example :sl(2)
I 1=x1, 2=x2, 3=x3
I =
0
@
0x3x2
x3 0 x1
x2x1 0
1
A;ff;gg =

;rf rg

22. Examples, 2D
IEvery closed oriented surface is symplectic, hence Poisson.
I(Radko) Classication oftopologically stablePoisson
structures on surfaces (tensor vanishes linearly on a disjoint
union of simple closed curves). Explicit determination of the
moduli space for 2{sphere.

23. Examples, 3D
I(Lickorish, Novikov, Zieschang) Every closed smooth oriented
3-manifold admits a foliation by surfaces, hence a regular rank
2 Poisson structure.
I(Evangelista-TorresOrozco-S.-Vera) Every closed smooth
oriented 3-manifold admits a generic rank 2 Poisson structure
with Bott-Morse singularities (circles and points).

24. Examples, 4D
I(GarcaNaranjo-S.-Vera) Every smooth closed orientable
4{manifold admits a generic rank 2 Poisson structure with
broken Lefschetz singularities.

25. Examples, 4D
I(S.-TorresOrozco) Every smooth closed orientable 4{manifold
admits a generic rank 2 Poisson structure withwrinkled
singularities.

26. Poisson Structures
IHamiltonian system:
_x=fH;xg;x2R
n
IHamiltonian vector elds:
Xh=idh
ICasimir Function,K2C
1
M
such that
XK= 0
IPoisson vector elds:
LZ = 0:

27. Modular Field
I(M;;) orientable Poisson manifold:
LX = divX
IModular Field:
Z:=h7!divXh
+
LZ = 0 y Z
f
=Z

X
lnjfj
Denition 1
An orientable Poisson manifold (M;) isunimodularif it admits a
volume form invariant under the ow of every Hamiltonian eld.

28. References
IPoisson structures on smooth 4-manifolds, Garca-Naranjo,S., Vera,
Lett. Math. Physics, 105, No.11, (2015) 1533{550.
IPoisson structures on wrinkled brations, Torres Orozco,S., Bull.
Mexican Math. Soc., 22, No.1 (2016), 263{280.
IOn Bott-Morse Foliations and their Poisson Structures in Dimension
3, Evangelista- Alvarado,S., Torres Orozco, Vera,
Jour. of Singularities, 19 (2019), 19{33.
IOn Computational Poisson Geometry I: Symbolic Foundations,
Evangelista-Alvarado, Ruz-Pantaleon,S.,
Jour. Geometric Mechanics 13(4), (2021) 607{628.
IOn Computational Poisson Geometry II: Numerical Methods,
Evangelista-Alvarado, Ruz-Pantaleon,S.,
Jour. Computational Dynamics 8(3) (2021) 273{307.