Koszul/Souriau Fisher Metric Spaces & Optimization by ... · Koszul proved that these metrics are invariant by all automorphisms of the convex cones. 5 / Paul LEVY (general use of

www.thalesgroup.com

Thales Air Systems Date

Koszul/Souriau Fisher Metric Spaces & Optimization by Maximum Entropy:

Hessian Information Geometry,Lie Group Thermodynamics

& Poincaré-Marle-Souriau EquationFrédéric BARBARESCO, THALES AIR SYSTEMS

Senior Scientist & Advanced Studies ManagerAdvanced Radar Concepts Dept.

21/11/2014

2 /2 / Preamble

Problems to be solved by THALES for Radar Applications

Problem 1 : How to define density of probability for covarian ce matrices of stationary time series (THPD matrix: To eplitz Hermitian Positive Definite matrix)

Problem 2 : How to define « Ordered Statistics » for covarianc e matrices, knowing that there is no « total orders » f or these matrices


( )Toeplitz

Definite Positive ,0det

Hermitian

??)/(

ξξξ

ξξξξ

>=

=+

p

nξξξξξξξ

≤≤≤>−⇒<.... :Order Global noBut

Definite Positive 0 :Order Local

21

1221

3 /3 / Preamble

Solution to Problem 1: Koszul/Souriau Solution of Maximum Entropy

Entropy as Legendre Transform of Generalized Characteristic Function (Laplace Transform on Convex Cone with Inner Product given by Cartan-Killing form)

Density of Probability as Souriau Covariant Solution of Maximum Entropy

Solution to Problem 2: Frechet Median Barycenter

Metric given by Koszul Hessian Geometry (hessian of Entropy) & Souriau Lie Group Thermodynamics (metric defined by symplectic cocycle and Geometric temperature)

Geometric Median by Fréchet barycenter in Metric space, solved by Karcher Flow


( )

( )∫

−

−

Θ−

Θ−

=

*Ω

ξ,

ξ,

dξe

ep

ξ

ξ

ξ ξ1

1

)()(1 ξ−Θ=x

dx

xdx

)()(

Φ=Θ=ξξξξξ ξ dp∫Ω

=*

)(.

∫Ω

−−=Φ*

,log)( ξξ dex x ( ))(, yxadadTryx θ=

∑=

=n

iimedian dMinArg

1

),( ξξξξ

2

,2

2

2*

log)(log

)(x

de

x

pExI

x

x

∂

∂=

∂∂−=

∫Ω

− ξξ

ξ

ξ

4 /4 /


Koszul-Vinberg Characteristic Function

François Massieu in 1869 demonstrated that some the rmal properties of physical systems could be derived fro m “characteristic functions”.

This idea was developed by Gibbs and Duhem with the notion of potentials in thermodynamics, and introduced by Poi ncaré in probability.

We will study generalization of this concept by

Jean-Louis Koszul in Mathematics

Jean-Marie Souriau in Statistical Physics.

The Koszul-Vinberg Characteristic Function (KVCF) on convex cones will be presented as cornerstone of “Informati on Geometry” theory:

defining Koszul Entropy as Legendre transform of minus the logarithm of KVCF (their gradients defining mutually inverse diffeomorphisms)

Fisher Information Metrics as hessian of these dual functions.

Koszul proved that these metrics are invariant by al l automorphisms of the convex cones.

5 /5 /

Paul LEVY(general use of characteristic function in

Probability)

Henri POINCARE(Introduction of characteristic function Ψ

in Probability)ψφφ logor == eψ

1869 François MASSIEU(introduction of characteristic function in Thermodynamic:

Gibbs-Duhem Potentials)

( )TTS

/1.

1

∂∂−= φφ

« je montre, dans ce mémoire, que toutes les propri étés d’un corps peuvent se déduire d’une fonction unique , que

j’appelle la fonction caractéristique de ce corps»

Koszul Characteristic: Massieu/Poincaré/Levy/Balian

Roger BALIAN(metric for quantum states by

hessian metric from Von-Neumann Entropy)

Roger Balian, 1986DISSIPATION IN MANY-BODY SYSTEMS:A GEOMETRIC APPROACH BASED ON

INFORMATION THEORY

XDXFDS ˆ,ˆ)ˆ()ˆ( −=

XTrXF ˆexpln)ˆ( =[ ]DdDdTrSdds ˆln.ˆ22 =−=

6 /6 / Thermodynamic Duhem-Massieu Potentials

Pierre Duhem Thermodynamic Potentials

Duhem P., « Sur les équations générales de la Thermodynamique », Annales Scientifiques de l’Ecole Normale Supérieure, 3e série, tome VIII, p. 231, 1891

“Nous avons fait de la Dynamique un cas particulier de la Thermodynamique, une Science qui embrasse dans des principes communs tous les changements d’état des corps, aussi bien les changements de lieu que les changements de qualités physiques “

four scientists were credited by Duhem with having carried out “the most important researches on that subject”:

F. Massieu had managed to derive Thermodynamics from a “characteristic function and its partial derivatives”

J.W. Gibbs had shown that Massieu’s functions “could play the role of potentials in the determination of the states of equilibrium” in a given system.

H. von Helmholtz had put forward “similar ideas”

A. von Oettingen had given “an exposition of Thermodynamics of remarkable generality” based on general duality concept in “Die thermodynamischen Beziehungen antithetisch entwickelt“, St. Petersbur g 1885

WTSEG +−=Ω )(

7 /7 /


Influence of Massieu on Poincaré

[M. Massieu showed that, if we make choice forindependent variables of v and T or of p and T,there is a function, moreover unknown, of whichthree functions of variables, p, U and S in the firstcase, v, U and S in the second, can be deductedeasily. M. Massieu gave to this function, the formof which depends on the choice of variables, nameof characteristic function.]

[Because from functions of M. Massieu, wecan deduct the other functions of variables, allthe equations of the Thermodynamics can bewritten not so as to contain more than thesefunctions and their derivatives; it will thus resultfrom it, in certain cases, a large simplification.We shall see soon an important application ofthese functions.]

2nd edition of Poincaré Lecture on « Thermodynamics »

8 /8 /

1908

Massieu Characteristic Function by Henri Poincaré

9 /9 /

1912

Characteristic Function by Henri Poincaré in Probability

H. Poincaré has introduced « Characteristic Function » in Probability with LAPLACE TRANSFORM

not with FOURIER TRANSFORM

10 /10 /


Koszul-Vinberg Characteristic Function

Jean-Marie Souriau has extended the Characteristic Function in Statistical Physics:

looking for other kinds of invariances through co-adjoint action of a group on its momentum space

defining physical observables like energy, heat and momentum as pure geometrical objects.

In covariant Souriau model, Gibbs equilibriums stat es are indexed by a geometric parameter, the Geometric Temperature , with values in the Lie algebra of the dynamical Galileo/Poincar é groups, interpreted as a space-time vector (a vector valued temperature of Planck), giving to the metric tensor a null Lie der ivative.

Fisher Information metric appears as the opposite o f the derivative of Mean “Moment map” by geometric temperature, equiva lent to a Geometric Capacity or Specific Heat .

We will synthetize the analogies between both Koszul and Souriau models, and will reduce their definitions to the ex clusive “Inner Product” selection using symmetric bilinear “Cartan-K illing form” (introduced by Elie Cartan in 1894).

11 /11 / Elie Cartan by Henri Poincaré


« A l’Exception d’Henri Poincaré qui écrivit peu avant sa mort un

rapport sur les travaux d’Elie Cartan à l’occasion de la

candidature de celui-ci à la Sorbonne, les mathématiciens

français ne voyaient pas l’importance de l’œuvre. »

Paulette LibermannLa géométrie différentielle d’Elie Cartan à Charles Ehresmann et

André LichnerowiczGéométrie au XXième siècle,

HERMANN, 2005

12 /12 / Cartanian Filiation: J.L Koszul & J.M. Souriau


2

2

2

2 log)(log

x

(x)

x

pE x

∂∂=

∂∂− Ωψξ

ξ

Ω∈∀= ∫Ω

−Ω xdex x )(

*

, ξψ ξ

( ) ( ) )(.,, 22121 1ZadQZZfZZf Z+=β

[ ]( ) ( ) ( )( ) [ ],..Im , , ,,, 212121 ββ βββ =∈∀∈∀= adZZZZfZZg g

*gg ∈

∂∂−=∈∈ − QQ

CfKereTemperatur CapacityHeat , , :)(β

ββ β

∫∈

−−=*Ωξ

xxx deep ξξ ξξ ,, /)(

GeDf toassociated cocycle with ))(( θθ=( ) [ ] ( )

MapMoment Souriau with

,,.)(),( 2121,2,1

µµξξσ ZZfZZZZ MM +=

( ) ( )InvolutionCartan , where

),( with )(,,

g

adadTryxByxByx yx

∈

=−=

ηη

βψ

∂∂−=

∂∂= Ω Q

x

(x)IFisher 2

2 log

Koszul FormsKoszul Characteristic Function

Koszul Hessian Metric

Jean-Louis Koszul

[ ] [ ]( ) [ ]( )2121 ,,,,, ZZfZZg βββ ββ =

(x)d Ω= ψα log

Ω== ψα log2dDg

Jean-Marie Souriau

Souriau Moment MapSouriau Geometric Temperature/Heat CapacitySouriau/Fisher Metric from Symplectic Cocycle

Elie CARTAN

Leçons sur les invariants intégraux, Hermann, 1922

La théorie des groupes finis et continus et la géométrie différentielle (Written by J. Leray)

13 /13 / Jean-Marie Souriau and his Lie Group Thermodynamics


J.M. Souriau, “Sur la Stabilité des Avions,” ONERA Publ., 62, vi+94, 1953

Engines could be positionned everywhereand a stable command could be defined

J.M. Souriau, Calcul linéaire, P.U.F., Paris, 1964.

Multilinear AlgebraLe Verrier-Souriau Algorithm

Computation of Matrix Characteristic Equation

J.M. Souriau, Structure des systèmes dynamiques, Dunod,

Paris, 1970Symplectic Geometry

Structure of Classical & Quantum Mechanics

- Moment Map- Geometric Noether

Theorem- General Barycentric

Theorem- Mass = Symplectic

cohomology of the action of the Galilean group (not for Poincaré Group in Relativity)

J.M. Souriau, Définition covariante des équilibres thermodynamiques, Supp. Nuov. Cimento,1,4, p.203-216, 1966J.M. Souriau, Thermodynamique et géométrie. In diff. Geo. methods in mathematical physics, II,

vol. 676 of Lecture Notes in Math., pages 369-397. Springer, 1978Lie Group Thermodynamics

(Gibbs Equilibrium is not covariant by Gallileo/Poi ncaré Groups)- Geometric Temperature (Vector in Lie Algebra of Dynamic al Group) & Geometric Entropy- FISHER METRIC DEFINED THROUGH SYMPLECTIC COCYCLE OF DYNAMICAL GROUP- FISHER METRIC = GEOMETRIC HEAT CAPACITY

AIRBUS BOEING

J.M. Souriau, Les groupes comme universaux, Géométrie au XXième siècle, Hermann, 2005J.M. Souriau, Grammaire de la Nature, 2007

Souriau theorem revisited by AIRBUS/BOEING

« La masse totale d’un système dynamique isolé est l a classe de cohomologie du défaut d’équivariance de l’applicati on moment »

14 /14 / Souriau Books


http://www.jmsouriau.com/structure_des_systemes_dynamiques.htm

Chapter 4 on « Statistical Mechanics »http://www.jmsouriau.com/Publications/JMSouriau-SSD-Ch4.pdf

15 /15 / Souriau Model for Gaussian Law


J.M. Souriau, Structure des systèmes dynamiques, Ch apitre 4 « Mécanique Statistique »

Souriau « Geometric Temperature » idea come from his book « Calcul Linéaire » (chap. on «Multilinear Algebra »

16 /16 / Souriau Mechanical Statistics


J.M. Souriau, Structure des systèmes dynamiques, Chapitre 4 « Mécanique

Statistique »

17 /17 / Covariant Souriau Statistical Mechanics


J.M. Souriau, Structure des systèmes dynamiques, Chapitre 4 « Mécanique

Statistique »

Classical Gibbs Equilibrium is not covariant according to

Dynamic Group of Mechanics (Gallileo Group and Poincaré

Group) !!!

18 /18 /

Hessian Geometry by J.L. Koszul

Hirohiko Shima Book, « Geometry of Hessian Structures », world Scientific Publishing 2007, dedicated to Jean-Louis Koszul

Hirohiko Shima Keynote Talk at GSI’13

http://www.see.asso.fr/file/5104/download/9914

Prof. M. Boyom tutorial :

http://repmus.ircam.fr/_media/brillouin/ressources/ une-source-de-nouveaux-invariants-de-la-geometrie-de-l -information.pdf

Jean-Louis Koszul

J.L. Koszul, « Sur la forme hermitienne canonique de s espaces homogènes complexes », Canad. J. Math. 7, pp. 562-576 ., 1955J.L. Koszul, « Domaines bornées homogènes et orbites de groupes de transformations affines », Bull. Soc. Math. France 8 9, pp. 515-533., 1961J.L. Koszul, « Ouverts convexes homogènes des espace s affines », Math. Z. 79, pp. 254-259., 1962J.L. Koszul, « Variétés localement plates et convexi té », Osaka J. Maht. 2, pp. 285-290., 1965J.L. Koszul, « Déformations des variétés localement plates », .Ann Inst Fourier, 18 , 103-114., 1968

Jean-Louis Koszul and his Hessian Geometry

www.thalesgroup.com


Koszul Information Geometry

20 /20 / Projective Legendre Duality and Koszul Characteristic Function

LEGENDRE TRANSFORM FOURIER/LAPLACE TRANSFORM

ENTROPY=LEGENDRE(- LOG[LAPLACE])

Ψ=Φ−= 22 log ddg

INFORMATION GEOMETRY METRIC

Sddg 2*2* =Ψ=

∫Ω

−−=Φ−=Ψ*

,log)(log)( dyexx yx)(,)( *** xxxx Ψ−=Ψ

ξξξ dpp xx∫Ω

−=Ψ*

)(log)(*

Φ(x)x,ξ

Ω

ξ,xξ,xx edξeep

*

+−−− == ∫/)(ξ

∫Ω

=*

)(.* ξξξ dpx x

Legendre Transform of minus logarithm of

characteristic function (Laplace transform) =

ENTROPY !!!

ds2=d2ENTROPY ds2=-d2LOG[LAPLACE]

21 /21 / Koszul-Vinberg Characteristic Function/Metric of convex cone

J.L. Koszul and E. Vinberg have introduced an affinely invariant Hessian metric on a sharp convex cone through its c haracteristic function.

is a sharp open convex cone in a vector space of finite dimension on (a convex cone is sharp if it does not contain any full straight line).

is the dual cone of and is a sharp open conv ex cone.

Let the Lebesgue measure on dual spa ce of , the following integral:

is called the Koszul-Vinberg characteristic function

Ω ER

*Ω Ωξd *E E

Ω∈∀= ∫Ω

−Ω xdex x )(

*

, ξψ ξ

22 /22 / Koszul-Vinberg Characteristic Function/Metric of convex cone

Koszul-Vinberg Metric :

We can define a diffeomorphism by:

with

When the cone is symmetric, the map is a biject ion and an isometry with a unique fixed point (the manifold is a Riemannian Symmetric Space given by this isometry):

, and

is characterized by

is the center of gravity of the cross section of :

Ω= ψlog2dg

[ ] ( )

loglog

2

1loglog)(log

2u

2

22

∫∫∫∫

∫∫

∫−

+==dudv

dudvdd

du

duddudxd

vu

vuv

u

uu

u ψψ

ψψψψ

ψ

ψψψψ

)(log* xdx x Ω−=−= ψα

)()(),(0

tuxfdt

dxfDuxdf

tu +==

=

Ω xx α−=*

xx =** )( nxx =*, cstexx =ΩΩ )()( *

*ψψ

*x nyxyyx =Ω∈= ,,/)(minarg ** ψ

*x nyxy =Ω∈ ,,* *Ω

∫∫Ω

−

Ω

−=**

,,* /. ξξξ ξξ dedex xx

23 /23 / Koszul Entropy via Legendre Transform

we can deduce “Koszul Entropy” defined as Legendre T ransform of minus logarithm of Koszul-Vinberg characteristic function :

with and where

Demonstration: we set

Using

and

we can write:

and

)(,)( *** xxxx Φ−=Φ Φ= xDx* ** Φ=

xDx

)(log)( xx Ω−=Φ ψ

Ω∈∀= ∫Ω

−Ω xdex x )(

*

, ξψ ξ

∫∫Ω

−

Ω

−=**

,,* /. ξξξ ξξ dedex xx

∫∫Ω

−

Ω

−Ω −==−

**

,,* /,)(log, ξξξψ ξξ dedehxdhx xxh

∫∫Ω

−

Ω

−−=−**

,,,* /.log, ξξ ξξξ dedeexx xxx

∫∫∫∫

∫∫∫

Ω

−

Ω

−−

Ω

−

Ω

−

Ω

−

Ω

−

Ω

−−

−

=Φ

+−=Φ

****

***

,,,,,**

,,,,**

/.loglog.)(

log/.log)(

ξξξξ

ξξξ

ξξξξξ

ξξξξ

dedeededex

dededeex

xxxxx

xxxx

24 /24 / Koszul-Vinberg Characteristic Function Legendre Transform


−=Φ

=

−

=Φ

−=Φ

−

=Φ

+−=Φ−=Φ

∫∫∫

∫∫

∫∫

∫∫

∫

∫∫

∫

∫∫∫∫

∫∫∫

ΩΩ

−

−

Ω

−

−

ΩΩ

−

−

ΩΩ

−

−−

ΩΩ

−

−

Ω

−

ΩΩ

−

−−

Ω

−

Ω

−

Ω

−−

Ω

−

Ω

−

Ω

−

Ω

−

Ω

−−

*

**

*

*

*

*

*

*

*

*

*

*

****

***

,

,

,

,**

,

,

,

,,

,

,,**

,

,,,**

,,,,,**

,,,,***

log.)(

1 with .log.log)(

.loglog)(

/.loglog.)(

log/.log)(,)(

ξξξ

ξξ

ξξ

ξξ

ξ

ξξ

ξ

ξξξξ

ξξξ

ξ

ξ

ξ

ξ

ξ

ξ

ξ

ξξ

ξ

ξξ

ξ

ξξξ

ξξξξξ

ξξξξ

dde

e

de

ex

dde

ed

de

eed

de

edex

dde

eedex

dedeededex

dededeexxxx

x

x

x

x

x

x

x

xx

x

xx

x

xxx

xxxxx

xxxx

25 /25 /

[ ]( ) [ ]

[ ]∫Ω

Φ=Φ≥Φ⇒

Φ−≥Φ

Φ≤Φ⇒Φ

*

*

*

)()()()(

)(,)( :Transform Legendre

)(conv. :Ineq.Jensen

***

***

**

ξξξξ

ξξ

Edpx

xxxx

EE

x

Koszul Entropy via Legendre Transform

We can then consider this Legendre transform as an entropy, that we could named “ Koszul Entropy ”:

With

and

ξξξξξξ ξ

ξ

ξ

ξ

dppdde

e

de

exxx

x

x

x

∫∫

∫∫ Ω

Ω

−

−

ΩΩ

−

−

−=−=Φ*

*

*

*

)(log)(log,

,

,

,*

Φ(x)x,ξdξex,ξ

Ω

ξ,xξ,xx eedξeep *Ω

ξ,x

*

+−∫−−

−− ===−

∫log

/)(ξ

( )ξξξξξξ dedξ.edpΦDx

ξΦ

Ω

Φ(x)x,ξxx

*

*

−

Ω

+−

Ω∫∫∫ ====

**

.)(.*

[ ]

1log)()(

loglog)(

*

*

*

*

*

*

*

)()(

)()(,

=⇒−Φ=Φ

−=−=Φ

∫∫

∫∫

Ω

Φ−

Ω

Φ−

Ω

Φ+Φ−

Ω

−

ξξ

ξξ

ξξ

ξξ

dedexx

dedex xx

[ ] [ ]( )ξξ

ξξξξξξ

ξξξξξξ

ξξ ξξ

EE

dpdp

xdpdpp

eep

xx

xxx

xxx

**

**

***

*)()(,

)(or

)(.)()( ifonly and if

)()()()()(log

)(loglog)(log

**

**

*

Φ=Φ

Φ=Φ

Φ=Φ=−⇒

Φ−===

∫∫

∫∫

ΩΩ

ΩΩ

Φ−Φ+−

26 /26 / Barycentre & Koszul Entropy


[ ] [ ]( )ξξ EE *)(*

Φ=Φ

[ ])(,)( *** xxxSupxx

Φ−=Φ


ξΦ

Ω

Φ(x)x,ξxx

*

*

−

Ω

+−

Ω∫∫∫ ====

**

.)(.*

Φ=Φ ∫∫

ΩΩ **

)(.)()( ** ξξξξξξ dpdp xx

Φ(x)x,ξdξex,ξ

Ω


ξ,x

*

+−∫−−

−− ===−

∫log

/)(ξ

Barycenter of Koszul Entropy = Koszul Entropy of Bary center

27 /27 / Koszul metric & Fisher Metric

To make the link with Fisher metric given by matrix , we can observe that the second derivative of is given by:

We could then deduce the close interrelation betwee n Fisher metric and hessian of Koszul-Vinberg characteristic logarith m.

[ ]

2

2

2

2

2

2

2

2

2

2

2

2

*

log)(log)(

,)()(log

,)()()(log

x

(x)

x

(x)

x

pExI

x

(x)

x

xx

x

p

xxp

x

x

x

∂∂=

∂Φ∂−=

∂∂−=⇒

∂Φ∂=

∂−Φ∂

=∂

∂

−Φ=Φ−=

Ωψξ

ξξ

ξξξ

ξ

2

2

2

2 log)(log)(

x

(x)

x

pExI x

∂∂=

∂∂−= Ωψξ

ξ

FISHER METRIC (Information Geometry) = KOSZUL HESSIAN METRIC (Hessian Geometry)

)(xI)(log ξxp

28 /28 / Koszul Metric and Fisher Metric as Variance

We can also observed that the Fisher metric or hess ian of KVCF logarithm is related to the variance of :


ξ

∫∫

∫−

−− −=

∂∂

⇒=*

*

* Ω

ξ,x

Ω

ξ,xΩ

Ω

ξ,xΩ dξe

dξex

(x)Ψdξe(x)Ψ .

1logloglog ξ

+−

−=

∂∂

∫∫∫∫

−−−

−

2

222

2

...1log

***

*

Ω

ξ,x

Ω

ξ,x

Ω

ξ,x

Ω

ξ,x

Ω dξedξedξe

dξex

(x)Ψ ξξ

2

2

2

22

2

)(.)(...log

−=

−=∂

∂∫∫∫

∫∫

∫−

−

−

−

***

*

*

*

Ω

x

Ω

x

Ω

Ω

ξ,x

ξ,x

Ω

Ω

ξ,x

ξ,xΩ dξpdξpdξ

dξe

edξ

dξe

e

x

(x)Ψ ξξξξξξ

[ ] [ ] )(log)(log

)( 222

2

2

2

ξξξψξξξξ VarEE

x

(x)

x

pExI x =−=

∂∂

=

∂∂

−= Ω

29 /29 / New Definition of Maximum Entropy Density

How to replace by mean value of , in :

with

Legendre Transform will do this inversion by invers ing

We then observe that Koszul Entropy provides density of Maximum Entropy with this general definition of den sity:

with and

where and


( )

( )∫

−

−

Θ−

Θ−

=

*Ω

ξ,

ξ,

dξe

ep

ξ

ξ

ξ ξ1

1

)( )(1 ξ−Θ=xdx

xdx

)()(

Φ=Θ=ξ

ξξξξ ξ dp∫Ω

=*

)(. ∫Ω

−−=Φ*

,log)( ξξ dex x

x ξ )( *x=ξ

∫Ω

−

−

=

*

,

,

)(ξ

ξ ξ

ξ

de

ep

x

x

x ∫Ω

=*

)(. ξξξξ dpx

dx

xd )(Φ=ξ

30 /30 /


Cartan-Killing Form and Invariant Inner Product

It is not possible to define an ad(g)-invariant inn er product for any two elements of a Lie Algebra, but a symmetric bili near form, called “ Cartan-Killing form ”, could be introduced (Elie Cartan PhD 1894)

This form is defined according to the adjoint endom orphism of that is defined for every element of with the help of the Lie bracket:

The trace of the composition of two such endomorphi sms defines a bilinear form, the Cartan-Killing form :

The Cartan-Killing form is symmetric:

and has the associativity property:

given by:

xadg x g[ ]yxyad x ,)( =

( )yxadadTryxB =),(),(),( xyByxB =

[ ]( ) [ ]( )zyxBzyxB ,,,, =

[ ]( ) [ ]( ) [ ]( )[ ]( ) [ ]( ) [ ]( )zyxBadadadTrzyxB

adadadTradadTrzyxB

zyx

zyxzyx

,,,,,

,,, ,

==

==

31 /31 /



Elie Cartan has proved that if is a simple Lie algebra (the Killing form is non-degenerate) then any invariant symmetric bilinear form on is a scalar multiple of the Cartan-Killing form.

The Cartan-Killing form is invariant under automorp hismsof the algebra :

To prove this invariance, we have to consider:

Then

g

g

)(gAut∈σg

( ) ( )yxByxB ,)(),( =σσ

[ ] [ ] [ ] [ ]1

)(

1

rewritten

),()(,)(

)(),(,

−

−

=

=⇒

==

σσ

σσσσ

σσσ

σ oo xx adad

zxzxyz

yxyx

( ) ( ) ( )( ) ( ) ),()(),(

)(),( 1)()(

yxBadadTryxB

adadTradadTryxB

yx

yxyx

==

== −

σσσσσσ σσ oo

32 /32 /



A natural G-invariant inner product could be introd uced by Cartan-Killing form:

Cartan Generating Inner Product: The following Inner product defined by Cartan-Killing form is invariant by auto morphisms of the algebra

where is a Cartan involution (An involut ion on is a Lie

algebra automorphism of whose square is equal to the

identity).

( ))(,, yxByx θ−=g∈θ g

θ g

33 /33 /


From Cartan-Killing Form to Koszul Information Metric

( )

( )InvolutionCartan , with

)(,,

Form KillingCartan

),(

g

yxByx

adadTryxB yx

∈−=

−

=

θθ Ω∈∀−=Φ ∫

Ω

− xdex x log)(

Function sticCharacteri Koszul

*

, ξξ

∫

∫

∫

−

−

Ω

Ω

=

=

−=Φ

Φ−=Φ

*Ω

ξ,x

ξ,x

x

x

xx

dξe

ep

dpx

dppx

xxxx

)(

Density Koszul

)(.with

)(log)()(

)(,)(

Entropy Koszul

*

*

*

**

***

ξ

ξξξ

ξξξ

2

,2

2

2

2

2

*

log

)(

)(log)(

Metric Koszul

x

de

x

(x)xI

x

pExI

x

x

∂

∂=

∂Φ∂−==

∂∂−=

∫Ω

− ξ

ξ

ξ

ξ

34 /34 / Koszul Density: Application for SPD matrices

We can then named this new density as “ Koszul Density ”:

With

Φ(x)x,ξdξex,ξ

Ω


ξ,x

*

+−∫−−

−− ===−

∫log

/)(ξ


ξΦ

Ω

Φ(x)x,ξxx

*

*

−

Ω

+−

Ω∫∫∫ ====

**

.)(.*

( ) ( )[ ] ( )∫

Ω

−−++−

===−

*

1

).(. with det)( 1detlog2

1

ξξξξξαξ ξξααξdpeep x

Trxn

xTr

x

( )

+=+=−==

==

∈∀=

−Ω

+−

Ω=Ω=

Ω

−Ω ∫

1*

2

1

dual-self )(,

,

2

1detlog

2

1log

)(det)(

)(,,,

**

xn

xdn

dx

Ixdex

RSymyxxyTryx

n

n

xyTryx

x

n

ψξ

ψξψ ξ

35 /35 / Relation of Koszul density with Maximum Entropy Principle

The density from Maximum Entropy Principle is given by:

If we take such that:

Then by using the fact that with equality if and only if , we find the following:


=

=

−

∫

∫∫

Ω

Ω

Ω*(.)

*

*

* )(.

1)(

such )(log)(xdp

dp

dppMaxx

x

xxpx ξξξ

ξξξξξ

∫−−−−

−

== ∫*Ω

ξ,x

*

dξex,ξ

Ω

ξ,xξ,xx edξeeq

log

/)(ξ

−−==

==

∫

∫∫∫

Ω

−∫−−

−−

−

*

,log

log,log)(log

1/).(

ξξξ

ξ

ξ dexeq

dξedξedξq

xdξex,ξ

x

Ω

ξ,x

Ω

ξ,x

Ω

x

*Ω

ξ,x

***

( )11log −−≥ xx1=x

ξξξξξ

ξξξ d

p

qpd

q

pp

x

xx

x

xx ∫∫

ΩΩ

−−≤−

** )(

)(1)(

)(

)(log)(

36 /36 / Relation of Koszul density with Maximum Entropy Principle

We can then observe that:

because

We can then deduce that:

If we develop the last inequality, using expression of :


0)()()(

)(1)(

***

=−=

− ∫∫∫

ΩΩΩ

ξξξξξξξξ dqdpd

p

qp xx

x

xx

1)()(**

== ∫∫ΩΩ

ξξξξ dqdp xx

ξξξξξξξξξξ dqpdppd

q

pp xxxx

x

xx ∫∫∫

ΩΩΩ

−≤−⇒≤−***

)(log)()(log)(0)(

)(log)(

)(ξxq

ξξξξξξξ ξ ddexpdpp xxxx ∫ ∫∫

Ω Ω

−

Ω

−−−≤−

* **

,log,)()(log)(

∫∫∫Ω

−

ΩΩ

+≤−***

,log)(.,)(log)( ξξξξξξξ ξ dedpxdpp xxxx

)(,)(log)( *

*

xxxdpp xx Φ−≤− ∫Ω

ξξξ )()(log)( **

*

xdpp xx Φ≤− ∫Ω

ξξξ

www.thalesgroup.com


Souriau Lie Group Thermodynamics

www.thalesgroup.com


Covariant Definition of Gibbs Equilibrium by Souriau

39 /39 /


Covariant Definition of Thermodynamic Equilibriums

Jean-Marie Souriau , student of Elie Cartan at ENS Ulm in 1946, has

given a covariant definition of thermodynamic equilibriums

formulated statistical mechanics and thermodynamics in the framework of Symplectic Geometry

by use of symplectic moments and distribution-tensor concepts, giving a geometric status for:

Temperature

Heat

Entropy

This work has been extended by C. Vallée & G. de Sax cé, P. Iglésias and F. Dubois.

40 /40 /


Covariant Definition of Thermodynamic Equilibriums

The first general definition of the “moment map” (con stant of the motion for dynamical systems) was introduced by Jea n-Marie Souriau during 1970’s

with geometric generalization such earlier notions as the Hamiltonian and the invariant theorem of Emmy Noether describing the connection between symmetries and invariants (it is the moment map for a one-dimensional Lie group of symmetries).

In symplectic geometry the analog of Noether’s theorem is the statement that the moment map of a Hamiltonian action which preserves a given time evolution is itself conserve d by this time evolution .

The conservation of the moment of a Hamilotnian act ion was called by Souriau the “ Symplectic or Geometric Noether theorem ”

considering phases space as symplectic manifold, cotangent fiber of configuration space with canonical symplectic form, if Hamiltonian has Lie algebra, moment map is constant along system integral curves .

Noether theorem is obtained by considering independ ently each component of moment map

41 /41 /


Souriau Covariant Model

Let be a differentiable manifold with a conti nuous positive density and let E a finite vector space and a continuous function defined on with values in E. A con tinuous positive function solution of this problem with respect to calculus of variations:

is given by:

and

and

Entropy can be stationary only if

there exist a scalar and an element belo nging to the dual of E.

Entropy appears naturally as Legendre transform of :

Mωd )(ξU

M)(ξp

=

=

−=

∫

∫∫

QdpU

dp

dppsArgMin

M

M

Mp ωξξ

ωξωξξ

ξ )()(

1)(

such that )(log)()(

)(.)()( ξββξ Uep −Φ=

∫−−=Φ

M

U de ωβ ξβ )(.log)( ∫

∫−

−

=

M

U

M

U

de

deU

Qω

ωξ

ξβ

ξβ

)(.

)(.)(

∫−=M

dpps ωξξ )(log)(

Φ βΦ

)(.)( ββ Φ−= QQs

42 /42 /


Souriau Covariant Model

This value is a strict minimum of s, and the equation:

has a maximum of one solution for each value of Q.

The function is differentiable and we c an write

and identifying E with its dual:

Uniform convergence of

proves that and that is convex.

Then, and are mutually i nverse and differentiable, where .

Identifying E with its bidual:

)(.)( ββ Φ−= QQs

∫

∫−

−

=

M

U

M

U

de

deU

Qω

ωξ

ξβ

ξβ

)(.

)(.)(

)(βΦ Qdd .β=Φ

β∂Φ∂=Q

∫−⊗

M

U deUU ωξξ ξβ )(.)()(

02

2

>∂

Φ∂−β

)(βΦ−

)(βQ )(QβdQds .β=

Q

s

∂∂=β

43 /43 /


Souriau-Gibbs Canonical Ensemble

In statistical mechanics, a canonical ensemble is the statistical ensemble that is used to represent the possible states of a mechanical system that is being maintained in therm odynamic equilibrium .

Souriau has defined this Gibbs canonical ensemble o n Symplectic manifold M for a Lie group action on M

The seminal idea of Lagrange was to consider that a statistical state is simply a probability measure on the manifo ld of motions

In Jean-Marie Souriau approach, one movement of a d ynamical system (classical state) is a point on manifold of movements.

For statistical mechanics, the movement variable is replaced by a random variable where a statistical state is prob ability law on this manifold.

44 /44 /


Souriau-Gibbs Canonical Ensemble

Symplectic manifolds have a completely continuous m easure, invariant by diffeomorphisms: the Liouville measure

All statistical states will be the product of Liouv ille measure by the scalar function given by the generalized partition functiondefined by the generalized energy (the moment that is defined in dual of Lie Algebra of this dynamical group) and the geometric temperature , where is a normalizing constant such the mass of probability is equal to 1,

Jean-Marie Souriau generalizes the Gibbs equilibrium state to all Symplectic manifolds that have a dynamical group .

To ensure that all integrals could converge, the ca nonical Gibbs ensemble is the largest open proper subset (in Lie algebra) whe re these integrals are convergent . This canonical Gibbs ensemble is convex .

the mean value of the energy

a generalization of heat capacity

Entropy by Legendre transform

λ

Ue .β−Φ

U

β Φ∫

−−=ΦM

U de ωβ .log

β∂Φ∂=Q

β∂∂−= Q

K

Φ−= Qs .β

45 /45 /


Souriau Lie Group Thermodynamic

For the group of time translation , this is the classical thermodynamic

Souriau has observed that if we apply this theory fo r non-commutative group (Galileo or Poincaré groups):

the symmetry has been broken

Classical Gibbs equilibrium states are no longer invariant by this group

This symmetry breaking provides new equations, discovered by Jean-Marie Souriau.

For each temperature , Jean-Marie Souriau has i ntroduced a tensor , equal to the sum of cocycle and Heat coboundary (with [.,.] Lie bracket):

This tensor has the following properties:

is a symplectic cocycle

The following symmetric tensor , defined on all values of is positive definite:

ββf f

( ) ( ) [ ]21222121 ,)( with )(.,,11

ZZZadZadQZZfZZf ZZ =+=β

βfβf

ββ fKer ∈βg (.)βad

[ ] [ ]( ) [ ]( )2121 ,,,,, ZZfZZg βββ ββ =

46 /46 /


Souriau Lie Group Thermodynamic

Souriau equations are universal, because they are n ot dependent of the symplectic manifold but only of:

the dynamical group G

its symplectic cocycle

the temperature

the heat

Souriau called this model “ Lie Groups Thermodynamics ”:

“Peut-être cette thermodynamique des groups de Lie a-t-elle un intérêt mathématique”.

For dynamic Galileo group (rotation and translation ) with only one axe of rotation:

this thermodynamic theory is the theory of centrifuge where the temperature vector dimension is equal to 2 (sub-group of invariance of size 2)

these 2 dimensions for vector-valued temperature are “thermic conduction” and “viscosity”, unifying “heat conduction” and “viscosity”.

( ) ( ) [ ]21222121 ,)( with )(.,,11

ZZZadZadQZZfZZf ZZ =+=β

ββ fKer ∈ [ ] [ ]( ) [ ]( )2121 ,,,,, ZZfZZg βββ ββ =

βf

Q

47 /47 /

The Galileo group of an observer is the group of af fine maps

Matrix Form of Gallileo Group

Symplectic cocycles of the Galilean group: V. Bargman n (Ann. Math. 59, 1954, pp 1–46) has proven that the symplec ticcohomology space of the Galilean group is one-dimens ional.

Lie Algebra of Gallileo Group

Dynamic Gallileo Group


=

1100

10

1

'

'

t

x

e

wuR

t

xrrrr

)3(

, and ,

'

..'

3

SOR

ReRwux

ett

wtuxRx

∈∈∈

+=++=

+rrr

rrrr

×∈∈∈

+

xxso

RRrr

arr

rrrrr

ωωεγηε

γηω

:)3(

, and ,

000

003

48 /48 /


Fundamental Souriau Theorem

Let be the largest open proper subset of , Lie algebra of G, such that

and are convergent int egrals

this set is convex and is invariant under eve ry transformation , where is the adjoint representation of G, with:

where is the cocycle associated with the group G and the moment, and is the image under of the pr obability measure .

Rmq: is changed but with linear dependence to , then Fisher metric is unchanged by dynamical group:

Ω g

∫−

M

U de ωξβ )(. ∫−

M

U de ωξ ξβ )(..

Ωga

gaa a

)(ββ ga→( ) ( ) )(.1 βθβθ gaaa +Φ=−Φ→Φ −

ss →( ) )()( QaaQaQ **

gg θθ =+→

)(ςς +→ Maθ

)(ς+Ma Ma ςΦ β

( ) ( )[ ] ( )βββ

βθβ Ia

aI =∂

Φ∂−=∂−Φ∂−=

−

2

2

2

12

)(g

49 /49 / Fundamental Souriau Theorem


Gibbs canonicalGibbs canonicalGibbs canonicalGibbs canonicalensembleensembleensembleensemble

Ω *Ωg *

g

R R

( )βΦ

( ) ( ) )(. βθβ gaa+Φ ( ) ( )ββ Φ−= QQs .

Q)(Qa *

gθ

β)(βga

ς

)(ς+Ma

e

aG

50 /50 /


Souriau Lie Group definition of Fisher Metric

Let be the derivative of (symplectic cocy cle of G) at the identity element and let us define:

Then

is a symplectic cocycle of ,that is independent of the moment of G

There exists a symmetric tensor defined on the image of

such that:

and

that gives the structure of a positive Euclidean sp ace

f θ

( ) ( ) [ ]21222121 ,)( with )(.,, ,11

ZZZadZadQZZfZZf ZZ =+=Ω∈∀ ββ

βf g

( ) Ω∈∀= ββββ , 0,f

βg[ ]ββ .,(.) =ad

[ ]( ) ( ) ( )( ).Im , , ,,, 212121 βββ β adZZZZfZZg ∈∀∈∀= g

( ) ( )( ).Im, , 0, 2121 ββ adZZZZg ∈∀≥

51 /51 /


Koszul Information Geometry, Souriau Lie Group Thermodynamics

Koszul Information

Geometry Model

Souriau Lie Groups

Thermodynamics Model

Characteristic function Ω∈∀−=Φ ∫Ω

− xdex x log)(*

, ξξ g∈∀−=Φ ∫− βωβ ξβ log)( )(.

M

U de

Entropy ξξξ dppx xx∫Ω

−=Φ*

)(log)()( ** ∫−=M

dpps ωξξ )(log)(

Legendre Transform )(,)( *** xxxx Φ−=Φ )(.)( ββ Φ−= QQs

Density

of probability ∫

−

−

+−

=

=

*Ω

ξ,x

ξ,x

x

Φ(x)x,ξx

dξe

ep

ep

)(

)(

ξ

ξ

∫−

−

Φ+−

=

=

M

U

U

U

de

ep

ep

ωξ

ξ

ξβ

ξβ

β

βξββ

)(.

)(.

)()(.

)(

)(

Dual Coordinate Systems

** and Ω∈Ω∈ xx

∫

∫∫ −

Ω

−

Ω

==

*Ω

ξ,x

ξ,x

xdξe

de

dpx*

*

.

)(.*

ξξξξξ

*gg ∈∈ Q and β

∫

∫∫ −

−

==

M

U

M

U

M de

deU

dpUQω

ωξωξξ

ξβ

ξβ

β )(.

)(.)(

)().(

heat Geometricor

MapMoment Souriau ofMean :

mapMoment Souriau :

eTemperatur GeometricSouriau :

Q

U

β

Dual

Coordinate Systems x

xx

∂Φ∂= )(* and

*

** )(

x

xx

∂Φ∂=

β∂Φ∂=Q and

Q

s

∂∂=β

Hessian Metric )(22 xdds Φ−= ( )βΦ−= 22 dds

Fisher metric

2

,2

2

2

2

2

*

log

)(

)(log)(

x

de

x

(x)xI

x

pExI

x

x

∂

∂=

∂Φ∂−=

∂∂−=

∫Ω

− ξ

ξ

ξ

ξ

2

)(.2

2

2

2

2

log

)(

)(log)(

β

ω

βββ

βξ

β

ξβ

βξ

∂

∂=

∂Φ∂−=

∂∂

−=

∫−

M

U de)(

I

pEI

Capacity GeometricSouriau :

)(2

2

β

ββββ

∂∂−=

∂∂−=

∂Φ∂−=

QK

Q)(I

52 /52 /


Geometric heat Capacity / Specific heat

We observe that the Information Geometry metric cou ld be considered as a generalization of “Heat Capacity”. Sou riau called it the “Geometric Capacity”. This geometric capacity is related to calorific capacity.

is related to the mean, and is related to the variance of

β∂Φ∂=Q

ββββ

∂∂−=

∂Φ∂−= Q)(

I2

2

)(

T

Q

kTTkT

T

QQK

∂∂=

∂

∂

∂∂−=

∂∂−=

2

11

βkT

1=β

Q K U

[ ] [ ]2

222 )().()(.)()(

−=−=

∂∂−= ∫∫

MM

dpUdpUUEUEQ

I ωξξωξξβ

β ββξξ

[ ]UEdpUβ

ΦQ

M

ξβ ωξξ ==∂∂= ∫ )().(

53 /53 / FOURIER HEAT EQUATION & HEAT CAPACITY

DCT

Q.=

∂∂

T

Q

T

QIFisher ∂

∂=∂∂−=

2

1)(

ββ [ ] 121

1

)(.

−−−

∆=∂

∂⇒∆=

∂∂ βββκβκ

FisherIt

TDCt

T

54 /54 / Θερµός : Souriau « Lie Group » Thermometer


Souriau has built a thermometer (θερµός) device principle that could measure the

Geometric Temperatureusing “Relative Ideal Gas Thermometer” based on

a theory ofDynamical Group Thermometry

and has also recovered the(Geometric) Laplace barometric law

[ ]

=∂∂−=

Entropy Geometric :s

eTemperatur (Planck) Geometric :

Heat Geometric :

with UVar)( ββ

βQ

QIFisher

*g∈

∂Φ∂=β

Q( ) ( )ββ Φ−= QQs , g∈∂∂=Q

sβ

rgmerprrr ,)( β−∝

55 /55 /


Koszul Information Geometry, Souriau Lie Group Thermodynamics

Koszul Information Geometry Model Souriau Lie Groups Thermodynamics Model

Convex Cone Ω∈x

Ω convex cone

Ω∈β

Ω convex cone: largest open subset of g , Lie algebra of G, such that ∫

−

M

U de ωξβ )(.

and ∫−

M

U de ωξ ξβ )(.. are convergent integrals

Transformation ( )Ω∈→ Autggxx with )(ββg

a→

Transformation of Potential (non invariant)

( )gxgxx detlog)()()( +Φ=Φ→Φ ΩΩΩ ( ) ( ) ( ) ( )βθβββ 1)( −−Φ=Φ→Φ aa

g

Transformation of Entropy (invariant)

( ) ( )********

)(x

x

gxx

ΩΩ

ΩΩΦ=

∂Φ∂Φ→Φ

x

xx

∂Φ∂= Ω )(

with *

( ) ( )

( )( )( )

( ) ( )

( ) ( )( ) ( ) ( )βθβββ

θβ

βθβ

ββ

ββ

1''

)()(

'

''

'

.with

)(.'''.''

−−Φ=Φ=Φ=Φ

+=∂

+Φ∂=

∂Φ∂=

=

=Φ−=Φ−=→

aa

aQaa

aaQ

a

QsQQQsQs

g

g

g

g

g

*

Information Geometry Metric (invariant)

( ) ( )[ ] ( )xIx

x

x

gxgxI =

∂Φ∂−=

∂+Φ∂

−= ΩΩ2

2

2

2 )(detlog)( ( ) ( ) ( )[ ] ( ) ( )ββ

ββ

βθββ Ia

aI =∂Φ∂−=

∂−Φ∂−=

−

2

2

2

12

)(g

56 /56 /


Invariance of Fisher Metric

In both Koszul and Souriau models, the Information Geo metry Metric and the Entropy are invariant respectively t o:

the automosphisms of the convex cone

to adjoint representation of Dynamical group G acting on , the convex cone considered as largest open subset of , Lie algebra of G, such that

and are convergent integrals.

g Ωga Ω

g

∫−

M

U de ωξβ )(.

∫−

M

U de ωξ ξβ )(..

( ) ( )[ ] ( )xIx

x

x

gxgxI =

∂Φ∂−=

∂+Φ∂

−= ΩΩ2

2

2

2 )(detlog)(

( ) ( ) ( )[ ] ( ) ( )ββ

ββ

βθββ Ia

aI =∂Φ∂−=

∂−Φ∂−=

−

2

2

2

12

)(g

( )Ω∈→ Autggxx with

)(ββ ga→

www.thalesgroup.com


Lie Group Action on Symplectic Manifold with Modern notation

(cf. Charles-Michel Marle)

58 /58 / Notation

Lie Algebra of Lie Group

Let a Lie Group and tangent space of at its neutralelement

Adjoint representation of

with

Tangent application of at neutral element of

For with

Curve from tangent to :

and transform by :


G GTe Ge

( )geg

e

iTAdGg

GTGLGAd

=∈→a

:1: −ghghig a

G

Ad

Ad

ad[ ]YXYadGTYXGTEndGTAdTad Xeeee ,)(, )(: =∈→= a

e G

)(KGLG n= CRK or =)(KMGT ne = 1)( ),( −=∈∈ gXgXAdGgKMX gn

[ ]YXYXXYYAdTYadKMYX XeXn ,)()()( )(, =−==∈)0(cIe d == )1(cX =

)exp()( tXAdt =γ)exp()( tXtc =

Ad

YXXYtXYtXdt

dYt

dt

dYAdTYad

ttXeX −====

=

−

= 0

1

0

)exp()exp()()()()( γ

59 /59 / Action of a Lie Group on a Symplectic Manifold

Let be an action of Lie Group G on differentiable manifold M, the fundamental field associated to an element of Lie algebra of group G is the vectors field on M:

is hamiltonian on a Symplectic Manifold , if is symplectic and if for all , the fundamental field is globally hamiltonian

There exist linear application from to d ifferential function on

We can then associate a differentiable application ,called moment of action :


g

MMG →×Φ :X

MX

( )( )0

,exp)(=

−Φ=t

M xtXdt

dxX

( ) ),(),(, 2121 xggxgg Φ=ΦΦ xxe =Φ ),(with and

Φ M Φg∈X MX

XJX

RMC

→→ ∞ ),(g

XJ g M

ΦJ

g

g*

∈=→

XXxJxJxJx

MJ

X ,),()(such that )(

:

a


We associate a bilinear and anti-symmetric form , Symplectic Cocycle of Lie algebra :

If with constant , then:

[ ] YXYX JJJYX ,),( , −=Θ

Θ

BracketPoisson :.,.with

[ ] [ ] [ ] 0),,(),,(),,( =Θ+Θ+Θ YXZXZYZYXwith

g

µ+= JJ ' *g∈µ

[ ]YXYXYX ,,),(),(' µ+Θ=ΘWith cobord of [ ]YXYX ,,),( µµ =∂ g


Equivariance of moment

There exist a unique affine action such that line ar part iscoadjoint representation

and that induce equivariance of moment

is called Cocycle associated to

The differential of 1-cocyle associated to a t neutralelement :

If then :

a

)(),(

:* gAdga

Ga

g θξξ +=→× **gg with XAdXAd gg 1

* ,, −= ξξ

J

( ) ( ) )()())(,(),( * gxJAdxJgaxgJ g θ+==ΦJ*

g→G:θ

eθeT θ J

[ ] YXYXe JJJYXYXT ,),(),( , −=Θ=θµ+= JJ ' [ ]YXYXYX ,,),(),(' µ+Θ=Θ

µµθθ *)()(' gAdgg −+=Where is cobord of Gµµ *

gAd−

www.thalesgroup.com


From Euler-Poincaré(-Tchesayev) Equation of Geometric Mechanics

to Poincaré-Marle-Souriau Equation of Geometric

Thermodynamics

63 /63 / Seminal Paper of Poincaré 1901, revisited by Marle

[1] Henri Poincaré, Sur une forme nouvelle des équations de la Mécanique, C. R. Acad. Sci. Paris, T. CXXXII, n. 7, p. 369–371., 1901

Heni Poincaré proved that when a Lie algebra acts l ocally transitively on the configuration space of a Lagrangian mechanic al system, the Euler-Lagrange equations are equivalent to a new sy stem of differential equations defined on the product of th e configuration space with the Lie algebra

[2] C.-M. Marle, On Henri Poincaré’s note on ”Sur une forme nouvelle des équations de la Mécanique ”, Journal of Geometry and Symmetries in Physics, JGSP 29, pp.1-38, 2013

Marle has written the Euler-Poincaré equations, un der an intrinsic form, without any reference to a particular system of local coordinates

Marle has proven that they can be conveniently expr essed in terms of the Legendre and momentum maps of the lift to the c otangent bundle of the Lie algebra action on the configuration spac e


64 /64 / Seminal Paper of Poincaré


« Ayant eu l’occasion de m’occuper du mouvement de ro tation d’un corps solide creux, dont la cavité est remplie de liquide, j’ai été conduit à m ettre les équations générales de la mécanique

sous une forme que je crois nouvelle et qu’il peut être intéressant de faire connaître »Henri Poincaré, CRAS, 18 Février 1901

« Elles sont surtout intéressantes dans le cas où U é tant nul, T ne dépend que des ηηηη »Henri Poincaré

65 /65 /

Lagrangian Mechanical system with M is smooth configuration space

Poincaré assumes that a finite dimensional Lie algebra acts on the configuration manifold M

New expression of the functional

If is a parametrized continuous, piecewise differenti able curve in M, and any lift of to M × g, we have:

EULER-POINCARE EQUATION by Lift of admissible Curve


[ ] RTMLMttdtdt

tdLI

t

t

→→

= ∫ : and ,: with )(

)( 10

1

0

γγγ

g

)(),(),(

:

xXXxXx

TMM

M=→×

ϕϕ

a

g

map)moment Souriau tod(associate

: * *g×→ MMTtϕ

[ ]( )

: with and : with ,

)(),(

: and

),(

,: with )()( 10

1

0

ggg

gg

gg →×=→×==

→×==

×→= ∫

MpVpMMppV(t)dt

dγ

xXLXx

RMLL

Vt

MttdttLI

MM

M

t

t

γγγ

ϕγγ

γγγ

a

o

ao

γγ γ

)()( γγ II =

66 /66 / MARLE Intrinsic Expression of EULER-POINCARE EQUATION

J.M. Marle intrinsic expression (independant of any ch oice of local coordinates) of the Euler-Poincaré equation:

Poincaré “This equation is useful mainly when only depends on its second variable (L agrangian reduction)


( )( ) ( )*

g gg→×=Ω

Ω=

−

MLdp

tVttVtLdaddt

d

t

tV

:

)(),()(),(

1*

2*

)(

ooϕ

γγ

[ ] [ ]gg

gg

*

**

∈∈−=

→==

XVXadXad

adX,V-V,XXad

VV

VV

and , , ,,

such that : and *

*

ξξξ

variable2nd its and1st its respect to with :

function theof alsdifferenti partial thebe :

:*

2

*1

RML

MLd

MTMLd

→×

→×

→×

g

gg

g

*:* gg*

g→×Mp

RML →×g:g∈X

67 /67 / EULER-POINCARE-MARLE EQUATION

J.M. Marle has given the Euler-Poincaré Equation in terms of the Legendre and the Momentum Maps

Souriau Moment Map:

The Legendre Map


*g→MTJ *:

g∈∈= XMTXXJ MM , , )(,),( *ξξπξξ o

( )XX MMM ),()( ξπϕξπ =o

( ))(),()(

: *

ξξπξϕξϕ

J

MMT

Mt

t

=×→

a

*g ( )J

pJ

Mt

t

,πϕ

ϕ

=

= o*g

L

) aldifferenti (vertical * LdMTTM vert→:L

ϕ

ϕϕϕ

oo

oooo

L

L

JLd

pJpLd tt

=⇒

==

2

2 with **gg

68 /68 / EULER-POINCARE-MARLE REDUCTION

Euler-Poincaré Equation with Legendre and Moment Map s:

The Euler-Poincaré Equation and Reduction: Following the remark made by Poincaré at the end of his note, let us now assume that the map only depends on its 2nd variable

The Euler-Poincaré Equation in Hamiltonian Formalism


( )( ) ( )

( ))(),()(

)(),()(),( 1*

)(

tVtdt

td

tVtLdJtVtJaddt

dtV

γϕγ

γγϕ

=

=

− oooL

RML →×g: g∈X

( )( ) 0)(*)( =

− tVLdaddt

dtV

( )

RMTH

MTTM

MTLH

→

→

∈−= −−

*

*

*11

:

:

, )()(,)(

L

LL ξξξξξ

69 /69 / POINCARE-MARLE-SOURIAU EQUATION ON DUAL LIE ALGEBRA

Euler-Poincaré Equation in the Framework of Souriau Lie Group Thermodynamics

Souriau Lie Group Thermodynamics:

Poincaré-Marle-Souriau Equation:


( )( ) 0)(*)( =

− tVLdaddt

dtV

0* =

∂Φ∂

−ββad

dt

dQad

dt

dQ *β=

( ) ( ) ( ))(),(, 11 QQQQQs −− ΘΦ−Θ=Φ−= ββ

( )

Θ=

∈∂

Φ∂=Θ=

− )(

)(

1 Q

Q

ββββ *

gg∈

∂∂=

Q

Qs )(β

70 /70 / POINCARE-MARLE-SOURIAU EQUATION ON DUAL LIE ALGEBRA

Recall that has a natural Poisson structure called Kiri llov-Kostant-Souriau structure, which allow to associate to any smoothfunction its Hamiltonian vector field :

If there exist a smooth function such that , the parametrized curve satisfies the

Hamilton differential equation on :

In Souriau Lie Group Thermodynamics:


*g

Rh →*g:

*g∈−= ξξξχ ξ , )( *

)(dhh adRh →*

g: JhH o=[ ] *

g→= 10,: ttJ ςξ o

*g

( ))()( *

)( taddt

tddh ξξ

ξ−=

( ) ( )ββ Φ−= QQs ,

( ))()( *

))(( tQaddt

tdQtQds−=

dQds β=

71 /71 / INFORMATION GEOMETRY BASED ON GEOMETRIC MECHANICS


Élie CARTAN est le fils de Joseph CARTAN maréchal-f errant.Elie CARTAN le FORGERON

(du Latin FABER) l’Homme qui bat la matière sur l’ ENCLUME, pour lui imprimer la COURBURE pour la mettre en FORME)

GEOMETRIC MECHANICSH. POINCARE

-SYMPLECTIC MECHANICS

J.M. SOURIAU

LIE GROUP THERMODYNAMICS

J.M. SOURIAU-

HESSIAN GEOMETRY OF INFORMATIONJ.L. KOSZUL

Texte de Bergson - Homo faber"En ce qui concerne l'intelligence humaine, on n'a p as assez remarqué que l'invention mécanique

a d'abord été sa démarche essentielle … . Si nous pouvions nous dépouiller de tout orgueil, si, pour définir notre espèce, nous nous en tenions strictement à ce que l'histoire et la préhistoire nous

présentent comme la caractéristique constante de l'homme et de l'intelligence, nous ne dirions peut-être pas Homo sapiens, mais Homo faber. En définitive, l'intelligence, env isagée dans ce qui en paraît

être la démarche originelle, est la faculté de fabr iquer des objets artificiels, en particulier des outils à faire des outils et d'en varier indéfinime nt la fabrication ."

Henri Bergson, L’ Évolution créatrice (1907), Éd. P UF, coll. "Quadrige", 1996, chap. II, pp.138-140

NEW FOUNDATION OF INFORMATION THEORY (Sapiens) by G EOMETRIC MECHANICS (Faber)

www.thalesgroup.com

Legendre Transform & Minimal Surface:

Seminal Paper of Legendre

73 /73 /


Legendre Transform and Minimal Surface

Legendre has introduced « Legendre Transform » to solve Minimal Surface Problem

Classical “Legendre transform” with our previous nota tions:

( ) ( ) ( )

( )( )

=

=

Φ=

Φ

Φ

=

=

=

=

=

==Φ

Φ−=Φ−=

dQ

ds

dQ

dsdQ

ds

d

d

d

dd

d

Q

y

x

q

p

Q

QQ

qpQs

yxz

QQQs

2

1

2

1

2

1

2

1

and , ),(

),(with

,.

β

ββ

β

ββ

βω

β

ββββ

dq

dy

dp

dx

qpyqxpyxz

ωωω

==

−+=

and with

),(..),(

74 /74 /


Legendre Transform and Minimal Surface

In the following relation, we recover the definitio n of Entropy :

But also relation with Mean Curvature

=

=⇒

==

=+

dQ

ds

dsdQ

dq

dy

dp

dx

ddqydpx

β

βωω

ω .

and

..

Φ=

−−=

⇒

−=

⇒>>

−=⇒<<

=

Φ+

Φ

=

+

Φ

Φ

Φ

β

βββ

β

β

βββ

d

dQ

Q

Q

QHI

Q

Q

QQQ

d

dQ

Q

Q

d

dQ

HIQ

H

d

d

d

d

d

d

Q

Q

d

d

with

1.2)(..

1

2)(1

.2

11

2

2

2

2

22

2nd Principle of Thermodynamics

www.thalesgroup.com

References

76 /76 / ReferencesKoszul References

[1] Koszul J.L., Variétés localement plates et conve xité, Osaka J. Math. , n °2, p.285-290, 1965

[2] Koszul J.L., Domaines bornées homogènes et orbit es de groupes de transformations affines, Bull. Soc . Math. France 89, pp. 515-533., 1961

[3] Koszul J.L., Ouverts convexes homogènes des espa ces affines, Math. Z. 79, pp. 254-259., 1962

[4] Koszul J.L., Déformations des variétés localemen t plates, .Ann Inst Fourier, 18 , 103-114., 1968

Souriau References

[5] Souriau J.M., Définition covariante des équilib res thermodynamiques, Suppl. Nuov. Cimento, n °1, pp. 203–216, 1966

[6] Souriau J.M., Structure of dynamical systems, v olume 149 of Progress in Mathematics. Birkhäuser Bo ston Inc., Boston, MA, 1997. A symplectic view of physics

[7] Souriau J.M., Thermodynamique et géométrie. In Differential geometrical methods in mathematical ph ysics, II, vol. 676 of Lecture Notes in Math., pages 369-397. Springer, Berlin, 19 78

[8] Souriau J.M., Géométrie Symplectique et physiqu e mathématique, CNRS 75/P.785, Dec. 1975

[9] Souriau J.M., Mécanique classique et géométrie symplectique, CNRS CPT-84/PE-1695, Nov. 1984

[10] Souriau J.M., On Geometric Mechanics, Discrete and continuous Dynamical systems, V.ol 19, N. 3, p p. 595–607, Nov. 2007

Author References

[11] Barbaresco F., Information Geometry of Covaria nce Matrix: Cartan-Siegel Homogeneous Bounded Domai ns, Mostow/Berger Fibration and Fréchet Median, Matrix Information G eometry, Springer, 2013

[12] Barbaresco F., Eidetic Reduction of Informatio n Geometry through Legendre Duality of Koszul Chara cteristic Function andEntropy, Geometric Theory of Information, Springer, 2014

[13] Barbaresco F., Koszul Information Geometry and Souriau Geometric Temperature/Capacity of Lie Group Thermodynamics,, MDPI Entropy, n °16, pp. 4521-4565, August 2014. http://www.mdpi.com/1099-4300/16/8/4521/pdf

Other References

[14] Shima H. , Geometry of Hessian Structures, wor ld Scientific Publishing 2007

[15] Gromov M., IHES Six Lectures on Probabiliy, Sy mmetry, Linearity. October 2014, Jussieu.

http://www.ihes.fr/~gromov/PDF/probability-huge-Lec ture-Nov-2014.pdf

https://www.youtube.com/watch?v=hb4D8yMdov4&index=5 &list=PLx5f8IelFRgGo3HGaMOGNAnAHIAr1yu5W

www.thalesgroup.com

Past Conferences (GSI’13, MaxEnt’14) and

Announcement (GSI’15 & Brillouin Seminar)

78 /78 / Past Conferences

French/Indian Workshop on « Matrix Information Geometry », Ecole Polytechnique & Thales Research & Technology, 23-25th February 2011 (with Prof. Rajendra Bhatia)

http://www.lix.polytechnique.fr/~schwander/resources/mig/slides/

SMAI’11 Congress, Mini-Symposium on « Information Geometry », 23-27th Mai 2011

http://smai.emath.fr/smai2011/programme_detaille.php

Symposium on « Information Geometry & Optimal Transport », hosted at Institut Henri Poincaré, 12th February 2012 (with GDR CNRS MIA)

https://www.ceremade.dauphine.fr/~peyre/mspc/mspc-thales-12/

SMF/SEE Conference on « Geometric Science of Information », Ecole des Mines de Paris, August 2013

http://www.see.asso.fr/gsi2013

SMF/SEE Conference on MaxEnt with special Issue "I nformation, Entropy and their Geometric Structures", Clos Lucé in Amboise, sponso red by Jaynes Foundation, 21-26th Sept. 2014

https://www.see.asso.fr/node/10784

Leon Brillouin Seminar on « Geometric Science of Information », Institut Henri Poincaré & IRCAM, launched by THALES since December 2009, with Ecole Polytechnque & INRIA/IRCAM

http://repmus.ircam.fr/brillouin/past-events

79 /79 / GSI’13 Proceedings


http://www.springer.com/computer/image+processing/book/978-3-642-40019-3

80 /80 / MaxEnt’14, Amboise, Clos Lucé, September 2014

34th International Workshop onBayesian Inference and Maximum Entropy

Methods in Science and EngineeringChâteau Clos Lucé, Parc Leonardo Da Vinci,

Amboise, France

https://www.see.asso.fr/maxent14http://djafari.free.fr/MaxEnt2014/Program_MaxEnt2014.html

Sponsored by Jaynes Foundation

Keynote speakers:Misha Gromov (IHES, Abel Prize 2009)Daniel Bennequin (Institut Mathématique de Jussieu)Roger Balian (CEA, French Academy of science)Stefano Bordoni (Bologna University)https://www.see.asso.fr/node/10784

81 /81 / GSI’15 « Geometric Science of Information » 28th -30th October 2015

Ecole Polytechnique


http://www.gsi2015.org

Authors will be solicited to submit a paper in a special Issue "Differential Geometrical Theory

of Statistics” in ENTROPY Journal, an international and interdisciplinary open access

journal of entropy and information studies published monthly online by MDPI.

http://www.mdpi.com/journal/entropy/special_issues/entropy-statistics

82 /82 / Léon Nicolas Brillouin Seminar

Corpus : « GeometricGeometricGeometricGeometric Science of InformationScience of InformationScience of InformationScience of Information »

Hosting lab: IRCAM, Salle Igor Stravinsky

Projet INRIA/CNRS/Ircam MuSync (Arshia Cont)

Animation : Arshia Cont (IRCAM & INRIA), Frank Nielse n (Ecole Polytechnique), F. Barbaresco (Thales)

Web Site: http://repmus.ircam.fr/brillouin/home

Abstracts, Videos & Slides: http://repmus.ircam.fr/brillouin/past-events

http://repmus.ircam.fr/brillouin/home

83 /83 / Thank you for your attention


Nous avouerons qu’une des prérogatives dela géométrie est de contribuer à rendrel’esprit capable d’attention: mais on nousaccordera qu’il appartient aux lettres del’étendre en lui multipliant ses idées, del’orner, de le polir, de lui communiquer ladouceur qu’elles respirent, et de faire servirles trésors dont elles l’enrichissent, àl’agrément de la société.

Joseph de Maistre

Si on ajoute que la critique qui accoutumel’esprit, surtout en matière de faits, à recevoirde simples probabilités pour des preuves, est,par cet endroit, moins propre à le former, quene le doit être la géométrie qui lui faitcontracter l’habitude de n’acquiescer qu’àl’évidence; nous répliquerons qu’à la rigueuron pourrait conclure de cette différence même,que la critique donne, au contraire, plusd’exercice à l’esprit que la géométrie: parceque l’évidence, qui est une et absolue, le fixeau premier aspect sans lui laisser ni la libertéde douter, ni le mérite de choisir; au lieu queles probabilités étant susceptibles du plus etdu moins, il faut, pour se mettre en état deprendre un parti, les comparer ensemble, lesdiscuter et les peser. Un genre d’étude quirompt, pour ainsi dire, l’esprit à cette opération,est certainement d’un usage plus étendu quecelui où tout est soumis à l’évidence; parceque les occasions de se déterminer sur desvraisemblances ou probabilités, sont plusfréquentes que celles qui exigent qu’onprocède par démonstrations: pourquoi nedirions –nous pas que souvent elles tiennentaussi à des objets beaucoup plus importants ?

Joseph de Maistre

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