lepage forms from lepage's idea to the variational sequence 7 decades between lepage and krupka...

Lepage formsfrom Lepage's idea to the variational sequence

7 decades between Lepage and Krupka

Colloquium on Variations, Geometry and PhysicsOlomouc, 25. 8. 2007

Michal Lenc and Jana MusilováInstitute of Theoretical Physics and Astrophysics

Masaryk University

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Richard P. Feynman

Such principles are fascinating and it is always worth while to try to see how general they are.(The Feynman lectures on physics, II-19)

Motto:(importance of thevariational principle)

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About what is this lecture? Lepage as a name Lepage as a personality Lepage and his original idea Dedecker’s contribution Krupka’s idea of Lepage equivalents of

Lagrangians Variational sequence and its representation by

differential forms Lepage forms as a “product” of the variational

sequence Examples Forms in physics education – Krupka’s

contribution

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Name LEPAGE

Minor planet (Nr. 2795) Lepage a=2.296 AU, e=0.0288, P=3.48 year 16.12.1979 La Silla (H.Debehogne; E. R. Netto)

… from 5 130 000 results on Google the most interesting is

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Personality LEPAGE Théophile Lepage Docteur en Sciences Université de Liège

1924 Student of E. Cartan (?) 11 students and 168 descendants 19 scientific papers 1929-1942 Dean of Faculté des Sciences de l’Université

Libre de Bruxelles 1953-1955 Curiosity: He had introduced a symplectic

analog of Hodge theory before the Hodge theory itself.

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Lepage’s key paper Th. H. J. Lepage: Sur les champs géodésiques du calcul

des variations I, II.

Bull. Acad. Roy. Belg. Cl. des Sciences 22

(1936), 716-739, 1036-1046.

7Lepage forms - Olomouc - 25. 8. 2007

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Original Lepage’s idea - I First note already 1933 Comptes rendus des séances de l’Académie

des sciences > séance 18. décembre 1933: Note de M. Th. H. J. Lepage présentée par M. Élie Cartan

A toute forme quadratique extérieure Ω=A dp dy+B dx dp+C dq dy+D dx dy+E dp dq on peut adjoindre une forme quadratique Ω1 covariante de

Ω relativement à toute transformation de contact effectuée sur les x,y,z,p,q, et telle que l’on ait

Ω’1=0 (mod dz – p dx – q dy). z=z(x,y), p=∂z/∂x, q= ∂z/∂y, A=A(x,y,z,p,q),….,E=E(x,y,z,p,q)

A contact 1-form

Original Lepage’s idea - II Studies of a double integral

Lepage congruencies

1 1 1( ) ( , ; ,..., , ,..., ; ,..., )

( , )...unknown functions, ,

i n n n

i ii i i i

I z f x y z z p p q q dx dy

z zz z x y p qx y

1 1

, 1,

(mod ,..., ), 0 (mod ,..., )( )

i i

i i i i

i i i i i ij i

n n

q p i ij i j

dz p dx q dy i nf dx dy X dx Y dy A

f dx dy df dx dy f dx f dy A

Contact 1-forms

∫λ=Ldx ^ dy

Lepage equivalent Θλ

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Original Lepage’s idea - III Application For a vector field [pi , qi],

pi(x,y,z1,…, zn), qi(x,y,z1,…, zn), denote [Ω]=Ω(x,y,zi, pi(x,y,z1,…, zn), qi(x,y,z1,…, zn))

Definition: A field [pi , qi] is called geodesic with respect to the form Ω, if d[Ω]= 0.

Proposition: A field [pi , qi] is geodesic with respect to the form Ω, iff

[ ] 0

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Dedecker’s paper P. Dedecker

A property of differential forms in the calculus of variations.

Pac. J. Math. 7 (1957), 1545-1549.

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Dedecker’s contribution ( , , ) , ,

mod , 0 mod

i i i i i ii

i i

LI L t q q dt L dt dq q dtq

L dt d

ωi … “predecessor” of contact formsθ … semi-basic form (contains only dt and dqi)ω … unique semi-basic form with dω=0 mod ωi

“relative integral invariant of E. Cartan” in terminology of Paul Dedecker special case of Lepage congruence “predecessor” of Lepage equivalent of L

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Krupka’s key paper Demeter Krupka:

Some geometric aspects of variational problems in fibred manifolds.

Folia Facultatis Scientiarum Naturalium Universitatis Purkynianae Brunensis, XIV (1973), 10, pp 65.

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Lepage forms after Krupka Basic structure

Horizontal and „pseudovertical“ forms

Lepage n-forms

( , , ), ( , , ), dim , dimrrY X J Y X Y m n X n

, 1

*,

, ( )( ,..., ) 0 0

, 0, ( )

rx

r rq X x q ij

r rq c

W j T

W j

11, 1

*1 1,

, ( ) , ... vertical,

, ( ) ( ( ) ), ( )

r rn n Y r

rn r r

W h d W

W i h i h

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Lepage equivalents after Krupka – first version Lepage equivalent of a Lagrangian

Example for n=1 (mechanics)

1 1, ,

* 22,1 1,

, :

( ) , ( )n X n Y

n Y

W W

h h d W

, , ( )LLdt L dt dq q dtq

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Krupka’s lecture note Demeter Krupka:

The Geometry of Lagrange structures. Lecture note for advanced course New Perspectives

in Field Theory held 1997 in Levoča, Slovakia

Preprint Series in Global Analysis GA 7/97, Silesian University, Opava 1997.

17

Lepage forms after Krupka Lepage n-forms on JrY - definition

1

1

1 1 1

1

1 1

1 1

11 1,

11,0

, ...*1, 0 0 ... 2

0

, ... , ...01

...

, ...0

...

( )

( ) ( ) 0 , ... vertical

( )

0, sym ( ... )

k

k

k k k

k

r r

r

rn Y

rr

ri j j

r r j j i contk

p j j j j jp k

j j

j j j

j j

a p d W

b hi d VJ Y

c f f

f d f f j jy

ff

y

1 10, sym ( ... )rj j

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Lepage forms after Krupka Lepage n-forms on JrY – theorem An n-form on JrY is Lepage form iff it holds

1 1

1 1

*1, ( 1) 2

00 0 ...

0 0 ... , ...

10 0

( 1) ...

... , ( / )

l k

k l

r r cont n cont

r r kl

p p j j ik l j j p p i

n ii

d

ff d dy

dx dx i x

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Lepage equivalents: Krupka

Lepage equivalent of a Lagrangian

Examples of Lepage equivalents mechanics (unique Lepage equivalent)

field theory (non-uniqueness, it depends on the order of Lagrangian)

, 0 0, , , , 2 1r sn X nW L W f L s r

1 1

( )0 1 ( 1)

( 1)lr r k

lkl

k l k l

d LLdtdt q

(2r-1)th orderFor rth orderLagrangian

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Examples of LE:field theory Poincaré-Cartan (cont≤1, unique)

Fundamental LE (for 1st order Lagrangian)

2nd order Lagrangian

0PC ii

LLy

11

1 11

1

1... ...!( )!

0

... ......

k k n

k k nk

k

kni i

j j i ik n kk j j

L dx dxy y

0 p i j ii pi ji

L L LL dy y y

dΘλ =0 iff Eλ =0

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Role of Lepage equivalents

variational function

Euler-Lagrange function

first variational formula (ξ…π-projectable)

* *0( ) ,r rJ J L

1

1

1 00 ...

( 1) ...k

k

rr

j jk j j

Lp d d dy

* * *r r r

r r rJ J J

J J i d d J i

Eλ(L)

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0 0r d

1r d ...d d

Pr

2r d r

P+1 rN

...d d d 0

Variational sequence

00 d1r d ...d d

2r

Pr d

0

0

2r

2r/

F2

0

0

1r

1r/

F1

0

0

Pr

Pr/

FP

...

E0 EP

E2E1 EP- 1

, 1,r r rq q c q cd

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„Physical“ part of VS Ε: λ → Ελ Η: E → HE

trivialLagrangians

Lagrangians

n-forms (n+1)-forms (n+2)-forms

dynamical forms E-L forms H-S forms

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Representation of VS - I Problem:

representation of variational sequences by differential forms (finite jet prolongations of fibered manifolds)

Variational bicomplex: infinite order of jets of fibered manifolds I. M. Anderson: Introduction to the variational bicomplex. Contemporary Mathematics 132 (1992), 51-73. A. M. Vinogradov and Vinogradov’s school (I. S. Krasilschik, V. V. Lychagin)

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Representation of VS - II Variational sequence – finite order: D. Krupka: Variational sequences on finite order

jet spaces. In: DGA Proc. Conf. Brno 1989. World Scientific, Singapore 1990, 236-254. representation for field theory (n > 1), special case of k-forms for k=n, n+1, n+2, (Lagrangians, E-L forms, H-S forms)

Krupka’s school (Kašparová, Krbek, Musilová,Šeděnková with Krupka, Štefánek…) field theory, k-forms for k=n, n+1, n+2 … general case, mechanics (n=1) … general case, all k

Other authors (Vitolo and Palese, Grigore) k=n, n+1, n+2, alternative approaches

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Representation of VS - III General solution of

the representation problem (field theory, rth order, all columns of VS)

M. Krbek, J. Musilová: Representation of the variational sequence by differential forms. Acta Applicandae Mathematicae 88 (2005), 177-199 Inspiration: Anderson’s expression for interior Euler operator. New concepts and results: Lie derivative with respect to vector fields along maps proofs appropriate for finite order problem generalization of integration by parts

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Representation of VS - IV Basic steps of the general solution

Step 1: Integration by parts:

appropriate decomposition of k-contact component of an (n+k)-form

Step 2: Construction of Euler operator: Linearity condition applied to the previous decomposition leads to (linear) interior Euler operator assigning to a form (class of forms in the variational sequence) its representative.

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Representation of VS - V Step 1 – Integration by parts

| | 0

| |

| | 0

( , , ), ( , , ),dim , dim

( , ), ( , ), ,

( ) ( ), ( ) ( 1)

r r

ri r J

n k k JJ

rJ J

k k k JJ

Y X J Y X Y m n X n

V x y V p

p I p dp R I d

ρ: (n+k)-form, R(ρ): local k-contact (n+k-1)-form

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Representation of VS - VI

Step 2: Construction of Euler operator – main theorem

There exists a unique decomposition of the above

mentioned type such that I(ρ) is R-linear.

| |

| | 0

1( ) ( 1)r

JJ k

J J

I d pk y

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Representation of VS - VII Properties of Euler operator

W … open subset of Y, ρ … (n+k)-form on JrW, 1≤ k ≤ N-n, N … dim JrY.

2 1, * 2 1

2 4 3,2 1 *

( ) ( ) ( )( ) ( ( )) 0

( ) ( ) ( ) ( )

( ) ker ( )

r r rn k

k k

r r

rn k

a I Wb I p dp R

c I I

d I W

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Generalized Lepage forms Lepage forms as a “product” of VSAn (n+k)-form ρ on JrY is called Lepage form, if following equivalent conditions hold.

1 1 1( ), ( ) 0k k kp d I d p dR p d

For mechanics see D. Krupka and J. Šeděnková, Proc. of DGA 2004, Charles University, Prague 2005

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Examples of LE – a particle Lagrangians for geodesics

Lepage equivalents

2 21 2

( , , ), ( , ), (, ), dim 1,0 3

1, ( )2 ( )

Y X V x X

g x xL mc g x x L m c

1

2 22

12

g xmc dx

g x x

g x x g xm c d dx

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Examples of LE – a string I Lagrangian – standard

Lepage equivalents … ρPC=Θλ,fundamental

21

( , , ), ( , ), ( , ; ),0 3,dim 2

( ) ( )( ) det

Y X V x X

L T g x x g x x g x x T h

1 det( )

( )det

T h d dT g g g g

dx x x x d x x x dh

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Examples of LE – a string II Lagrangian – for Polyakov action

Lepage equivalents … ρPC=Θλ,fundamental

2

( , , ), ( , ), ( , ; ),0 3,dim 2

det , 0 , 12

iji j

Y X V x XTL f f g x x i j

2

0 1

det2det ( )

iji j

i ii

T f g f x x d d

T f g x dx f d f d

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Demo Krupka as a teacher Main courses and seminars on

Masaryk University Courses in theoretical physics (QM, EM, TSP) General relativity Mathematics for QM and relativity Group theory in physics Mathematical analysis (theory of integrals) Algebra (basic and advanced) Variational calculus Analysis on manifolds

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Forms in physics education Integrating differential forms after

Spivak, general Stokes theorem

Student’s comment: “This is a self-production of Jacobians!”

*

[0,1]kc

c

d

Michael Spivak: Calculus on manifolds.Perseus Books, Cambridge, Massachusetts,1998,27-th edition. (1-st edition 1965)

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Concluding theorem

Theorem

Excellent scientist andenthusiastic teacher

successful students