Snímek 1

Report
Colloquium on Variations, Geometry and Physics
Olomouc, 25. 8. 2007
Michal Lenc and Jana Musilová
Institute of Theoretical Physics and Astrophysics
Masaryk University
Lepage forms
from Lepage's idea
to the variational sequence
7 decades between Lepage
and Krupka
Motto:
(importance of the
variational principle)
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)
Lepage forms - Olomouc - 25. 8. 2007
2
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
Lepage forms - Olomouc - 25. 8. 2007
3
Name LEPAGE
… from 5 130 000 results on Google the
most interesting is

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)
Lepage forms - Olomouc - 25. 8. 2007
4
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.
Lepage forms - Olomouc - 25. 8. 2007
5
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.
Lepage forms - Olomouc - 25. 8. 2007
6
Lepage forms - Olomouc - 25. 8. 2007
7
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).
A contact 1-form
z=z(x,y), p=∂z/∂x, q= ∂z/∂y, A=A(x,y,z,p,q),….,E=E(x,y,z,p,q)
Lepage forms - Olomouc - 25. 8. 2007
8
Original Lepage’s idea - II


Studies of a double integral
∫λ=Ldx ^ dy
I ( zi )   f ( x, y; z1 ,..., zn , p1 ,..., pn ; q1 ,..., qn ) dx dy
zi
zi
zi  zi ( x, y )...unknown functions, pi 
, qi 
x
y
Lepage congruencies
Contact 1-forms
i  dzi  pi dx  qi dy, 1  i  n
  f dx dy  ii , i  X i dx  Yi dy  Aiji
  f dx dy (mod 1 ,..., n ), d   0 (mod 1 ,..., n ) 
  f dx dy  ( f qi dx  f pi dy ) i  Aij i  j
Lepage equivalent Θλ
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
Lepage forms - Olomouc - 25. 8. 2007

10
Dedecker’s paper

P. Dedecker
A property of differential forms
in the calculus of variations.
Pac. J. Math. 7 (1957), 1545-1549.
Lepage forms - Olomouc - 25. 8. 2007
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Dedecker’s contribution
L i i
i
i
I   L(t , q , q ) dt ,   L dt  i  ,   dq  q dt
q
i
i
  L dt mod  , d   0 mod 
i
i
ωi … “predecessor” of contact forms
θ … 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
Lepage forms - Olomouc - 25. 8. 2007
12
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.
Lepage forms - Olomouc - 25. 8. 2007
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Lepage forms after Krupka

Basic structure
(Y ,  , X ), ( J rY ,  r , X ), dimY  m  n, dim X  n

Horizontal and „pseudovertical“ forms
  W ,  ( j  )(1 ,..., q )  0  T j    i  0
r
q, X
r
x
r
x
  W , j    0,  ( )
r
q ,c

r *
Lepage n-forms
   W , h (d  )  
r
n
r 1
n 1,Y
W ,  ... r 1 vertical,
  rn 1W , i ( )  h(i( ) r*1,r ), h ( )  
Lepage forms - Olomouc - 25. 8. 2007
14
Lepage equivalents after
Krupka – first version

Lepage equivalent of a Lagrangian
 
1
n, X
W,   W :
1
n ,Y
*
h(  )   2,1
 , h (d  )   n2 1,YW

Example for n=1 (mechanics)
L  


  L dt ,   L dt    ,   (dq  q dt )
q
Lepage forms - Olomouc - 25. 8. 2007
15
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.
Lepage forms - Olomouc - 25. 8. 2007
16
Lepage forms after Krupka

Lepage n-forms on JrY - definition
(a ) p1 d    rn11,Y W
(b) hi ( ) d   0    VJ r 1Y ,  ... r 1,0  vertical
r
(c)  r*1,r   f 00   fi , j1 ... jk  j1 ... jk  i  cont  2
k 0
f 0
p , j1 ... jk
jk , j1 ... jk 1

d
f

f
 0, sym ( j1... jk )
p 


y j1 ... jk
f 0
jr 1 , j1 ... jr

f
 0, sym ( j1... jr 1 )


y j1 ... jr 1
17
Lepage forms after Krupka

Lepage n-forms on JrY – theorem
An n-form on JrY is Lepage form iff it holds

*
r 1, r
    d  cont ( n1)  cont 2
 r k


f
l

0
  f 00     (1) d p1 ...d pl 
  j1 ... jk  i

y j1 ... jk , p1 ... pl i 
k 0  l 0
1
n
i
0  dx  ...  dx , i  i( / x ) 0
r
Lepage forms - Olomouc - 25. 8. 2007
18
Lepage equivalents: Krupka

Lepage equivalent of a Lagrangian
  W ,   L0 ,   W , f0  L, s  2r 1
r
n, X

s
n
Examples of Lepage equivalents
 mechanics (unique Lepage equivalent)
l
 r  k 1
d
L
l
  L dt     (1) l 

dt q( k l 1)
k  0  l 1
r 1
 
 ( k )

(2r-1)th order
For rth order
Lagrangian
 field theory (non-uniqueness, it depends
on the order of Lagrangian)
Lepage forms - Olomouc - 25. 8. 2007
19
Examples of LE:field theory
 Poincaré-Cartan (cont≤1, unique)
 PC
L 
 L 0     i
yi
 Fundamental LE (for 1st order Lagrangian)
k

L
k
ik 1
in
1
1
   k !( nk )! 1  k  j1 ... jk ik1...in   ...    dx  ...  dx
y j1 ...y jk
k 0
dΘ =0 iff E =0
n
λ
λ
 2nd order Lagrangian
 L
L
  L0     d p 
 y
y pi
 i
 
L 
   i    j  i
y ji

Lepage forms - Olomouc - 25. 8. 2007
20
Role of Lepage equivalents

variational function
 ( )   J     J   ,   L0
r *
r *



Euler-Lagrange function
Eλ(L)
L 
p1d    (1) d j1 ...d jk    0
y j1 ... jk
k 0
r
r

first variational formula (ξ…π-projectable)
J   J r    J  iJ r  d    d J  i J r   
r
*
r
*
Lepage forms - Olomouc - 25. 8. 2007
r
*
21
Variational sequence
0

    d
0
d
r
1

d
2r
d
...
d
Pr
d
r d
0

d

d
...
d

d
0
0
0
r
q

r
1
r
2
F1
E0
 /
r
1
r
1
0
F2
E1
 /
r
2
r
2
0
r
P
FP
E2
r
q ,c
r
q 1,c
0
r d
P+1

...
d
r d
N

0
EP
...E P- 1 r /r
P
P
0
Lepage forms - Olomouc - 25. 8. 2007
22
„Physical“ part of VS
Ε: λ → Ελ
trivial
Lagrangians
dynamical forms
Η: E → HE
E-L forms
H-S forms
Lagrangians
n-forms
(n+1)-forms
(n+2)-forms
Lepage forms - Olomouc - 25. 8. 2007
23
Representation of VS - I

Problem:

Variational bicomplex:
representation of variational sequences by
differential forms (finite jet prolongations of
fibered manifolds)
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)
Lepage forms - Olomouc - 25. 8. 2007
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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
Lepage forms - Olomouc - 25. 8. 2007
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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
Lepage forms - Olomouc - 25. 8. 2007
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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.
Lepage forms - Olomouc - 25. 8. 2007
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Representation of VS - V
 Step 1 – Integration by parts
(Y ,  , X ), ( J rY ,  r , X ),dim Y  m  n, dim X  n
r
(V , ),   ( xi , y ),    r n  kV , pk    J  J
| J |0
r
pk   I (  )  pk dpk R(  ), I (  )      ( 1)|J | d JJ
| J |0
ρ: (n+k)-form, R(ρ): local k-contact (n+k-1)-form
Lepage forms - Olomouc - 25. 8. 2007
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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.
r
 
1 
|J |
I (  )     (1) d J  
k
| J | 0
 yJ
Lepage forms - Olomouc - 25. 8. 2007

pk  

29
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.
(a)
(
(b)
I ( pk dpk R (  ))  0
(c )
I (  )  (
(d )
ker I (  )  
2
)   I ( )  
2 r 1, r *
2 r 1
nk
W
) I ()
4 r  3,2 r 1 *
r
nk
W
Lepage forms - Olomouc - 25. 8. 2007
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Generalized Lepage forms

Lepage forms as a “product” of VS
An (n+k)-form ρ on JrY is called Lepage form, if
following equivalent conditions hold.
pk 1 d   I (d  ), pk 1 dR( pk 1 d  )  0
For mechanics see D. Krupka and J. Šeděnková,
Proc. of DGA 2004, Charles University, Prague 2005
Lepage forms - Olomouc - 25. 8. 2007
31
Examples of LE – a particle

Lagrangians for geodesics
(Y ,  , X ), (V , ),   (, x ), dim X  1, 0    3

 

g
x
1
  x
 
2 2
L1  mc g x x , L2   
  ( )m c 
2   ( )

Lepage equivalents
g x
1  mc
dx
g x x 
 



g x
1 g x x
2 2
2  
  m c  d 
dx
2



Lepage forms - Olomouc - 25. 8. 2007
32
Examples of LE – a string I

Lagrangian – standard
(Y ,  , X ),(V , ),  ( ,  ; x  ),0    3,dim X  2
L1  T ( g x x )2  ( g x x )( g x x )  T  det h

Lepage equivalents … ρPC=Θλ,fundamental
1  T  det h d  d 

T ( g g   g g )
 det h
dx   ( x x x d  x x x d )
Lepage forms - Olomouc - 25. 8. 2007
33
Examples of LE – a string II

Lagrangian – for Polyakov action
(Y ,  , X ),(V , ),  ( ,  ; x  ),0    3,dim X  2
T
L2  
 det f f ij g  xi xj , 0  i, j  1
2

Lepage equivalents … ρPC=Θλ,fundamental
T
2 
 det f g  f ij xi xj d  d 
2


0i
1i
T  det f g  xi dx  ( f d  f d )
Lepage forms - Olomouc - 25. 8. 2007
34
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
Lepage forms - Olomouc - 25. 8. 2007
35
Forms in physics education

Integrating differential forms after
Spivak, general Stokes theorem
Michael Spivak:
Calculus on manifolds.
Perseus Books,
Cambridge, Massachusetts,1998,
27-th edition.
(1-st edition 1965)

 
c

c*
[0,1]k
 d   


Student’s comment:
“This is a self-production of Jacobians!”
Lepage forms - Olomouc - 25. 8. 2007
36
Concluding theorem
Theorem
Excellent scientist and
enthusiastic teacher
successful students
Lepage forms - Olomouc - 25. 8. 2007
37

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