Colloquium on Variations Geometry and Physics Olomouc 25

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 8. 2007 Lepage forms - Olomouc - 25.

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

§ § § 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 About what is this lecture? § § Lepage forms - Olomouc - 25. 8. 2007 3

Name LEPAGE … from 5 130 000 results on Google the most interesting is n 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 n Théophile Lepage ü 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. ü ü ü Lepage forms - Olomouc - 25. 8. 2007 5

Lepage’s key paper n 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

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Original Lepage’s idea - I First note already 1933 n ü 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 ü 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 Ω 1 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

Studies of a double integral Original Lepage’s idea - II n ∫λ=Ldx ^ dy n Lepage congruencies Contact 1 -forms Lepage equivalent Θλ

Original Lepage’s idea - III n Application ü For a vector field [pi , qi], pi(x, y, z 1, …, zn), qi(x, y, z 1, …, zn), denote [Ω]=Ω(x, y, zi, pi(x, y, z 1, …, zn), qi(x, y, z 1, …, zn)) ü Definition: A field [pi , qi] is called geodesic with respect to the form Ω, if d[Ω]= 0. ü Proposition: A field [pi , qi] is geodesic with respect to the form Ω, iff Lepage forms - Olomouc - 25. 8. 2007 10

Dedecker’s paper n 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 11

Dedecker’s contribution 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 13

Lepage forms after Krupka n Basic structure n Horizontal and „pseudovertical“ forms n Lepage n-forms Lepage forms - Olomouc - 25. 8. 2007 14

Lepage equivalents after n Lepage equivalent of a Lagrangian Krupka – first version n Example for n=1 (mechanics) 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 n Lepage n-forms on Jr. Y - definition 17

Lepage forms after Krupka n Lepage n-forms on Jr. Y – theorem An n-form on Jr. Y is Lepage form iff it holds Lepage forms - Olomouc - 25. 8. 2007 18

n Examples of Lepage equivalents ü mechanics (unique Lepage equivalent) Lepage equivalents: Krupka ü field theory (non-uniqueness, it depends on the order of Lagrangian) (2 r-1)th order For rth order Lagrangian Lepage forms - Olomouc - 25. 8. 2007 19

§ Fundamental LE (for 1 st order Lagrangian) Examples of LE: field theory § 2 nd order Lagrangian dΘλ =0 iff Eλ =0 Lepage forms - Olomouc - 25. 8. 2007 20

Role of Lepage equivalents n variational function n Euler-Lagrange function n first variational formula (ξ…π-projectable) Lepage forms - Olomouc - 25. 8. 2007 Eλ(L) 21

Variational sequence 0 d r d 0 0 d r 1 r 2 d . . . d r. P d d r 1 d . . . d d r 2 F 1 E 0 / r 1 0 0 r 1 F 2 E 1 / r 2 0 r 2 r P FP E 2 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 n Problem: n 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 24

Representation of VS - II n Variational sequence – finite order: Ø 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) Ø 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 Ø Other authors (Vitolo and Palese, Grigore) k=n, n+1, n+2, alternative approaches Lepage forms - Olomouc - 25. 8. 2007 25

Representation of VS - III n 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: Ø Lie derivative with respect to vector fields along maps Ø proofs appropriate for finite order problem Ø generalization of integration by parts Lepage forms - Olomouc - 25. 8. 2007 26

Representation of VS - IV n 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 27

Representation of VS - V § Step 1 – Integration by parts ρ: (n+k)-form, R(ρ): local k-contact (n+k-1)-form Lepage forms - Olomouc - 25. 8. 2007 28

Representation of VS - VI n 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. Lepage forms - Olomouc - 25. 8. 2007 29

Representation of VS - VII n Properties of Euler operator W … open subset of Y, ρ … (n+k)-form on Jr. W, 1≤ k ≤ N-n, N … dim Jr. Y. Lepage forms - Olomouc - 25. 8. 2007 30

Generalized Lepage forms n Lepage forms as a “product” of VS An (n+k)-form ρ on Jr. Y is called Lepage form, if following equivalent conditions hold. 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 n Lagrangians for geodesics n Lepage equivalents Lepage forms - Olomouc - 25. 8. 2007 32

Examples of LE – a string I n Lagrangian – standard n Lepage equivalents … ρPC=Θλ, fundamental Lepage forms - Olomouc - 25. 8. 2007 33

Examples of LE – a string II n Lagrangian – for Polyakov action n Lepage equivalents … ρPC=Θλ, fundamental Lepage forms - Olomouc - 25. 8. 2007 34

Demo Krupka as a teacher n Main courses and seminars on Masaryk University ü 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 ü ü ü ü Lepage forms - Olomouc - 25. 8. 2007 35

Forms in physics education n 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) n 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