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Lepage forms

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.

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Lepage forms

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  1. 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

  2. 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 the variational principle) Lepage forms - Olomouc - 25. 8. 2007

  3. 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

  4. Name LEPAGE … from 5 130 000 results on Google the most interesting is • Minor planet (Nr. 2795) Lepagea=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

  5. 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

  6. Lepage’s keypaper • 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

  7. Lepage forms - Olomouc - 25. 8. 2007

  8. 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 Lepage forms - Olomouc - 25. 8. 2007

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

  10. 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 Lepage forms - Olomouc - 25. 8. 2007

  11. 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

  12. 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

  13. 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

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

  15. Lepage equivalents after Krupka – first version • Lepage equivalent of a Lagrangian • Example for n=1 (mechanics) Lepage forms - Olomouc - 25. 8. 2007

  16. 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

  17. Lepage forms after Krupka • Lepage n-forms on JrY - definition

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

  19. 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) (2r-1)th order For rth order Lagrangian Lepage forms - Olomouc - 25. 8. 2007

  20. Examples of LE:field theory • Poincaré-Cartan (cont≤1, unique) • Fundamental LE (for 1st order Lagrangian) • 2nd order Lagrangian dΘλ =0 iff Eλ=0 Lepage forms - Olomouc - 25. 8. 2007

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

  22. 0 0 0 P+1 N d d d ... d d 0 0 d d d ... d d d ... d d 0  0 / / / E E E E               r r r r r r r r r r r r r r 1 0 P 2 2 2 P 2 1 1 2 1 1 P P P E ...  r P- 1 F F F 0 P 1 2 0 0 0 Variational sequence Lepage forms - Olomouc - 25. 8. 2007

  23. „Physical“ part of VS Ε: λ → ΕλΗ: E → HE dynamical forms E-L forms H-S forms trivial Lagrangians Lagrangians n-forms (n+1)-forms (n+2)-forms Lepage forms - Olomouc - 25. 8. 2007

  24. 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) Lepage forms - Olomouc - 25. 8. 2007

  25. 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

  26. 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

  27. 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) interiorEuler operator assigning to a form (class of forms in the variational sequence) its representative. Lepage forms - Olomouc - 25. 8. 2007

  28. 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

  29. 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. Lepage forms - Olomouc - 25. 8. 2007

  30. 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. Lepage forms - Olomouc - 25. 8. 2007

  31. 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. For mechanics see D. Krupka and J. Šeděnková, Proc. of DGA 2004, Charles University, Prague 2005 Lepage forms - Olomouc - 25. 8. 2007

  32. Examples of LE – a particle • Lagrangians for geodesics • Lepage equivalents Lepage forms - Olomouc - 25. 8. 2007

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

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

  35. 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

  36. Forms in physics education • Integrating differential forms after Spivak, general Stokes theorem • Student’s comment: “This is a self-production of Jacobians!” Michael Spivak: Calculus on manifolds. Perseus Books, Cambridge, Massachusetts,1998, 27-th edition. (1-st edition 1965) Lepage forms - Olomouc - 25. 8. 2007

  37. Concluding theorem Theorem Excellent scientist and enthusiastic teacher successful students Lepage forms - Olomouc - 25. 8. 2007

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