Integration by parts and vector differential forms in higher order variational calculus onfibred manifolds

Roman Ya. Matsyuk

Infinitesimal variation of Action functional in classical (non-quantum) field theory with higher derivatives is presented in terms of well-defined intrinsic geometric objects independent of the particular field which varies. ``Integration by parts'' procedure for this variation consists in application of nonlinear Green formula to the vertical differential of the Lagrangian. Euler-Lagrange expressions and the Green operator are calculated by simple pull-backs of certain vector bundle valued differential forms associated with the given variational problem.

1. N.Bourbaki,Variety Diffrentielles et Analitiques.Fasciculedersultats,Hermann,Paris, 1971.

2. P.E.J.Dhooghe,Contact transformations, contact algebras,and lifts on jet bundles, Ann. Math. Pura ed Appl. 131 (1982),291-300.

3. M.Ferraris and M.Francaviglia, Energy-momentum tensors and stress tensors in geometric field theories, J.Math.Phys. 26 (1985), no.6, 1243-1252.

4. H. Goldschmidt and Sh.Sternberg, The Hamilton-Cartan formalism in the calculus of variations, Ann. Inst. Fourier 23 (1973),no.1,203-267.

5. M. Horak and I.Kolar, On the higher order Poincare-Cartan forms, Czechoslovak Math. J. 33 (1983), no. 3, 467-475.

6. I.Kolar, Lie derivatives and higher-order Lagrangians, Proceedings of the Conference (CSSR- GDR-Poland) on Differential Geometry and its Applications, September, 1980,U niv. Karlova, Praha, 1981,pp. 117-123.

7. I.Kolar, On the Euler-Lagrange differential infibred manifolds, Repts Math. Phys. 12 (1977), no.3, 301-305.

8. J.L.Koszul, Lectures on Fibre Bundles and Differential Geometry, Tata Institute of Fundamental Research, Bombay, 1960.

9. D.Krupka, Some Geometric Aspects of Variational Problems in Fibred Manifolds, Folia Facultatis Scientiarum Naturalium Universitatis Purkynianae Brunensis. Physica 15,tomus XIV, opus 10, Universita J. E. Purkyne, Brno, 1973,65pp.

10. R.Ya.Matsyuk, Poincare-invariant Equations of Motion in Lagrangian Mechanics With Higher Derivatives, Thesis, Institute for Applied Problems in Mechanics and Mathematics, Academy of Science. Ukraine - L'viv, 1984, pp.140.(Russian)

11. R.Ya.Matsyuk, Exterior differential equations in generalized mechanics and symmetry properties, Methods for Studying Differential and Integral Operators, Naukova Dumka, Keyive, 1989,pp.153-160.(Russian)

12. R.Ya.Matsyuk., Geometrical meaning of integration by parts in variational calculus, Lobachevskij and the Contemporary Geometry, An International Scientic Conference. Kazan', August18-22. Theses. Part I., Kazan' State Univ., Kazan', 1992, pp.60-11.(Russian)

13. R.Ya.Matsyuk, Symmetries of vector exterior differential systems and the inverse problem in second-order Ostrohrads'kyj mechanics, Symmetry in Nonlinear Mathematical Physics, Proc. conf.held at Keyive, Ukraine, July 3-8, 1995, Journal of Nonlinear Mathematical Physics 4 (1997), no.1/2, 89-97.

14. Jaime Munoz Masque, Poincare-Cartan forms in higher order variational calculus onfibred manifolds, Revista Matematica Iberoamericana 1 (1985),no.4,85-126. 15.

15. A.Trautman, Invariance of Lagrangian systems, General Relativity. Papers in honour of J.L.Synge (L.O'Raifeartaigh,ed.), Clarendon Press, Oxford, 1975, pp.85-99.

16. W.M.Tulczyjew, The Euler-Lagrange resolution, Differential Geometrical Methods in Mathematical Physics. Proc. conf. 1979, Lecture Notes in Mathematics, 836, Springer-Verlag, Berlin e.a, 1980, pp.22-48.

17. F.A.E. Pirani, D.C. Robinson and W.F. Shadwick, Local Jet Bundle Formulation of Backlund Transformations, D.Reidel, Dordrecht, 1979.

18. J.F.Pommaret, Systems of Partial Differential Equations and Lie Pseudogroups, Gordon & Breach, New York, 1978.

19. Sh.Sternberg, Lecture Notes on Differential Geometry, Prentice Hall, Englewood Cliffs, N.J., 1964.

	Integration by parts and vector differential forms in higher order variational calculus onfibred manifolds
Author	Roman Ya. Matsyuk wlanc@litech.lviv.ua Institute for Applied Problems in Mechanics and Mathematics, L'viv
Abstract	Infinitesimal variation of Action functional in classical (non-quantum) field theory with higher derivatives is presented in terms of well-defined intrinsic geometric objects independent of the particular field which varies. ``Integration by parts'' procedure for this variation consists in application of nonlinear Green formula to the vertical differential of the Lagrangian. Euler-Lagrange expressions and the Green operator are calculated by simple pull-backs of certain vector bundle valued differential forms associated with the given variational problem.
Keywords	Action functional, variational calculus, fibred manifold
DOI	doi:10.30970/ms.11.1.85-107
Reference	1. N.Bourbaki,Variety Diffrentielles et Analitiques.Fasciculedersultats,Hermann,Paris, 1971. 2. P.E.J.Dhooghe,Contact transformations, contact algebras,and lifts on jet bundles, Ann. Math. Pura ed Appl. 131 (1982),291-300. 3. M.Ferraris and M.Francaviglia, Energy-momentum tensors and stress tensors in geometric field theories, J.Math.Phys. 26 (1985), no.6, 1243-1252. 4. H. Goldschmidt and Sh.Sternberg, The Hamilton-Cartan formalism in the calculus of variations, Ann. Inst. Fourier 23 (1973),no.1,203-267. 5. M. Horak and I.Kolar, On the higher order Poincare-Cartan forms, Czechoslovak Math. J. 33 (1983), no. 3, 467-475. 6. I.Kolar, Lie derivatives and higher-order Lagrangians, Proceedings of the Conference (CSSR- GDR-Poland) on Differential Geometry and its Applications, September, 1980,U niv. Karlova, Praha, 1981,pp. 117-123. 7. I.Kolar, On the Euler-Lagrange differential infibred manifolds, Repts Math. Phys. 12 (1977), no.3, 301-305. 8. J.L.Koszul, Lectures on Fibre Bundles and Differential Geometry, Tata Institute of Fundamental Research, Bombay, 1960. 9. D.Krupka, Some Geometric Aspects of Variational Problems in Fibred Manifolds, Folia Facultatis Scientiarum Naturalium Universitatis Purkynianae Brunensis. Physica 15,tomus XIV, opus 10, Universita J. E. Purkyne, Brno, 1973,65pp. 10. R.Ya.Matsyuk, Poincare-invariant Equations of Motion in Lagrangian Mechanics With Higher Derivatives, Thesis, Institute for Applied Problems in Mechanics and Mathematics, Academy of Science. Ukraine - L'viv, 1984, pp.140.(Russian) 11. R.Ya.Matsyuk, Exterior differential equations in generalized mechanics and symmetry properties, Methods for Studying Differential and Integral Operators, Naukova Dumka, Keyive, 1989,pp.153-160.(Russian) 12. R.Ya.Matsyuk., Geometrical meaning of integration by parts in variational calculus, Lobachevskij and the Contemporary Geometry, An International Scientic Conference. Kazan', August18-22. Theses. Part I., Kazan' State Univ., Kazan', 1992, pp.60-11.(Russian) 13. R.Ya.Matsyuk, Symmetries of vector exterior differential systems and the inverse problem in second-order Ostrohrads'kyj mechanics, Symmetry in Nonlinear Mathematical Physics, Proc. conf.held at Keyive, Ukraine, July 3-8, 1995, Journal of Nonlinear Mathematical Physics 4 (1997), no.1/2, 89-97. 14. Jaime Munoz Masque, Poincare-Cartan forms in higher order variational calculus onfibred manifolds, Revista Matematica Iberoamericana 1 (1985),no.4,85-126. 15. 15. A.Trautman, Invariance of Lagrangian systems, General Relativity. Papers in honour of J.L.Synge (L.O'Raifeartaigh,ed.), Clarendon Press, Oxford, 1975, pp.85-99. 16. W.M.Tulczyjew, The Euler-Lagrange resolution, Differential Geometrical Methods in Mathematical Physics. Proc. conf. 1979, Lecture Notes in Mathematics, 836, Springer-Verlag, Berlin e.a, 1980, pp.22-48. 17. F.A.E. Pirani, D.C. Robinson and W.F. Shadwick, Local Jet Bundle Formulation of Backlund Transformations, D.Reidel, Dordrecht, 1979. 18. J.F.Pommaret, Systems of Partial Differential Equations and Lie Pseudogroups, Gordon & Breach, New York, 1978. 19. Sh.Sternberg, Lecture Notes on Differential Geometry, Prentice Hall, Englewood Cliffs, N.J., 1964.
Pages	85-107
Volume	11
Issue	1
Year	1999
Journal	Matematychni Studii
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Table of content of issue	html

Integration by parts and vector differential forms in higher order variational calculus on fibred manifolds

Integration by parts and vector differential forms in higher order variational calculus onfibred manifolds