Lie algebras of conservation laws of variational partial differential equations
-
Emanuele Fiorani
Abstract
We establish a version of Noether’s first Theorem according to which the (equivalence classes of) conserved quantities of given Euler–Lagrange equations in several independent variables are in one-to-one correspondence with the (equivalence classes of) vector fields satisfying an appropriate pair of geometric conditions, namely: (a) they preserve the class of vector fields tangent to holonomic submanifolds of a jet space; (b) they leave invariant the action from which the Euler–Lagrange equations are derived, modulo terms identically vanishing along holonomic submanifolds. Such a bijective correspondence Φ͠ between equivalence classes comes from an explicit (non-bijective) linear map Φ from vector fields into conserved differential operators, and not from a map into divergences of conserved operators as it occurs in other proofs of Noether’s Theorem. Where possible, claims are given a coordinate-free formulation and all proofs rely just on basic differential geometric properties of finite-dimensional manifolds.
Acknowledgements
We are grateful to Franco Cardin and Juha Pohjanpelto for very useful discussions on various aspects of this paper.
Funding: This research was partially supported by the Project MIUR “Real and Complex Manifolds: Geometry, Topology and Harmonic Analysis” and by GNSAGA of INdAM.
References
[1] S. C. Anco, J. Pohjanpelto, Classification of local conservation laws of Maxwell’s equations. Acta Appl. Math. 69 (2001), 285–327. MR1885280 Zbl 0989.3512610.1023/A:1014263903283Search in Google Scholar
[2] I. M. Anderson, The variational bicomplex. Utah State University preprint (1989), http://math.uni.lu/~jubin/seminar/bicomplex.pdf.Search in Google Scholar
[3] A. V. Bocharov, V. N. Chetverikov, S. V. Duzhin, N. G. Khor’ kova, I. S. Krasil’ shchik, A. V. Samokhin, Y. N. Torkhov, A. M. Verbovetsky, A. M. Vinogradov, Symmetries and conservation laws for differential equations of mathematical physics, volume 182 of Translations of Mathematical Monographs. Amer. Math. Soc. 1999. MR1670044 Zbl 0911.00032Search in Google Scholar
[4] E. Fiorani, A. Spiro, Lie algebras of conservation laws of variational ordinary differential equations. J. Geom. Phys. 88 (2015), 56–75. MR3293396 Zbl 1308.7005010.1016/j.geomphys.2014.11.005Search in Google Scholar
[5] S. Germani, Leggi di conservazione e simmetrie: un approccio geometrico al Teorema di Noether. Tesi di Laurea Magistrale, Università di Camerino, Camerino, 2012.Search in Google Scholar
[6] M. Gromov, Partial differential relations. Springer 1986. MR864505 Zbl 0651.5300110.1007/978-3-662-02267-2Search in Google Scholar
[7] D. Joyce, On manifolds with corners. In: Advances in geometric analysis, volume 21 of Adv. Lect. Math. (ALM), 225–258, Int. Press, Somerville, MA 2012. MR3077259 Zbl 1317.58001Search in Google Scholar
[8] Y. Kosmann-Schwarzbach, The Noether theorems. Springer 2011. MR2761345 Zbl 1216.0101110.1007/978-0-387-87868-3Search in Google Scholar
[9] I. S. Krasil’ shchik, V. V. Lychagin, A. M. Vinogradov, Geometry of jet spaces and nonlinear partial differential equations, volume 1 of Advanced Studies in Contemporary Mathematics. Gordon and Breach Science Publishers, New York 1986. MR861121 Zbl 0722.35001Search in Google Scholar
[10] B. A. Kupershmidt, Geometry of jet bundles and the structure of Lagrangian and Hamiltonian formalisms. In: Geometric methods in mathematical physics (Proc. NSF-CBMS Conf., Univ. Lowell, Lowell, Mass., 1979), volume 775 of Lecture Notes in Math., 162–218, Springer 1980. MR569303 Zbl 0439.5801610.1007/BFb0092026Search in Google Scholar
[11] A. Kushner, V. Lychagin, V. Rubtsov, Contact geometry and non-linear differential equations, volume 101 of Encyclopedia of Mathematics and its Applications. Cambridge Univ. Press 2007. MR2352610 Zbl 1122.5304410.1017/CBO9780511735141Search in Google Scholar
[12] J. M. Lee, Introduction to smooth manifolds. Springer 2013. MR2954043 Zbl 1258.53002Search in Google Scholar
[13] V. V. Lyčagin, Contact geometry and second-order nonlinear differential equations (Russian). Uspekhi Mat. Nauk34 (1979), no. 1(205), 137–165. MR525652 Zbl 0427.58002Search in Google Scholar
[14] J. Milnor, Morse theory. Princeton Univ. Press 1963. MR0163331 Zbl 0108.1040110.1515/9781400881802Search in Google Scholar
[15] E. Noether, Invariante Variationsprobleme. Nachr. Königl. Ges. Wissensch. Göttingen, Math.-Phys. Klasse (1918), 235–257. Zbl 46.0770.0110.1007/978-3-642-39990-9_13Search in Google Scholar
[16] P. J. Olver, Applications of Lie groups to differential equations. Springer 1986. MR836734 Zbl 0588.2200110.1007/978-1-4684-0274-2Search in Google Scholar
[17] P. J. Olver, Noether’s theorems and systems of Cauchy–Kovalevskaya type. In: Nonlinear systems of partial differential equations in applied mathematics, Part 2 (Santa Fe, N.M., 1984), volume 23 of Lectures in Appl. Math., 81–104, Amer. Math. Soc. 1986. MR837699 Zbl 0656.58039Search in Google Scholar
[18] P. J. Olver, Book review of [8], Bull. Amer. Math. Soc. (N.S.) 50 (2013), 161–167. MR299499910.1090/S0273-0979-2011-01364-7Search in Google Scholar
[19] A. Spiro, Cohomology of Lagrange complexes invariant under pseudogroups of local transformations. Int. J. Geom. Methods Mod. Phys. 4 (2007), 669–705. MR2343431 Zbl 1153.7000910.1142/S0219887807002223Search in Google Scholar
[20] F. Takens, A global version of the inverse problem of the calculus of variations. J. Differential Geom. 14 (1979), 543–562 (1981). MR600611 Zbl 0463.5801510.4310/jdg/1214435235Search in Google Scholar
[21] K. Yamaguchi, Contact geometry of higher order. Japan. J. Math. (N.S.) 8 (1982), 109–176. MR722524 Zbl 0548.5800210.4099/math1924.8.109Search in Google Scholar
[22] K. Yamaguchi, Geometrization of jet bundles. Hokkaido Math. J. 12 (1983), 27–40. MR689254 Zbl 0561.5800210.14492/hokmj/1381757789Search in Google Scholar
© 2018 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Negative refraction and tiling billiards
- Classification of degree two curves in the symmetric square with positive self-intersection
- On lattice coverings by simplices
- On CMC hypersurfaces in 𝕊n+1 with constant Gauß–Kronecker curvature
- Fully truncated simplices and their monodromy groups
- Lie algebras of conservation laws of variational partial differential equations
- Feet in orthogonal-Buekenhout–Metz unitals
- Pseudo-metric 2-step nilpotent Lie algebras
Articles in the same Issue
- Frontmatter
- Negative refraction and tiling billiards
- Classification of degree two curves in the symmetric square with positive self-intersection
- On lattice coverings by simplices
- On CMC hypersurfaces in 𝕊n+1 with constant Gauß–Kronecker curvature
- Fully truncated simplices and their monodromy groups
- Lie algebras of conservation laws of variational partial differential equations
- Feet in orthogonal-Buekenhout–Metz unitals
- Pseudo-metric 2-step nilpotent Lie algebras
