## Abstract

We will pose the inverse problem question within the Krupka variational sequence framework. In particular, the interplay of inverse problems with symmetry and invariance properties will be exploited considering that the cohomology class of the variational Lie derivative of an equivalence class of forms, closed in the variational sequence, is trivial. We will focalize on the case of symmetries of globally defined field equations which are only locally variational and prove that variations of local Noether strong currents are variationally equivalent to global canonical Noether currents. Variations, taken to be generalized symmetries and also belonging to the kernel of the second variational derivative of the local problem, generate canonical Noether currents - associated with variations of local Lagrangians - which in particular turn out to be conserved along any section. We also characterize the variation of the canonical Noether currents associated with a local variational problem.

## References

[1] G. Allemandi, M. Francaviglia, M. Raiteri: Covariant charges in Chern-Simons AdS3 gravity. Classical Quantum Gravity 20 (3) (2003) 483-506.

[2] I. M. Anderson, T. Duchamp: On the existence of global variational principles. Amer. Math. J. 102 (1980) 781-868.

[3] D. Bashkirov, G. Giachetta, L. Mangiarotti, G. Sardanashvily: Noether's second theorem for BRST symmetries. J. Math. Phys. 46 (5) (2005). 053517, 23 pp.

[4] D. Bashkirov, G. Giachetta, L. Mangiarotti, G. Sardanashvily: Noether's second theorem in a general setting reducible gauge theories. J. Phys. A38 (2005) 5329-5344.

[5] D. Bashkirov, G. Giachetta, L. Mangiarotti, G. Sardanashvily: The antifield Koszul-Tate complex of reducible Noether identities. J. Math. Phys. 46 (10) (2005). 103513, 19 pp.

[6] E. Bessel-Hagen: Über die Erhaltungssätze der Elektrodynamik. Math. Ann. 84 (1921) 258-276.

[7] A. Borowiec, M. Ferraris, M. Francaviglia, M. Palese: Conservation laws for non-global Lagrangians. Univ. Iagel. Acta Math. 41 (2003) 319-331.

[8] J. Brajerèík, D. Krupka: Variational principles for locally variational forms. J. Math. Phys. 46 (5) (2005). 052903, 15 pp

[9] F. Cattafi, M. Palese, E. Winterroth: Variational derivatives in locally Lagrangian field theories and Noether-Bessel-Hagen currents. Int. J. Geom. Methods Mod. Phys. 13 (8) (2016). 1650067

[10] P. Dedecker, W. M. Tulczyjew: Spectral sequences and the inverse problem of the calculus of variations. In: Lecture Notes in Mathematics. Springer-Verlag (1980) 498-503.

[11] D. J. Eck: Gauge-natural bundles and generalized gauge theories. Mem. Amer. Math. Soc. 247 (1981) 1-48.

[12] M. Ferraris, M. Francaviglia, M. Raiteri: Conserved Quantities from the Equations of Motion (with applications to natural and gauge natural theories of gravitation). Class.Quant.Grav. 20 (2003) 4043-4066.

[13] M. Ferraris, M. Palese, E. Winterroth: Local variational problems and conservation laws. Diff. Geom. Appl 29 (2011) S80-S85.

[14] M. Francaviglia, M. Palese, R. Vitolo: Symmetries in finite order variational sequences. Czech. Math. J. 52 (1) (2002) 197-213.

[15] M. Francaviglia, M. Palese, R. Vitolo: The Hessian and Jacobi Morphisms for Higher Order Calculus of Variations. Diff. Geom. Appl. 22 (1) (2005) 105-120.

[16] M. Francaviglia, M. Palese, E. Winterroth: Locally variational invariant field equations and global currents: Chern-Simons theories. Commun. Math. 20 (1) (2012) 13-22.

[17] M. Francaviglia, M. Palese, E. Winterroth: Variationally equivalent problems and variations of Noether currents. Int. J. Geom. Meth. Mod. Phys. 10 (1) (2013). 1220024

[18] M. Francaviglia, M. Palese, E. Winterroth: Cohomological obstructions in locally variational field theories. Jour. Phys. Conf. Series 474 (2013). 012017

[19] G. Giachetta, L. Mangiarotti, G. Sardanashvily: Lagrangian supersymmetries depending on derivatives. Global analysis and cohomology. Comm. Math. Phys. 259 (1) (2005) 103-128.

[20] Y. Kosmann-Schwarzbach: The Noether Theorems; translated from French by Bertram E. Schwarzbach. Sources and Studies in the History of Mathematics and Physical Sciences, Springer, New York (2011).

[21] D. Krupka: Some Geometric Aspects of Variational Problems in Fibred Manifolds. Folia Fac. Sci. Nat. UJEP Brunensis 14 (1973) 1-65.

[22] D. Krupka: Variational Sequences on Finite Order Jet Spaces. In: J. Janyška and D. Krupka: Differential Geometry and its Applications, Proc. Conf., Brno, Czechoslovakia. World Scientific (1989) 236-254.

[23] D. Krupka, O. Krupková, G. Prince, W. Sarlet: Contact symmetries of the Helmholtz form. Differential Geom. Appl. 25 (5) (2007) 518-542.

[24] E. Noether: Invariante Variationsprobleme. Nachr. Ges. Wiss. Gött., Math. Phys. Kl. II (1918) 235-257.

[25] M. Palese, O. Rossi, E. Winterroth, J. Musilová: Variational sequences, representation sequences and applications in physics. SIGMA 12 (2016). 045, 45 pages

[26] M. Palese, E. Winterroth: Covariant gauge-natural conservation laws. Rep. Math. Phys. 54 (3) (2004) 349-364.

[27] M. Palese, E. Winterroth: Global Generalized Bianchi Identities for Invariant Variational Problems on Gauge-natural Bundles. Arch. Math. (Brno) 41 (3) (2005) 289-310.

[28] M. Palese, E. Winterroth: Noether Theorems and Reality of Motion. In: Proc. Marcel Grossmann Meeting 2015. World Scientific (2016). to appear

[29] M. Palese, E. Winterroth: Variational Lie derivative and cohomology classes. AIP Conf. Proc. 1360 (2011) 106-112.

[30] M. Palese, E. Winterroth: Topological obstructions in Lagrangian field theories, with an application to 3D Chern-Simons gauge theory. preprint submitted

[31] G. Sardanashvily: Noether conservation laws issue from the gauge invariance of an Euler-Lagrange operator, but not a Lagrangian. arXiv:math-ph/0302012 (2003).

[32] G. Sardanashvily: Noether identities of a differential operator. The Koszul-Tate complex. Int. J. Geom. Methods Mod. Phys. 2 (5) (2005) 873-886.

[33] G. Sardanashvily: Noether's theorems. Applications in mechanics and field theory. Atlantis Press, Paris (2016). xvii+297 pp.

[34] F. Takens: A global version of the inverse problem of the calculus of variations. J. Diff. Geom. 14 (1979) 543-562.

[35] W. M. Tulczyjew: The Lagrange Complex. Bull. Soc. Math. France 105 (1977) 419-431.

[36] A. M. Vinogradov: On the algebro-geometric foundations of Lagrangian field theory. Soviet Math. Dokl. 18 (1977) 1200-1204.