World Scientific
Skip main navigation

Cookies Notification

We use cookies on this site to enhance your user experience. By continuing to browse the site, you consent to the use of our cookies. Learn More
×
Our website is made possible by displaying certain online content using javascript.
In order to view the full content, please disable your ad blocker or whitelist our website www.worldscientific.com.

System Upgrade on Mon, Jun 21st, 2021 at 1am (EDT)

During this period, the E-commerce and registration of new users may not be available for up to 6 hours.
For online purchase, please visit us again. Contact us at [email protected] for any enquiries.

The Noether–Bessel-Hagen symmetry approach for dynamical systems

    The Noether–Bessel-Hagen theorem can be considered a natural extension of Noether Theorem to search for symmetries. Here, we develop the approach for dynamical systems introducing the basic foundations of the method. Specifically, we establish the Noether–Bessel-Hagen analysis of mechanical systems where external forces are present. In the second part of the paper, the approach is adopted to select symmetries for a given systems. In particular, we focus on the case of harmonic oscillator as a testbed for the theory, and on a cosmological system derived from scalar–tensor gravity with unknown scalar-field potential V(φ). We show that the shape of potential is selected by the presence of symmetries. The approach results particularly useful as soon as the Lagrangian of a given system is not immediately identifiable or it is not a Lagrangian system.

    AMSC: 70H33, 58E30, 58D19, 53A15

    References

    • 1. S. Capozziello, R. De Ritis, C. Rubano and P. Scudellaro , Noether symmetries in cosmology, Riv. Nuovo Cimento 19(4) (1996) 1. Crossref, ISIGoogle Scholar
    • 2. K. F. Dialektopoulos and S. Capozziello , Noether symmetries as a geometric criterion to select theories of gravity, Int. J. Geom. Methods Modern Phys. 15(Supp. 01) (2018) 1840007. Link, ISIGoogle Scholar
    • 3. E. Bessel-Hagen , Uber die Erhaltungssatze der Electrodynamik, Math. Ann. 84 (1921) 258–276. CrossrefGoogle Scholar
    • 4. Y. Kossmann-Schwarzbach , The Noether Theorems (Springer-Verlag, New York, 2011). CrossrefGoogle Scholar
    • 5. D. Krupka , Introduction to Global Variational Geometry, Atlantis Studies in Variational Geometry, Vol. 1 (Atlantis Press, Amsterdam–Beijing–Paris, 2015). CrossrefGoogle Scholar
    • 6. J. Brajerčík and D. Krupka , Variational principles for locally variational forms, J. Math. Phys. 46 (2005) 052903. Crossref, ISIGoogle Scholar
    • 7. J. Brajerčík and D. Krupka , Cohomology and local variational principles, in Proc. Conf. XV Int. Workshop Geometry and Physics, Puerto de la Cruz, Canary Islands, Spain, 11–16 September 2006, Publ. de la RSME, 2007, pp. 119–124. Google Scholar
    • 8. Z. Urban and J. Brajerčík , The fundamental Lepage form in variational theory for submanifolds, Int. J. Geom. Methods Modern Phys. 15(6) (2018) 1850103. Link, ISIGoogle Scholar
    • 9. F. Cattafi, M. Palese and E. Winterroth , Variational derivatives in locally Lagrangian field theories and Noether–Bessel-Hagen currents, Int. J. Geom. Methods Modern Phys. 13(8) (2016) 1650067. Link, ISIGoogle Scholar
    • 10. M. Palese and E. Winterroth , Topological obstructions in Lagrangian field theories, with an application to 3D Chern–Simons gauge theory, J. Math. Phys. 58 (2017) 023502. Crossref, ISIGoogle Scholar
    • 11. M. Francaviglia, M. Palese and E. Winterroth , Variationally equivalent problems and variations of Noether currents, Int. J. Geom. Methods Modern Phys. 10(1) (2013) 1220024. Link, ISIGoogle Scholar
    • 12. D. Bashkirov et al., Noether’s second theorem in a general setting: Reducible gauge theories, J. Phys. A: Math. Gen. 38 (2005) 5329. CrossrefGoogle Scholar
    • 13. E. Noether , Invariante Variationsprobleme, Nachr. Konig. Gessell. Wissen. Gottingen, Math.-Phys. Kl. 1918 (1918) 235–257. Google Scholar
    • 14. D. Krupka , Invariant variational structures on fibered manifolds, Int. J. Geom. Methods Modern Phys. 12 (2015) 1550020. Link, ISIGoogle Scholar
    • 15. A. Trautman, Invariance of Lagrangian systems, in General Relativity, Papers in Honour of J. L. Synge (Oxford, Clarendon Press, 1972), pp. 85–99. Google Scholar
    • 16. A. Trautman , Noether equations and conservation laws, Comm. Math. Phys. 6 (1967) 248–261. CrossrefGoogle Scholar
    • 17. D. Krupka , A geometric theory of ordinary first order variational problems in fibered manifolds. II. Invariance, J. Math. Anal. Appl. 49 (1975) 469–476. Crossref, ISIGoogle Scholar
    • 18. G. Sardanashvily , Noether’s Theorems, Applications in Mechanics and Field Theory, Atlantis Studies in Variational Geometry, Vol. 3 (Atlantis Press, Amsterdam, 2016). CrossrefGoogle Scholar
    • 19. P. J. Olver , Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, Vol. 107 (Springer-Verlag, New York, 1986). CrossrefGoogle Scholar
    • 20. G. W. Bluman and S. Kumei , Symmetries and Differential Equations (Springer-Verlag, New York, 1989). CrossrefGoogle Scholar
    • 21. D. Krupka , Variational forces, Lepage Research Institute Library 6 (2018) 1–38. Google Scholar
    • 22. N. Chien, T. Honein and G. Herrmann , Dissipative systems, conservation laws and symmetries, Internat. J. Solids Structures 33(20–22) (1996) 2959–2968. Crossref, ISIGoogle Scholar
    • 23. T. Honein, N. Chien and G. Herrmann , On conservation laws for dissipative systems, Phys. Lett. A 155 (1991) 223–224. Crossref, ISIGoogle Scholar
    • 24. S. Capozziello, A. Stabile and A. Troisi , Spherically symmetric solutions in f(R)-gravity via Noether Symmetry Approach, Class. Quantum Grav. 24 (2007) 2153. Crossref, ISIGoogle Scholar
    • 25. A. Paliathanasis, S. Basilakos, E. N. Saridakis, S. Capozziello, K. Atazadeh, F. Darabi and M. Tsamparlis , New Schwarzschild-like solutions in f(T) gravity through Noether symmetries, Phys. Rev. D 89 (2014) 104042. Crossref, ISIGoogle Scholar
    • 26. F. Bajardi, K. F. Dialektopoulos and S. Capozziello , Higher dimensional static and spherically symmetric solutions in extended Gauss–Bonnet gravity, Symmetry 12(3) (2020) 372. Crossref, ISIGoogle Scholar
    • 27. S. Capozziello, N. Frusciante and D. Vernieri , New spherically symmetric solutions in f(R)-gravity by Noether symmetries, Gen. Relativ. Gravit. 44 (2012) 1881. Crossref, ISIGoogle Scholar
    • 28. S. Capozziello and A. De Felice , f(R) cosmology by Noether’s symmetry, J. Cosmol. Astropart. Phys. 0808 (2008) 016. Crossref, ISIGoogle Scholar
    • 29. S. Capozziello, G. Marmo, C. Rubano and P. Scudellaro , Noether symmetries in Bianchi universes, Int. J. Modern Phys. D 6 (1997) 491. Link, ISIGoogle Scholar
    • 30. S. Capozziello, M. De Laurentis and S. D. Odintsov , Hamiltonian dynamics and Noether symmetries in extended gravity cosmology, Eur. Phys. J. C 72 (2012) 2068. Crossref, ISIGoogle Scholar
    • 31. K. Atazadeh and F. Darabi , f(T) cosmology via Noether symmetry, Eur. Phys. J. C 72 (2012) 2016. Crossref, ISIGoogle Scholar
    • 32. D. Krupka, Z. Urban and J. Volná , Variational submanifolds of Euclidean spaces, J. Math. Phys. 59(3) (2018) 032903. Crossref, ISIGoogle Scholar
    • 33. Z. Urban and J. Volná , On a global Lagrangian construction for ordinary variational equations on 2-manifolds, J. Math. Phys. 60(9) (2019) 092902. Crossref, ISIGoogle Scholar
    • 34. J. Volná and Z. Urban , First-order variational sequences in field theory, in The Inverse Problem of the Calculus of Variations, Local and Global Theory, ed. D. Zenkov (Atlantis Press, Amsterdam–Beijing–Paris, 2015), pp. 215–284. CrossrefGoogle Scholar
    • 35. D. Krupka, O. Krupková and D. Saunders , Cartan–Lepage forms in geometric mechanics, Internat. J. Non-Linear Mech. 47 (2012) 1154–1160. Crossref, ISIGoogle Scholar
    • 36. L. D. Landau and E. M. Lifshitz , Mechanics, Course of Theoretical Physics, Vol. 1 (Pergamon Press, Oxford, 1969). Google Scholar
    • 37. M. B. Green, J. H. Schwarz and E. Witten , Superstring Theory. Vol. 1: Introduction, Cambridge Monographs On Mathematical Physics (Cambridge University Press, 1987). Google Scholar
    • 38. M. B. Green, J. H. Schwarz and E. Witten , Superstring Theory. Vol. 2: Loop Amplitudes, Anomalies and Phenomenology, Cambridge Monographs On Mathematical Physics (Cambridge University Press, 1987). Google Scholar
    • 39. J. Polchinski , String Theory. Vol. 1: An Introduction to the Bosonic String (Cambridge University Press, 1998). Google Scholar
    • 40. J. Polchinski , String Theory. Vol. 2: Superstring Theory and Beyond (Cambridge University Press, 1998). Google Scholar
    • 41. K. Becker, M. Becker and J. H. Schwarz , String Theory and M-theory: A Modern Introduction (Cambridge University Press, 2006). CrossrefGoogle Scholar
    • 42. T. Clifton, P. G. Ferreira, A. Padilla and C. Skordis , Modified gravity and cosmology, Phys. Rep. 513 (2012) 1. Crossref, ISIGoogle Scholar
    • 43. T. Han, J. D. Lykken and R. J. Zhang , On Kaluza–Klein states from large extra dimensions, Phys. Rev. D 59 (1999) 105006. Crossref, ISIGoogle Scholar
    • 44. C. Rovelli and F. Vidotto , Covariant Loop Quantum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam Theory (Cambridge University Press, 2014). CrossrefGoogle Scholar
    • 45. A. Ashtekar and P. Singh , Loop quantum cosmology: A status report, Class. Quantum. Grav. 28 (2011) 213001. Crossref, ISIGoogle Scholar
    • 46. C. Rovelli , Loop quantum gravity, Living Rev. Relativ. 1 (1998) 1. CrossrefGoogle Scholar
    • 47. E. Kiritsis and G. Kofinas , Horava–Lifshitz cosmology, Nuclear Phys. B 821 (2009) 467. Crossref, ISIGoogle Scholar
    • 48. R. G. Cai, L. M. Cao and N. Ohta , Topological black holes in Horava–Lifshitz gravity, Phys. Rev. D 80 (2009) 024003. Crossref, ISIGoogle Scholar
    • 49. T. P. Sotiriou , Horava–Lifshitz gravity: A status report, J. Phys. Conf. Ser. 283 (2011) 012034. CrossrefGoogle Scholar
    • 50. S. Mukohyama , Horava–Lifshitz cosmology: A review, Class. Quantum Grav. 27 (2010) 223101. Crossref, ISIGoogle Scholar
    • 51. N. Arkani-Hamed, S. Dimopoulos, G. Dvali and G. Gabadadze, Nonlocal modification of gravity and the cosmological constant problem, preprint (2002), arXiv:hep-th/0209227. Google Scholar
    • 52. L. Modesto and S. Tsujikawa , Non-local massive gravity, Phys. Lett. B 727 (2013) 48. Crossref, ISIGoogle Scholar
    • 53. L. Modesto , Super-renormalizable gravity, in The Thirteenth Marcel Grossmann Meeting, eds. R. T. JantzenK. RosquistR. Ruffini, Proc. Conf. MG13 Meeting on General Relativity, 1–7 July 2012, Stockholm University, Stockholm, Sweden (World Scientific, 2015). https://doi.org/10.1142/9789814623995_0098 LinkGoogle Scholar
    • 54. T. P. Sotiriou and V. Faraoni , f(R) theories of gravity, Rev. Modern Phys. 82 (2010) 451. CrossrefGoogle Scholar
    • 55. A. De Felice and S. Tsujikawa , f(R) theories, Living Rev. Relativ. 13 (2010) 3. Crossref, ISIGoogle Scholar
    • 56. Y. F. Cai, S. Capozziello, M. De Laurentis and E. N. Saridakis , f(T) teleparallel gravity and cosmology, Rep. Progr. Phys. 79(10) (2016) 106901. Crossref, ISIGoogle Scholar
    • 57. R. T. Hammond , Torsion gravity, Rep. Progr. Phys. 65 (2002) 599. Crossref, ISIGoogle Scholar
    • 58. H. I. Arcos and J. G. Pereira , Torsion gravity: A reappraisal, Int. J. Modern Phys. D 13 (2004) 2193. Link, ISIGoogle Scholar
    • 59. S. Capozziello and F. Bajardi , Gravitational waves in modified gravity, Int. J. Modern Phys. D 28(5) (2019) 1942002. Link, ISIGoogle Scholar
    • 60. S. Capozziello and M. De Laurentis , Extended theories of gravity, Phys. Rep. 509 (2011) 167. Crossref, ISIGoogle Scholar
    • 61. S. Capozziello and M. Francaviglia , Extended theories of gravity and their cosmological and astrophysical applications, Gen. Relativ. Gravit. 40 (2008) 357. Crossref, ISIGoogle Scholar
    • 62. A. H. Guth, The inflationary universe: A possible solution to the horizon and flatness problems, Phys. Rev. D 23 (1981) 347. [Adv. Ser. Astrophys. Cosmol. 3 (1987) 139]. Google Scholar
    • 63. A. H. Guth and S. Y. Pi , Fluctuations in the new inflationary universe, Phys. Rev. Lett. 49 (1982) 1110. Crossref, ISIGoogle Scholar
    • 64. A. D. Linde, A new inflationary universe scenario: A possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems, Phys. Lett. B 108 (1982) 389. [Adv. Ser. Astrophys. Cosmol. 3 (1987) 149]. Google Scholar
    • 65. E. J. Copeland, M. Sami and S. Tsujikawa , Dynamics of dark energy, Int. J. Modern Phys. D 15 (2006) 1753. Link, ISIGoogle Scholar
    • 66. F. L. Bezrukov and M. Shaposhnikov , The Standard Model Higgs boson as the inflaton, Phys. Lett. B 659 (2008) 703. Crossref, ISIGoogle Scholar
    • 67. S. Capozziello and R. de Ritis , Relation between the potential and nonminimal coupling in inflationary cosmology, Phys. Lett. A 177 (1993) 1. Crossref, ISIGoogle Scholar
    • 68. S. Capozziello and R. de Ritis , Noether’s symmetries and exact solutions in flat nonminimally coupled cosmological models, Class. Quantum Grav. 11 (1994) 107. Crossref, ISIGoogle Scholar
    • 69. A. Paliathanasis, M. Tsamparlis, S. Basilakos and S. Capozziello , Scalar–tensor gravity cosmology: Noether symmetries and analytical solutions, Phys. Rev. D 89(6) (2014) 063532. Crossref, ISIGoogle Scholar
    • 70. S. Capozziello, R. de Ritis and P. Scudellaro , Noether’s symmetries in (n + 1)-dimensional nonminimally coupled cosmologies, Int. J. Modern Phys. D 2 (1993) 463. Link, ISIGoogle Scholar
    • 71. A. Borowiec, S. Capozziello, M. De Laurentis, F. S. N. Lobo, A. Paliathanasis, M. Paolella and A. Wojnar , Invariant solutions and Noether symmetries in hybrid gravity, Phys. Rev. D 91(2) (2015) 023517. Crossref, ISIGoogle Scholar
    • 72. E. Piedipalumbo, M. De Laurentis and S. Capozziello , Noether symmetries in interacting quintessence cosmology, Phys. Dark Univ. 27 (2020) 100444. Crossref, ISIGoogle Scholar
    Published: 20 October 2020
    Remember to check out the Most Cited Articles!

    Check out new Mathematical Physics books in our Mathematics 2021 catalogue
    Featuring authors Bang-Yen Chen, John Baez, Matilde Marcolli and more!