Skip to main content Accessibility help
×
Home

CONSERVED QUANTITIES ON MULTISYMPLECTIC MANIFOLDS

  • LEONID RYVKIN (a1), TILMANN WURZBACHER (a2) and MARCO ZAMBON (a3)

Abstract

Given a vector field on a manifold $M$ , we define a globally conserved quantity to be a differential form whose Lie derivative is exact. Integrals of conserved quantities over suitable submanifolds are constant under time evolution, the Kelvin circulation theorem being a well-known special case. More generally, conserved quantities are well behaved under transgression to spaces of maps into $M$ . We focus on the case of multisymplectic manifolds and Hamiltonian vector fields. Our main result is that in the presence of a Lie group of symmetries admitting a homotopy co-momentum map, one obtains a whole family of globally conserved quantities. This extends a classical result in symplectic geometry. We carry this out in a general setting, considering several variants of the notion of globally conserved quantity.

Copyright

Corresponding author

References

Hide All
[1] Audin, M., Torus Actions on Symplectic Manifolds, Progress in Mathematics (Birkhäuser, Basel, 2004).
[2] Callies, M., Fregier, Y., Rogers, C. L. and Zambon, M., ‘Homotopy moment maps’, Adv. Math. 303 (2016), 9541043.
[3] Cantrijn, F., Ibort, A. and De Leon, M., ‘On the geometry of multisymplectic manifolds’, J. Aust. Math. Soc. Ser. A 66 (1999), 303330.
[4] Chorin, A. J. and Marsden, J. E., A Mathematical Introduction to Fluid Mechanics, 3rd edn, Texts in Applied Mathematics, 4 (Springer, New York, 1993).
[5] Frégier, Y., Laurent-Gengoux, C. and Zambon, M., ‘A cohomological framework for homotopy moment maps’, J. Geom. Phys. 97 (2015), 119132.
[6] Hélein, F. and Kouneiher, J., ‘Covariant Hamiltonian formalism for the calculus of variables with several variables’, Adv. Theor. Math. Phys. 8 (2004), 565601.
[7] Madsen, T. B. and Swann, A., ‘Closed forms and multi-moment maps’, Geom. Dedicata 165 (2013), 2552.
[8] Rogers, C. L., ‘ L -algebras from multisymplectic geometry’, Lett. Math. Phys. 100(1) (2012), 2950.
[9] Rogers, C. L., ‘2-plectic geometry, Courant algebroids, and categorified prequantization’, J. Symplectic Geom. 11(1) (2013), 5391.
[10] Ryvkin, L. and Wurzbacher, T., ‘Existence and unicity of co-moments in multisymplectic geometry’, Differential Geom. Appl. 41 (2015), 111.
[11] Schreiber, U., ‘Differential cohomology in a cohesive infinity-topos’, Preprint, 2013,arXiv:1310.7930.
[12] Zambon, M., ‘ L -algebras and higher analogues of Dirac structures and Courant algebroids’, J. Symplectic Geom. 10(4) (2012), 563599.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

Related content

Powered by UNSILO

CONSERVED QUANTITIES ON MULTISYMPLECTIC MANIFOLDS

  • LEONID RYVKIN (a1), TILMANN WURZBACHER (a2) and MARCO ZAMBON (a3)

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.