Relative Critical Points

dc.contributor.authorLewis, D.
dc.date.accessioned2019-02-19T18:29:11Z
dc.date.available2019-02-19T18:29:11Z
dc.date.issued2013
dc.description.abstractRelative equilibria of Lagrangian and Hamiltonian systems with symmetry are critical points of appropriate scalar functions parametrized by the Lie algebra (or its dual) of the symmetry group. Setting aside the structures – symplectic, Poisson, or variational – generating dynamical systems from such functions highlights the common features of their construction and analysis, and supports the construction of analogous functions in non-Hamiltonian settings. If the symmetry group is nonabelian, the functions are invariant only with respect to the isotropy subgroup of the given parameter value. Replacing the parametrized family of functions with a single function on the product manifold and extending the action using the (co)adjoint action on the algebra or its dual yields a fully invariant function. An invariant map can be used to reverse the usual perspective: rather than selecting a parametrized family of functions and finding their critical points, conditions under which functions will be critical on specific orbits, typically distinguished by isotropy class, can be derived. This strategy is illustrated using several well-known mechanical systems – the Lagrange top, the double spherical pendulum, the free rigid body, and the Riemann ellipsoids – and generalizations of these systems.uk_UA
dc.description.sponsorshipThis paper is a contribution to the Special Issue “Symmetries of Dif ferential Equations: Frames, Invariants and Applications”. The full collection is available at http://www.emis.de/journals/SIGMA/SDE2012.html. The author is indebted to the referees for their many valuable suggestions and corrections. Their insightful contributions greatly improved this work.uk_UA
dc.identifier.citationRelative Critical Points / Lewis D. // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 53 назв. — англ.uk_UA
dc.identifier.issn1815-0659
dc.identifier.other2010 Mathematics Subject Classification: 37J15; 53D20; 58E09; 70H33
dc.identifier.otherDOI: http://dx.doi.org/10.3842/SIGMA.2013.038
dc.identifier.urihttps://nasplib.isofts.kiev.ua/handle/123456789/149195
dc.language.isoenuk_UA
dc.publisherІнститут математики НАН Україниuk_UA
dc.relation.ispartofSymmetry, Integrability and Geometry: Methods and Applications
dc.statuspublished earlieruk_UA
dc.titleRelative Critical Pointsuk_UA
dc.typeArticleuk_UA

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