Clebsch Variational Principles in Field Theories and Singular Solutions of Covariant Epdiff Equations
نویسندگان
چکیده
منابع مشابه
Clebsch Variational Principles in Field Theories and Singular Solutions of Covariant Epdiff Equations
This paper introduces and studies a field theoretic analogue of the Clebsch variational principle of classical mechanics. This principle yields an alternative derivation of the covariant Euler-Poincaré equations that naturally includes covariant Clebsch variables via multisymplectic momentum maps. In the case of diffeomorphism groups, this approach gives a new interpretation of recently derived...
متن کاملSergiy Vasylkevych Research Statement 1 Singular Solutions of Epdiff Equations 1.1 Epdiff Equations
Since completion of the Ph.D. program at California Institute of Technology, my research has been focused on application of methods of differential geometry and global analysis to the study of nonlinear partial differential equations. Specifically, it concerns • Euler and Euler-α equations of ideal fluid flow (both fixed and moving boundary case), • Camassa-Holm equation, • EPDiff equations (al...
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The EPDiff equation for geodesic flow on the diffeomorphisms admits a remarkable ansatz for its singular solutions, called “diffeons.” Because this solution ansatz is a momentum map, the diffeons evolve according to canonical Hamiltonian equations. We examine diffeon solutions on Einstein spaces that are “mostly” symmetric, i.e., whose quotient by a subgroup of the isometry group is 1-dimension...
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The Clebsch method provides a unifying approach for deriving variational principles for continuous and discrete dynamical systems where elements of a vector space are used to control dynamics on the cotangent bundle of a Lie group via a velocity map. This paper proves a reduction theorem which states that the canonical variables on the Lie group can be eliminated, if and only if the velocity ma...
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Metric independent σ models are constructed. These are field theories which generalise the membrane idea to situations where the target space has fewer dimensions than the base manifold. Instead of reparametrisation invariance of the independent variables, one has invariance of solutions of the field equations under arbitrary functional redefinitions of the field quantities. Among the many inte...
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ژورنال
عنوان ژورنال: Reports on Mathematical Physics
سال: 2013
ISSN: 0034-4877
DOI: 10.1016/s0034-4877(13)60032-4