Hamiltonian and Lagrangian Flows on Center Manifolds: With Applications to Elliptic Variational Problems
The theory of center manifold reduction is studied in this monograph in the context of (infinite-dimensional) Hamil- tonian and Lagrangian systems. The aim is to establish a "natural reduction method" for Lagrangian systems to their center manifolds. Nonautonomous problems are considered as well assystems invariant under the action of a Lie group ( including the case of relative equilibria). The theory is applied to elliptic variational problemson cylindrical domains. As a result, all bounded solutions bifurcating from a trivial state can be described by a reduced finite-dimensional variational problem of Lagrangian type. This provides a rigorous justification of rod theory from fully nonlinear three-dimensional elasticity. The book will be of interest to researchers working in classical mechanics, dynamical systems, elliptic variational problems, and continuum mechanics. It begins with the elements of Hamiltonian theory and center manifold reduction in order to make the methods accessible to non-specialists, from graduate student level.
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Notations and basic facts on center manifolds
The linear theory
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Hamiltonian and Lagrangian Flows on Center Manifolds: with Applications to ...
No preview available - 1991
According action analysis analytic applications associated assume Banach space bounded bracket calculation called canonical coordinates center manifold Chapter classical close completely condition consider constant construct corresponding cross-section defined depend developed differential equation domain eigenvalues elliptic energy equation equilibrium exactly example existence expansion exterior derivative fact function G-invariant given gives Hamiltonian system Hence holds implies integral invariant invertible Lagrangian flow Lagrangian system Lemma Lie group linear locally mapping means modelled Moreover natural reduction neighborhood Note obtain original particular periodic Poisson positive possible Proceedings Proof properties provides question reduced Hamiltonian reduced Lagrangian regular relation respect restriction result reversibility satisfies shown smooth solution submanifold subspace symmetry symplectic form symplectic structure takes Theorem theory transform two-form unique variable variational problem vector field yields