1,721,046 research outputs found
A bifurcation result for semi-Riemannian trajectories of the Lorentz force equation
No full textAbstractWe obtain a bifurcation result for solutions of the Lorentz equation in a semi-Riemannian manifold; such solutions are critical points of a certain strongly indefinite functionals defined in terms of the semi-Riemannian metric and the electromagnetic field. The flow of the Jacobi equation along each solution preserves the so-called electromagnetic symplectic form, and the corresponding curve in the symplectic group determines an integer valued homology class called the Maslov index of the solution.We study electromagnetic conjugate instants with symplectic techniques, and we prove at first, an analogous of the semi-Riemannian Morse Index Theorem (see (Calculus of Variations, Prentice-Hall, Englewood Cliffs, NJ, USA, 1963)). By using this result, together with recent results on the bifurcation for critical points of strongly indefinite functionals (see (J. Funct.Anal. 162(1) (1999) 52)), we are able to prove that each non-degenerate and non-null electromagnetic conjugate instant along a given solution of the semi-Riemannian Lorentz force equation is a bifurcation point
A note on the regularity and the existence of Riemannian k-splines
No full textIn this paper, we present a comprehensive proof concerning the regularity of critical points for the spline energy functional on Riemannian manifolds, even for the general higher-order case. Although this result is widely acknowledged in the literature, a detailed proof was previously absent. Our proof relies on a generalization of the Lemma of DuBois-Reymond. Furthermore, we establish the existence of minimizers for the spline energy functional in cases where multiple interpolation points are prescribed alongside just one velocity
Deforming solutions of geometric variational problems with varying symmetry groups
No full textWe prove an equivariant implicit function theorem for variational problems that are invariant under a varying symmetry group (corresponding to a bundle of Lie groups). Motivated by applications to families of geometric variational problems lack- ing regularity, several non-smooth extensions of the result are discussed. Among such applications is the submanifold problem of deforming the ambient metric preserving a given variational property of a prescribed family of submanifolds, e.g., constant mean curvature, up to the action of the corresponding ambient isometry groups
Multiple brake orbits in m-dimensional disks
Get PDFLet (M,g) be a (complete) Riemannian surface, and let Ω⊂M be an open subset whose closure is homeomorphic to a disk. We prove that if ∂Ω is smooth and it satisfies a strong concavity assumption, then there are at least two distinct orthogonal geodesics in Ω⋃∂Ω. Using the results given in Giambò et al. (Adv Differ Eq 10:931–960, 2005), we then obtain a proof of the existence of two distinct brake orbits for a class of Hamiltonian systems. In our proof we shall use recent deformation results proved in Giambò et al. (Nonlinear Anal Ser A Theory Methods Appl 73:290–337, 2010)
Functions on the sphere with critical points in pairs and orthogonal geodesic chords
Get PDFUsing an estimate on the number of critical points for a Morse-even function on the sphere S^m, m ≥ 1, we prove a multiplicity result for orthogonal geodesic chords in Riemannian manifolds with boundary that are diffeomorphic to Euclidean balls. This yields also a multiplicity result for brake orbits in a potential well
Multiple orthogonal geodesic chords in nonconvex Riemannian disks using obstacles
Get PDFWe use nonsmooth critical point theory and the theory of geodesics with obstacle to show a multiplicity result about orthogonal geodesic chords in a Riemannian manifold (with boundary) which is homeomorphic to an N-disk. This applies to brake orbits in a potential well of a natural Hamiltonian system, providing a further step towards the proof of a celebrated conjecture by Seifert (Math Z 51:197–216, 1948)
On the semi-Riemannian bumpy metric theorem.
No full textWe prove the semi-Riemannian bumpy metric theorem using equivariant variational genericity.
The theorem states that, on a given compact manifold M, the set of semi-Riemannian metrics
that admit only nondegenerate closed geodesics is generic relatively to the Ck-topology, k =
2, . . . ,∞, in the set of metrics of a given index on M. A higher-order genericity Riemannian
result of Klingenberg and Takens is extended to semi-Riemannian geometr
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