Solutions of elliptic equations with indefinite nonlinearities via Morse theory and linking
Solutions périodiques de systèmes hamiltoniens
Some constructions of biharmonic maps on the warped product manifolds
In this paper, we characterize a class of biharmonic maps from and between product manifolds in terms of the warping function. Examples are constructed when one of the factors is either Euclidean space or sphere.
Some geometric aspects of the calculus of variations in several independent variables
This paper describes some recent research on parametric problems in the calculus of variations. It explains the relationship between these problems and the type of problem more usual in physics, where there is a given space of independent variables, and it gives an interpretation of the first variation formula in this context in terms of cohomology.
Some results on connecting orbits for a class of Hamiltonian systems.
Some results on the calculus of variations on jet spaces
Sur le problème inverse du calcul des variations : existence de lagrangiens associés à un spray dans le cas isotrope
En utilisant la version de Spencer-Goldschmidt du théorème de Cartan-Kähler nous étudions les conditions nécessaires et suffisantes pour qu’un système d’équations différentielles ordinaires du second ordre soit le système d’Euler-Lagrange associé à un lagrangien régulier. Dans la thèse de Z. Muzsnay cette technique a été déjà appliquée pour donner une version moderne du papier classique de Douglas qui traite le cas de la dimension 2. Ici nous considérons le cas où la dimension est arbitraire, nous...
Surfaces of revolution in the Heisenberg group and the spectral generalization of the Willmore functional.
Symmetries and currents in nonholonomic mechanics
In this paper we derive general equations for constraint Noethertype symmetries of a first order non-holonomic mechanical system and the corresponding currents, i.e. functions constant along trajectories of the nonholonomic system. The approach is based on a consistent and effective geometrical theory of nonholonomic constrained systems on fibred manifolds and their jet prolongations, first presented and developed by Olga Rossi. As a representative example of application of the geometrical theory...
Symmetries in finite order variational sequences
We refer to Krupka’s variational sequence, i.e. the quotient of the de Rham sequence on a finite order jet space with respect to a ‘variationally trivial’ subsequence. Among the morphisms of the variational sequence there are the Euler-Lagrange operator and the Helmholtz operator. In this note we show that the Lie derivative operator passes to the quotient in the variational sequence. Then we define the variational Lie derivative as an operator on the sheaves of the variational sequence. Explicit...
Tensorial version of the calculus of variations.
The brachistochrone problem with frictional forces
The brachistochrone problem with frictional forces
In this paper we show the existence of the solution for the classical brachistochrone problem under the action of a conservative field in presence of frictional forces. Assuming that the frictional forces and the potential grow at most linearly, we prove the existence of a minimizer on the travel time between any two given points, whenever the initial velocity is great enough. We also prove the uniqueness of the minimizer whenever the given points are sufficiently close.
The calculus of variations on jet bundles as a universal approach for a variational formulation of fundamental physical theories
As widely accepted, justified by the historical developments of physics, the background for standard formulation of postulates of physical theories leading to equations of motion, or even the form of equations of motion themselves, come from empirical experience. Equations of motion are then a starting point for obtaining specific conservation laws, as, for example, the well-known conservation laws of momenta and mechanical energy in mechanics. On the other hand, there are numerous examples of physical...
The concentration-compactness principle in the calculus of variations. The locally compact case, part 1
The Dirichlet problem with sublinear nonlinearities
We investigate the existence of solutions of the Dirichlet problem for the differential inclusion for a.e. y ∈ Ω, which is a generalized Euler-Lagrange equation for the functional . We develop a duality theory and formulate the variational principle for this problem. As a consequence of duality, we derive the variational principle for minimizing sequences of J. We consider the case when G is subquadratic at infinity.
The Euler and Helmholtz operators on fibered manifolds with oriented bases
We study naturality of the Euler and Helmholtz operators arising in the variational calculus in fibered manifolds with oriented bases.
The total divergence equation.
The Weinstein Conjecture in P X Cl.
Three solutions for a nonlinear Neumann boundary value problem
The aim of this paper is to establish the existence of at least three solutions for the nonlinear Neumann boundary-value problem involving the p(x)-Laplacian of the form in Ω, on ∂Ω. Our technical approach is based on the three critical points theorem due to Ricceri.