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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 Calabi functional on a ruled surface

Gábor Székelyhidi (2009)

Annales scientifiques de l'École Normale Supérieure

We study the Calabi functional on a ruled surface over a genus two curve. For polarizations which do not admit an extremal metric we describe the behavior of a minimizing sequence splitting the manifold into pieces. We also show that the Calabi flow starting from a metric with suitable symmetry gives such a minimizing sequence.

The calculus of variations on jet bundles as a universal approach for a variational formulation of fundamental physical theories

Jana Musilová, Stanislav Hronek (2016)

Communications in Mathematics

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 compact category and mulitple periodic solutions of Hamiltionian systems on symmetric starshaped energy surfaces.

Thomas Bartsch, Mónica Clapp (1992)

Mathematische Annalen

The computation of the index of a Morse function at a critical point.

Sakkalis, Takis (1988)

International Journal of Mathematics and Mathematical Sciences

The concentration-compactness principle in the calculus of variations. The locally compact case, part 1

P. L. Lions (1984)

Annales de l'I.H.P. Analyse non linéaire

The configuration space of gauge theory on open manifolds of bounded geometry

Jürgen Eichhorn, Gerd Heber (1997)

Banach Center Publications

We define suitable Sobolev topologies on the space $𝒞_{P} (B_{k}, f)$ of connections of bounded geometry and finite Yang-Mills action and the gauge group and show that the corresponding configuration space is a stratified space. The underlying open manifold is assumed to have bounded geometry.

The Conley index in Hilbert spaces and its applications

K. Gęba, M. Izydorek, A. Pruszko (1999)

Studia Mathematica

We present a generalization of the classical Conley index defined for flows on locally compact spaces to flows on an infinite-dimensional real Hilbert space H generated by vector fields of the form f: H → H, f(x) = Lx + K(x), where L: H → H is a bounded linear operator satisfying some technical assumptions and K is a completely continuous perturbation. Simple examples are presented to show how this new invariant can be applied in searching critical points of strongly indefinite functionals having...

The Conley index over a space.

Thomas Bartsch (1992)

Mathematische Zeitschrift

The CR Yamabe conjecture the case $n = 1$

Najoua Gamara (2001)

Journal of the European Mathematical Society

Let $(M, θ)$ be a compact CR manifold of dimension $2 n + 1$ with a contact form $θ$ , and $L = (2 + 2 / n) Δ_{b} + R$ its associated CR conformal laplacien. The CR Yamabe conjecture states that there is a contact form $\tilde{θ}$ on $M$ conformal to $θ$ which has a constant Webster curvature. This problem is equivalent to the existence of a function $u$ such that $L u = u^{1 + 2 / n}$ , $u > 0$ on $M$ . D. Jerison and J. M. Lee solved the CR Yamabe problem in the case where $n \geq 2$ and $(M, θ)$ is not locally CR equivalent to the sphere $S^{2 n + 1}$ of $𝐂^{n}$ . In a join work with R. Yacoub, the CR Yamabe problem...

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Displaying 1 – 20 of 105

Tensorial version of the calculus of variations.

Teoria di Lusternik-Schnirelman su varietà con bordo negli spazi di Hilbert

The Approximation of Instantons.

The Blow-Up of p-Harmonic maps.

The Bonnet-Meyers theorem is true for Riemann Hilbert Manifolds.

The brachistochrone problem with frictional forces