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The brachistochrone problem with frictional forces

Roberto Giambò, Fabio Giannoni (2010)

ESAIM: Control, Optimisation and Calculus of Variations

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

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 Dirichlet problem with sublinear nonlinearities

Aleksandra Orpel (2002)

Annales Polonici Mathematici

We investigate the existence of solutions of the Dirichlet problem for the differential inclusion 0 ∈ Δ x ( y ) + ∂ x G ( y , x ( y ) ) for a.e. y ∈ Ω, which is a generalized Euler-Lagrange equation for the functional J ( x ) = ∫ Ω 1 / 2 | ∇ x ( y ) | ² - G ( y , x ( y ) ) d y . 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.

Three solutions for a nonlinear Neumann boundary value problem

Najib Tsouli, Omar Chakrone, Omar Darhouche, Mostafa Rahmani (2014)

Applicationes Mathematicae

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 - Δ p ( x ) u + a ( x ) | u | p ( x ) - 2 u = μ g ( x , u ) in Ω, | ∇ u | p ( x ) - 2 ∂ u / ∂ ν = λ f ( x , u ) on ∂Ω. Our technical approach is based on the three critical points theorem due to Ricceri.

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