Narrow Results

We will analyze two particular features of an exponential gravitational Lagrangian. On the one hand, while this choice of the Lagrangian density allows for two free parameters, only one scale, the cosmological constant, arises as fundamental when the proper Einsteinian limit is to be recovered. On the other hand, the vacuum energy arising from f(R) theories such that f(0) ≠ 0 needs a cancellation mechanism, by which the present value of the cosmological constant can be recast.

The displacement-traction problem of three-dimensional linearized elasticity can be posed as three different minimization problems, depending on whether the displacement vector field, or the stress tensor field, or the strain tensor field, is the unknown.

The objective of this paper is to put these three different formulations of the same problem in a new perspective, by means of Legendre–Fenchel duality theory. More specifically, we show that both the displacement and strain formulations can be viewed as Legendre–Fenchel dual problems to the stress formulation. We also show that each corresponding Lagrangian has a saddle-point, thus fully justifying this new duality approach to elasticity.

The existence or not of pathologies in the context of Lagrangian theory is studied with the aid of Machine Learning algorithms. Using an example in the framework of classical mechanics, we make a proof of concept, that the construction of new physical theories using machine learning is possible. Specifically, we utilize a fully-connected, feed-forward neural network architecture, aiming to discriminate between “healthy” and “nonhealthy” Lagrangians, without explicitly extracting the relevant equations of motion. The network, after training, is used as a fitness function in the concept of a genetic algorithm and new healthy Lagrangians are constructed. These new Lagrangians are different from the Lagrangians contained in the initial data set. Hence, searching for Lagrangians possessing a number of pre-defined properties is significantly simplified within our approach. The framework employed in this work can be used to explore more complex physical theories, such as generalizations of General Relativity in gravitational physics, or constructions in solid state physics, in which the standard procedure can be laborious.

In this study, we present a new analogue of Euler–Lagrange and Hamilton equations on an almost Kähler model of a Finsler manifold. Also, we give some corollaries about the related mechanical systems and equations.

We apply the method of Jacobi Last Multiplier to the fifty second-order ordinary differential equations of Painlevé type as given in Ince in order to obtain a Lagrangian and consequently solve the inverse problem of Calculus of Variations for those equations. The easiness and straightforwardness of Jacobi's method is underlined.

In this paper, we briefly review some of our recent results in the weak KAM theory for time-periodic positive definite Lagrangians, referring to the original papers^24,25 for further information.