Over the past decade, f(R) theories have been extensively studied as one of the simplest modifications to General Relativity. In this article we review various applications of f(R) theories to cosmology and gravity - such as inflation, dark energy, local gravity constraints, cosmological perturbations, and spherically symmetric solutions in weak and strong gravitational backgrounds. We present a number of ways to distinguish those theories from General Relativity observationally and experimentally. We also discuss the extension to other modified gravity theories such as Brans-Dicke theory and Gauss-Bonnet gravity, and address models that can satisfy both cosmological and local gravity constraints.

f(R) theories

The present paper, together with the previous one (Part 1: Theory, published in this journal) is intended to give an explicit method for computing all Lyapunov Characteristic Exponents of a dynamical system. After the general theory on such exponents developed in the first part, in the present paper the computational method is described (Chapter A) and some numerical examples for mappings on manifolds and for Hamiltonian systems are given (Chapter B).

Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; A method for computing all of them. Part 2: Numerical application

We search for spherically symmetric solutions of f(R) theories of gravity via the Noether symmetry approach. A general formalism in the metric framework is developed considering a point-like f(R) Lagrangian where spherical symmetry is required. Examples of exact solutions are given.

Spherically symmetric solutions in f(R) gravity via the Noether symmetry approach

A brief review of the Fermi-Pasta-Ulam (FPU) paradox is given, together with its suggested resolutions and its relation to other physical problems. We focus on the ideas and concepts that have become the core of modern nonlinear mechanics, in their historical perspective. Starting from the first numerical results of FPU, both theoretical and numerical findings are discussed in close connection with the problems of ergodicity, integrability, chaos and stability of motion. New directions related to the Bose-Einstein condensation and quantum systems of interacting Bose-particles are also considered.

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The Fermi-Pasta-Ulam problem: fifty years of progress.

The inverse problem in the calculus of variations and the geometry of the tangent bundle

New phenomenon in the stochastic transition of coupled oscillators

The concept of dynamical symmetry for constrained Lagrangian systems is analysed and generalized. Both the direct and converse Noether theorems are proved for singular Lagrangians.

Symmetries and constants of motion for constrained Lagrangian systems: a presymplectic version of the Noether theorem

We consider the celebrated model of Fermi, Pasta and Ulam and give a numerical estimate for its thresholds of stochasticity, thus determining a critical energy as a function of the frequency of the corresponding oscillators. The results turn out to be qualitatively similar to those already obtained for a chain of particles with nearest-neighbour Lennard-Jones interaction potential.

Stochasticity thresholds in the Fermi-Pasta-Ulam model

It is shown that whenever a (1, 1)-type tensor field, defined through two alternative Lagrangians, implies complete integrability for a second-order Lagrangian vector field Γ on the tangent bundle of a given configuration manifold, then a bundle atlas can be found in which both Γ and a class of equivalent Lagrangians are completely separated into a sum of second-order vector fields and Lagrangians, each one corresponding to a single degree of freedom

C. Ferrario

Papers

The inverse problem in the calculus of variations and the geometry of the tangent bundle

New phenomenon in the stochastic transition of coupled oscillators

Symmetries and constants of motion for constrained Lagrangian systems: a presymplectic version of the Noether theorem

Stochasticity thresholds in the Fermi-Pasta-Ulam model

Separability of completely integrable dynamical systems admitting alternative Lagrangian descriptions