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.

/pdf/the-fermi-pasta-ulam-problem-fifty-years-of-progress-35lqx769o1.pdf

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

We study the correlations and the isothermal compressibility of the Lennard-Jones fluid in the Percus-Yevick approximation in the critical region. The equation is solved numerically and the critical behavior is found similar to that for short-range forces: classical exponents but a nonclassical and nonuniversal scaling function that leads to an asymmetry between liquid and vapor with a true spinodal present only on the high-density side of the critical density.

Critical behavior of the Lennard-Jones fluid in the Percus-Yevick approximation.

We give a numerical estimate for the critical energy as a function of frequency, which describes the border of stochasticity for a system of coupled oscillators The results are given for the model of a chain of particles with nearest-neighbour Lennard-Jones interaction

/pdf/stochasticity-thresholds-for-systems-of-coupled-oscillators-3res58d89h.pdf

Stochasticity thresholds for systems of coupled oscillators

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.

G. Lo Vecchio

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

Critical behavior of the Lennard-Jones fluid in the Percus-Yevick approximation.

Stochasticity thresholds for systems of coupled oscillators

Stochasticity thresholds in the Fermi-Pasta-Ulam model