Nonlinear Pendulum: A Simple Generalization

In this work we solve the nonlinear second order differential equation of the simple pendulum with a general initial angular displacement ($\theta(0)=\theta_0$) and velocity ($\dot{\theta}(0)=\phi_0$), obtaining a closed-form solution in terms of the Jacobi elliptic function $\text{sn}(u,k)$, and of the the incomplete elliptical integral of the first kind $F(\varphi,k)$. Such a problem can be used to introduce concepts like elliptical integrals and functions to advanced undergraduate students, to motivate the use of Computer Algebra Systems to analyze the solutions obtained, and may serve as an exercise to show how to carry out a simple generalization, taking as a starting point the paper of Bel\'endez \emph{et al} \cite{belendez}, where they have considered the standard case $\dot{\theta}(0)=0$

Magnetic quantum phase transitions of the antiferromagnetic J_{1}-J_{2} Heisenberg model

We obtain the complete phase diagram of the antiferromagnetic $J_{1}$-$J_{2}$ model, $0\leq \alpha = J_2/J1 \leq 1$, within the framework of the $O(N)$ nonlinear sigma model. We find two magnetically ordered phases, one with N\' eel order, for $\alpha \leq 0.4$, and another with collinear order, for $\alpha\geq 0.6$, separated by a nonmagnetic region, for $0.4\leq \alpha \leq 0.6$, where a gapped spin liquid is found. The transition at $\alpha=0.4$ is of the second order while the one at $\alpha=0.6$ is of the first order and the spin gaps cross at $\alpha=0.5$. Our results are exact at $N\rightarrow\infty$ and agree with numerical results from different methods.Comment: 4 pages, 5 figure

Comments on "Growth of Covariant Perturbations in the Contracting Phase of a Bouncing Universe" by A. Kumar

A recent paper by Kumar (2012) (hereafter K12) claimed that in a contracting model, described by perturbations around a collapsing Friedmann model containing dust or radiation, the perturbations can grow in such a way that the linearity conditions would become invalid. This conclusion is not correct due to the following facts: first, it is claimed that the linearity conditions are not satisfied, but nowhere in K12 the amplitudes of the perturbations were in fact estimated. Therefore, without such estimates, the only possible conclusion from this work is the well known fact that the perturbations indeed grow during contraction, which, per se, does not imply that the linearity conditions become invalid. Second, some evaluations of the linearity conditions are incorrect because third other terms, instead of the appropriate second order ones, are mistakenly compared with first order terms, yielding artificially fast growing conditions. Finally, it is claimed that the results of K12 are in sharp contrast with the results of the paper by Vitenti and Pinto-Neto (2012) (hereafter VPN12), because the former was obtained in a gauge invariant way. However, the author of K12 did not realized that the evolution of the perturbations were also calculated in a gauge invariant way in VPN12, but some of the linearity conditions which are necessary to be checked cannot be expressed in terms of gauge invariant quantities. In the present work, the incorrect or incomplete statements of K12 are clarified and completed, and it is shown that all other correct results of K12 were already present in VPN12, whose conclusions remain untouched, namely, that cosmological perturbations of quantum mechanical origin in a bouncing model can remain in the linear regime all along the contracting phase and at the bounce itself for a wide interval of energy scales of the bounce. (Abstract abridged)Comment: 7 pages, revtex4-1, accepted for publication in PR

Spectra of primordial fluctuations in two-perfect-fluid regular bounces

We introduce analytic solutions for a class of two components bouncing models, where the bounce is triggered by a negative energy density perfect fluid. The equation of state of the two components are constant in time, but otherwise unrelated. By numerically integrating regular equations for scalar cosmological perturbations, we find that the (would be) growing mode of the Newtonian potential before the bounce never matches with the the growing mode in the expanding stage. For the particular case of a negative energy density component with a stiff equation of state we give a detailed analytic study, which is in complete agreement with the numerical results. We also perform analytic and numerical calculations for long wavelength tensor perturbations, obtaining that, in most cases of interest, the tensor spectral index is independent of the negative energy fluid and given by the spectral index of the growing mode in the contracting stage. We compare our results with previous investigations in the literature.Comment: 11 pages, 5 figure

Roughness correction to the Casimir force : Beyond the Proximity Force Approximation

We calculate the roughness correction to the Casimir effect in the parallel plates geometry for metallic plates described by the plasma model. The calculation is perturbative in the roughness amplitude with arbitrary values for the plasma wavelength, the plate separation and the roughness correlation length. The correction is found to be always larger than the result obtained in the Proximity Force Approximation.Comment: 7 pages, 3 figures, v2 with minor change

Large Adiabatic Scalar Perturbations in a Regular Bouncing Universe

It has been shown that a contracting universe with a dust-like ($w \approx 0$) fluid may provide an almost scale invariant spectrum for the gravitational scalar perturbations. As the universe contracts, the amplitude of such perturbations are amplified. The gauge invariant variable $\Phi$ develops a growing mode which becomes much larger than the constant one around the bounce phase. The constant mode has its amplitude fixed by Cosmic Background Explorer (COBE) normalization, thus the amplitude of the growing mode can become much larger than 1. In this paper, we first show that this is a general feature of bouncing models, since we expect that general relativity should be valid in all scales away from the bounce. However, in the Newtonian gauge, the variable $\Phi$ gives the value of the metric perturbation $\phi$, raising doubts on the validity of the linear perturbative regime at the bounce. In order to address this issue, we obtain a set of necessary conditions for the perturbative series to be valid along the whole history of the model, and we show that there is a gauge in which all these conditions are satisfied, for a set of models, if the constant mode is fixed by COBE normalization. As a by-product of this analysis, we point out that there are sets of solutions for the perturbation variables where some gauge-fixing conditions are not well defined, turning these gauges prohibited for those solutions.Comment: 10 pages, revtex4, minor revision, version to appear in PR