Hadamard’s variational formula in terms of stress and strain tensors

Starting from a Lagrangian action functional for two scalar fields we construct, by variational methods, the Laplacian Green function for a bounded domain and an appropriate stress tensor. By a further variation, imposed by a given vector field, we arrive at an interior version of the Hadamard variational formula, previously considered by P. Garabedian. It gives the variation of the Green function in terms of a pairing between the stress tensor and a strain tensor in the interior of the domain, this contrasting the classical Hadamard formula which is expressed as a pure boundary variation.

Topological Gravity of Chern-Simons-Antoniadis-Savvidy in 2+1 Dimensions

Pada artikel ini akan ditunjukan analisa dari perluasan gauge invariant exact dan metric independent untuk menkontruksi lower-rank field-strength tensor. Hasil ini akan digunakan untuk mengkontruski ulang Chern-Simons-Antoniadis-Savvidy formasi (2n+1) pada dimensi genap dengan menggunakan pendekatan diferensial geometri. Selanjutnya akan dianalisa bentuk topological gravitasi 2-dimensi yang merupakan perluasan dari teorema Chern-Weil yang telah dikembangkan oleh Izurieta-Munoz-Salgado. Hasil dari penelitian ini memperlihatkan bahwa aksi Lagrangian yang sama seperti pada topological gravitasi Chern-Simons forms pada dimensi (2n+1) invariant terhadap Poincare group SO(D−1,1) SO(D−1,2). This article determine and analyess of the extended gauge invariant exact and metric independent to construct the lower-rank field-strength tensor. These results used to construct Chern-Simons-Antoniadis-Savvidy (2n+1)-forms even dimensions using a differential geometry approach. This result analyzed 2-dimensional topological gravity forms that extended Chern-Weil theorem which has been developed by Izurieta-Munoz-Salgado. These results show similary topological gravity Lagrangian action of Chern-Simons forms (2n+1)-dimension invariant under Poincare group SO(D−1,1) SO(D−1,2).Keywords: Gauge theory, field-strength tensor, Chern-Weill theorem, Chern-Simons-Antoniadis-Savvidy forms, topological gravity

Dynamically Generated Inflationary ΛCDM

Our primary objective is to construct a plausible, unified model of inflation, dark energy and dark matter from a fundamental Lagrangian action first principle, wherein all fundamental ingredients are systematically dynamically generated starting from a very simple model of modified gravity interacting with a single scalar field employing the formalism of non-Riemannian spacetime volume-elements. The non-Riemannian volume element in the initial scalar field action leads to a hidden, nonlinear Noether symmetry which produces an energy-momentum tensor identified as the sum of a dynamically generated cosmological constant and dust-like dark matter. The non-Riemannian volume-element in the initial Einstein–Hilbert action upon passage to the physical Einstein-frame creates, dynamically, a second scalar field with a non-trivial inflationary potential and with an additional interaction with the dynamically generated dark matter. The resulting Einstein-frame action describes a fully dynamically generated inflationary model coupled to dark matter. Numerical results for observables such as the scalar power spectral index and the tensor-to-scalar ratio conform to the latest 2018 PLANCK data.

Dynamically Generated Inflationary Lambda-CDM

Our primary objective is to construct a plausible unified model of inflation, dark energy and dark matter from a fundamental Lagrangian action first principle, where all fundamental ingredients are systematically dynamically generated starting from a very simple model of modified gravity interacting with a single scalar field employing the formalism of non-Riemannian spacetime volume-elements. The non-Riemannian volume element in the initial scalar field action leads to a hidden nonlinear Noether symmetry which produces energy-momentum tensor identified as a sum of a dynamically generated cosmological constant and a dust-like dark matter. The non-Riemannian volume-element in the initial Einstein-Hilbert action upon passage to the physical Einstein-frame creates dynamically a second scalar field with a non-trivial inflationary potential and with an additional interaction with the dynamically generated dark matter. The resulting Einstein-frame action describes a fully dynamically generated inflationary model coupled to dark matter. Numerical results for observables such as the scalar power spectral index and the tensor-to-scalar ratio conform to the latest 2018 PLANCK data.

Minimal Boundaries in Tonelli Lagrangian Systems

We prove several new results concerning action minimizing periodic orbits of Tonelli Lagrangian systems on an oriented closed surface $M$. More specifically, we show that for every energy larger than the maximal energy of a constant orbit and smaller than or equal to the Mañé critical value of the universal abelian cover, the Lagrangian system admits a minimal boundary, that is, a global minimizer of the Lagrangian action on the space of smooth boundaries of open sets of $M$. We also extend the celebrated graph theorem of Mather in this context: in the tangent bundle $\textrm{T} M$, the union of the supports of all lifted minimal boundaries with a given energy projects injectively to the base $M$. Finally, we prove the existence of action minimizing simple periodic orbits on energies just above the Mañé critical value of the universal abelian cover. This provides in particular a class of nonreversible Finsler metrics on the two-sphere possessing infinitely many closed geodesics.

Syzygy sequences of the -center problem

The purpose of this paper is to consider the$N$-center problem with collinear centers, to identify its syzygy sequences that can be realized by minimizers of the Lagrangian action functional and to count the number of such syzygy sequences. In particular, we show that the number of such realizable syzygy sequences of length$\ell$greater than or equal to two for the 3-center problem is at least$F_{\ell +2}-2$, where$\{F_{n}\}$is the Fibonacci sequence. Moreover, with fixed length$\ell$, the density of such realizable syzygy sequences of length$\ell$for the$N$-center problem approaches one as$N$increases to infinity. Using reflection symmetry, the minimizers that we found can be extended to periodic solutions.

The Characterization of the Variational Minimizers for Spatial RestrictedN+1-Body Problems

We use Jacobi's necessary condition for the variational minimizer to study the periodic solution for spatial restrictedN+1-body problems with a zero mass on the vertical axis of the plane forNequal masses. We prove that the minimizer of the Lagrangian action on the anti-T/2 or odd symmetric loop space must be a nonconstant periodic solution for any2≤N≤472; hence the zero mass must oscillate, so that it cannot be always in the same plane with the other bodies. This result contradicts with our intuition that the small mass should always be at the origin.