Narrow Results

In this paper, we examine the dynamical evolution of flat FRW cosmological model in $f(R,L_m)$ gravity theory. We consider the general form of $f(R,L_m)$ defined as $f(R,L_m)=\mathrm{\Lambda }+\frac{\alpha }{2}R+\beta L_m^n$, where $\mathrm{\Lambda }$, $\alpha$, $\beta$, $n$ are model parameters, with the matter Lagrangian given by $L_m=-p$. We investigate the model through phase plane analysis, actively studying the evolution of cosmological solutions using dynamical systems techniques. To analyze the evolution equations, we introduce suitable transformations of variables and discuss the corresponding solutions by phase-plane analysis. The nature of critical points is analyzed and stable attractors are examined for $f(R,L_m)$ gravity cosmological model. We examine the linear and classical stabilities of the model and discuss it in detail. Further, we investigate the transition stage of the Universe, i.e. from the early decelerating stage to the present accelerating phase of the Universe by evolution of the effective equation of state, $\{r,s\}$ parameters and statefinder diagnostics for the central values of parameters $\alpha ,\beta$ and $n$ constrained using MCMC technique with cosmic chronometer data.

We characterize the special Lagrangian submanifolds of a generalized Calabi–Yau manifold, with vanishing Maslov class. Then, we carefully describe several examples, including a non-Kähler generalized Calabi–Yau manifold foliated by special Lagrangian submanifolds.

We present a novel $C^0$-characterization of symplectic embeddings and diffeomorphisms in terms of Lagrangian embeddings. Our approach is based on the shape invariant, which was discovered by Sikorav and Eliashberg, intersection theory and the displacement energy of Lagrangian submanifolds, and the fact that non-Lagrangian submanifolds can be displaced immediately. This characterization gives rise to a new proof of $C^0$-rigidity of symplectic embeddings and diffeomorphisms. The various manifestations of Lagrangian rigidity that are used in our arguments come from $J$-holomorphic curve methods. An advantage of our techniques is that they can be adapted to a $C^0$-characterization of contact embeddings and diffeomorphisms in terms of coisotropic (or pre-Lagrangian) embeddings, which in turn leads to a proof of $C^0$-rigidity of contact embeddings and diffeomorphisms. We give a detailed treatment of the shape invariants of symplectic and contact manifolds, and demonstrate that shape is often a natural language in symplectic and contact topology. We consider homeomorphisms that preserve shape, and propose a hierarchy of notions of Lagrangian topological submanifold. Moreover, we discuss shape-related necessary and sufficient conditions for symplectic and contact embeddings, and define a symplectic capacity from the shape.

Multi-Hamiltonian systems are investigated by using the Hamilton–Jacobi method. Integration of a set of total differential equations which includes the equations of motion and the action integral function is discussed. It is shown that this set is integrable if and only if the total variations of the Hamiltonians vanish. Two examples are studied.

We consider a homogeneous and isotropic Universe, described by the minisuperspace Lagrangian with the scale factor as a generalized coordinate. We show that the energy of a closed Universe is zero. We apply the uncertainty principle to this Lagrangian and propose that the quantum uncertainty of the scale factor causes the primordial fluctuations of the matter density. We use the dynamics of the early Universe in the Einstein–Cartan theory of gravity with spin and torsion, which eliminates the big-bang singularity and replaces it with a nonsingular bounce. Quantum particle production in highly curved spacetime generates a finite period of cosmic inflation that is consistent with the Planck satellite data. From the inflated primordial fluctuations we determine the magnitude of the temperature fluctuations in the cosmic microwave background, as a function of the numbers of the thermal degrees of freedom of elementary particles and the particle production coefficient which is the only unknown parameter.

In this paper, we investigate the behavior of the energy eigenvalues of the Schrödinger equation by using the canonical quantization method. We obtain the Hamiltonian of the Schrödinger equation by the Lagrangian in terms of the new coordinates. Then we calculate the partition function by the eigenvalues and the thermodynamic properties of the system in the superstatistics formalism for the modified Dirac delta and the Gamma distributions. All results in the limiting cases satisfy that of the harmonic oscillator. Furthermore, the effects of the all parameters in the problem of energy eigenvalues and thermodynamic properties are calculated and shown graphically.

In the one-dimensional stationary case, we construct a mechanical Lagrangian describing the quantum motion of a nonrelativistic spinless system. This Lagrangian is written as a difference between a function T, which represents the quantum generalization of the kinetic energy and which depends on the coordinate x and the temporal derivatives of x up the third order, and the classical potential V(x). The Hamiltonian is then constructed and the corresponding canonical equations are deduced. The function T is first assumed to be arbitrary. The development of T in a power series together with the dimensional analysis allow us to fix univocally the series coefficients by requiring that the well-known quantum stationary Hamilton–Jacobi equation be reproduced. As a consequence of this approach, we formulate the law of the quantum motion representing a new version of the quantum Newton law. We also analytically establish the famous Bohm relation outside the framework of the hydrodynamical approach and show that the well-known quantum potential, although it is a part of the kinetic term, plays really the role of an additional potential as assumed by Bohm.

Peierls brackets are part of the space-time approach to quantum field theory, and provide a Poisson bracket which, being defined for pairs of observables which are group invariant, is group invariant by construction. It is therefore well suited for combining the use of Poisson brackets and the full diffeomorphism group in general relativity. The present paper provides an introduction to the topic, with applications to gauge field theory.

After the last missing piece, the Higgs particle, has probably been identified, the Standard Model of the subatomic particles appears to be a quite robust structure, that can survive on its own for a long time to come. Most researchers expect considerable modifications and improvements to come in the near future, but it could also be that the Model will stay essentially as it is. This, however, would also require a change in our thinking, and the question remains whether and how it can be reconciled with our desire for our theories to be “natural”.

In this study we analyzed the deformation of the polymeric rod impacting on the rigid wall which is called "Taylor impact test."" We simulated three-dimensional Taylor impact test depending on the various polymeric materials using the explicit finite element method by employing DYNA3D code. In simulation, polymeric materials were modeled using viscoelastic constitutive relations with the relaxation time and shear modulus. We have carried out the numerical simulation for the transient deformation characteristics and discussed effects of the viscoelastic constants on the deformation of the polymeric rod under impact.

Variational Integrator (VI) is a numerical technique, in which the Lagrangian of the system is used as the action integral. It is a special type of numerical solution that preserves the energy and momentum of the system. In this paper, we retrieve numerical solutions for heat and wave equation with the help of all possible combinations of finite difference scheme like forward–forward, forward–backward, forward–centered, backward–forward, backward–backward, backward–centered, centered–forward, centered–backward, centered–centered. We also use Lagrangian approach along with the projection technique to obtain approximate solutions of these linear models. This approach provides the best approximate solutions as well as preserves the energy of the system while the finite difference scheme gives only the numerical solutions. We also draw a comparison of existing exact solution with all approximate solutions for both models and provide graphical representation of these solutions.

We study the principle of least (stationary) action for mem-elements. The least action principle allows us to derive relationships between the electrical variables for each of the six mem-elements. The principle of least action from modern physics is a natural environment to characterize mem-elements, including various one-period loops in the context of periodic circuits. The time-integrals of Lagrangian lead to the action and coaction quantities and a full characterization of mem-elements with periodic control variables.

In this paper, Lagrangian-based method has been proposed for tuning the parameters of fractional order $P{I}^{\alpha }{D}^{\beta }$ controller. In this method, the five parameters ($K_p$, $K_i$, $K_d$, $\alpha$ and $\beta$) of fractional order $P{I}^{\alpha }{D}^{\beta }$ controller (FOPID) are suitably optimized, and successfully applied to a benchmark stable second-order feedback system. To prove the performance of the proposed method, several state-of-the-art approaches were compared. The computational complexity, robustness and stability analysis has been performed to investigate the performance of all these algorithms. Moreover, the precision and flexibility analysis among all these approaches has also been carried out in this paper. The closed loop response of the second-order bench mark stable plant in Simulink has also been depicted in this paper.

We considered derivation of Vlasov–Einstein–Maxwell system of equations from the first principles, i.e. using classical Maxwell–Einstein–Hilbert action principle. We know many papers in which the theories indicated as Einstein–Vlasov, Vlasov–Maxwell–Einstein, Einstein–Maxwell–Boltzmann are discussed, and we discuss difficulties of usually used equations. We use another way of derivation and obtain an alternative version based on the generalized Fock–Weinberg form of equation of motion.

The dynamics of optical solitons, with dispersion management, propagating through a dense wavelength-division-multiplexed system with damping and amplification is studied. The adiabatic parameter dynamics of the solitons in the presence of perturbation terms have been obtained by virtue of the variational principle. In particular, the Gaussian and super-Gaussian pulses have been considered.

This paper obtains the parameter dynamics of super-sech optical solitons propagating through optical fibers. The variational principle is applied to obtain these parameter dynamics. Both polarization preserving fibers as well as birefringent fibers are studied in this paper. The study is then finally extended to the case of DWDM systems. The adiabatic parameter dynamics of soliton parameters is then obtained in the presence of perturbation terms. The numerical simulations are also obtained to complete the analysis.

In this work, I present a practical way to obtain the vibration-rotation kinetic energy operator for an N-atomic molecule in an arbitrary body-frame . The body-frame need not be orthogonal or rigid. In practice, I derive the explicit form of the measuring vectors associated with the body-frame components of the internal angular momentum. Their inner products with the vector derivatives of the shape coordinates give the "Coriolis" part of the metric tensor appearing in the Hamiltonian, and their inner products among themselves give the "rotational" part. As a simple example, the measuring vectors are explicitly derived in an oblique bond-vector body-frame. The metric tensor elements are also derived for a tetra-atomic pyramidal molecule, whose shape is parametrized in bond-angle coordinates.

The global objective of this work is to show the capabilities of the Eulerian–Lagrangian spray atomization (ELSA) model for the simulation of Diesel sprays in cold starting conditions. Our main topic is to focus in the analysis of spray formation and its evolution at low temperature 255 K (-18°C) and nonevaporative conditions. Spray behavior and several macroscopic properties, included the liquid spray penetration, and cone angle are also characterized. This study has been carried out using different ambient temperature and chamber pressure conditions. Additionally, the variations of several technical quantities, as the area coefficient and effective diameter are also studied. The results are compared with the latest experimental results in this field obtained in our institute. In the meantime, we also compare with the normal ambient temperature at 298 K (25°C) where the numerical validation of the model has shown a good agreement.

In computational fluid dynamics there have been many attempts to combine the advantages of having a fixed mesh, on which to carry out spatial calculations, with using particles moving according to the velocity field. These ideas in fact go back to particle-in-cell methods, proposed about 60 years ago. Of course, some procedure is needed to transfer field information between particles and mesh. There are many possible choices for this “assignment”, or “projection”. Several requirements may guide this choice. Two well-known ones are conservativity and stability, which apply to volume integrals of the fields. An additional one is here considered: preservation of information. This means that assignment from the particles onto the mesh and back should yield the same field values when the particles and the mesh coincide in position. The resulting method is termed “mass” assignment, due to its strong similarities with the finite element method. Several procedures are tested, including the well-known FLIP, on three scenarios: simple 1D convection, 2D convection of Zalesak’s disk, and a CFD simulation of the Taylor–Green periodic vortex sheet. Mass assignment is seen to be clearly superior to other methods.

The recently proposed color based tracking systems are unable to properly adapt the ellipse that represents an object to be tracked. This most likely leads to inaccurate descriptions of the object in the later application. This paper presents a Lagrangian based method in order to discover a regularizing component for the covariance matrix. Technically, we intend to reduce the residuals between the estimated probability distribution and the expected one. We argue that, by doing this, the shape of the ellipse can be properly adapted in the tracking stage. Experimental results show that the proposed method has favorable performance in shape adaption and object localization.