Search Results (9)

In the Moon–Earth elliptic restricted three-body problem, near-polar and near-circular lunar-type periodic orbits are numerically continued from Keplerian circular orbits using Broyden’s method with line search. The Hamiltonian system, expressed in Cartesian coordinates, is treated via the symplectic scaling method. The radii of the initial Keplerian circular orbits are then scaled and normalized. For cases in which the integer ratios $\{j/k\}$ of the mean motions between the inner and outer orbits are within the range $[9,150]$, some periodic orbits of the elliptic restricted three-body problem are investigated. For the middle-altitude cases with $j/k\in [38,70]$, the perturbations due to $J_2$ and $C_{22}$ are incorporated, and some new near-polar periodic orbits are computed. The orbital dynamics of these near-polar, near-circular periodic orbits are well characterized by the first-order double-averaged system in the Poincaré–Delaunay elements. Linear stability is assessed through characteristic multipliers derived from the fundamental solution matrix of the linear varational system. Stability indices are computed for both the near-polar and planar near-circular periodic orbits across the range $j/k\in [9,50]$. Full article

In this work, we will explore the effects of non-commutativity in fractional classical and quantum schemes using the flat Friedmann–Robertson–Walker (FRW) cosmological model coupled to a scalar field in the K-essence formalism. In previous work, we have obtained the commutative solutions in both regimes in the fractional framework. Here, we introduce non-commutative variables, considering that all minisuperspace variables $q_{nc}^i$ do not commute, so the symplectic structure was modified. In the quantum regime, the probability density presents a new structure in the scalar field corresponding to the value of the non-commutative parameter, in the sense that this probability density undergoes a shift back to the direction of the scale factor, causing classical evolution to arise earlier than in the commutative world. Full article

We deduce a non-linear commutator higher-spin (HS) symmetry algebra which encodes unitary irreducible representations of the AdS group—subject to a Young tableaux $Y(s_1,\dots ,s_k)$ with $k\ge 2$ rows—in a d-dimensional anti-de Sitter space. Auxiliary representations for a deformed non-linear HS symmetry algebra in terms of a generalized Verma module, as applied to additively convert a subsystem of second-class constraints in the HS symmetry algebra into one with first-class constraints, are found explicitly in the case of a $k=2$ Young tableaux. An oscillator realization over the Heisenberg algebra for the Verma module is constructed. The results generalize the method of constructing auxiliary representations for the symplectic $sp\left(2k\right)$ algebra used for mixed-symmetry HS fields in flat spaces [Buchbinder, I.L.; et al. Nucl. Phys. B 2012, 862, 270–326]. Polynomial deformations of the $su(1,1)$ algebra related to the Bethe ansatz are studied as a byproduct. A nilpotent BRST operator for a non-linear HS symmetry algebra of the converted constraints for $Y(s_1,s_2)$ is found, with non-vanishing terms (resolving the Jacobi identities) of the third order in powers of ghost coordinates. A gauge-invariant unconstrained reducible Lagrangian formulation for a free bosonic HS field of generalized spin $(s_1,s_2)$ is deduced. Following the results of [Buchbinder, I.L.; et al. Phys. Lett. B 2021, 820, 136470.; Buchbinder, I.L.; et al. arXiv 2022, arXiv:2212.07097], we develop a BRST approach to constructing general off-shell local cubic interaction vertices for irreducible massive higher-spin fields (being candidates for massive particles in the Dark Matter problem). A new reducible gauge-invariant Lagrangian formulation for an antisymmetric massive tensor field of spin $(1,1)$ is obtained. Full article

We propose explicit K-symplectic and explicit symplectic-like methods for the charged particle system in a general strong magnetic field. The K-symplectic methods are also symmetric. The charged particle system can be expressed both in a canonical and a non-canonical Hamiltonian system. If the three components of the magnetic field can be integrated in closed forms, we construct explicit K-symplectic methods for the non-canonical charged particle system; otherwise, explicit symplectic-like methods can be constructed for the canonical charged particle system. The symplectic-like methods are constructed by extending the original phase space and obtaining the augmented separable Hamiltonian, and then by using the splitting method and the midpoint permutation. The numerical experiments have shown that compared with the higher order implicit Runge-Kutta method, the explicit K-symplectic and explicit symplectic-like methods have obvious advantages in long-term energy conservation and higher computational efficiency. It is also shown that the influence of the parameter $\epsilon$ in the general strong magnetic field on the Runge-Kutta method is bigger than the two kinds of symplectic methods. Full article

We study new geometrical and topological aspects of polarized k-symplectic manifolds. In addition, we study the De Rham cohomology groups of the k-symplectic group. In this work, we pay particular attention to the problem of the orientation of polarized k-symplectic manifolds in a way analogous to symplectic manifolds which are all orientable. Full article

We study the complete, compact, locally affine manifolds equipped with a k-symplectic structure, which are the quotients of ${\mathbb{R}}^{n(k+1)}$ by a subgroup $\mathsf{\Gamma }$ of the affine group $A\left(n\right(k+1\left)\right)$ of ${\mathbb{R}}^{n(k+1)}$ acting freely and properly discontinuously on ${\mathbb{R}}^{n(k+1)}$ and leaving invariant the $\mathit{k}$-symplectic structure, then we construct and give some examples and properties of compact, complete, locally affine two-symplectic manifolds of dimension three. Full article

Many force–gradient explicit symplectic integration algorithms have been designed for the Hamiltonian $H=T\left(\mathbf{p}\right)+V\left(\mathbf{q}\right)$ with kinetic energy $T\left(\mathbf{p}\right)={\mathbf{p}}^2/2$ in the existing references. When a force–gradient operator is appropriately adjusted as a new operator, it is still suitable for a class of Hamiltonian problems $H=K(\mathbf{p},\mathbf{q})+V\left(\mathbf{q}\right)$ with integrable part $K(\mathbf{p},\mathbf{q})={\sum }_{i=1}^n{\sum }_{j=1}^n{a}_{ij}{p}_i{p}_j+{\sum }_{i=1}^n{b}_i{p}_i$, where $a_{ij}=a_{ij}\left(\mathbf{q}\right)$ and $b_i=b_i\left(\mathbf{q}\right)$ are functions of coordinates $\mathbf{q}$. The newly adjusted operator is not a force–gradient operator but is similar to the momentum-version operator associated to the potential V. The newly extended (or adjusted) algorithms are no longer solvers of the original Hamiltonian, but are solvers of slightly modified Hamiltonians. They are explicit symplectic integrators with symmetry or time reversibility. Numerical tests show that the standard symplectic integrators without the new operator are generally poorer than the corresponding extended methods with the new operator in computational accuracies and efficiencies. The optimized methods have better accuracies than the corresponding non-optimized counterparts. Among the tested symplectic methods, the two extended optimized seven-stage fourth-order methods of Omelyan, Mryglod and Folk exhibit the best numerical performance. As a result, one of the two optimized algorithms is used to study the orbital dynamical features of a modified Hénon–Heiles system and a spring pendulum. These extended integrators allow for integrations in Hamiltonian problems, such as the spiral structure in self-consistent models of rotating galaxies and the spiral arms in galaxies. Full article

In recent literature, one-loop tests of the higher-spin AdS ${}_{d+1}$ /CFT ${}_d$ correspondences were carried out. Here, we extend these results to a more general set of theories in $d>2$ . First, we consider the Type B higher spin theories, which have been conjectured to be dual to CFTs consisting of the singlet sector of N free fermion fields. In addition to the case of N Dirac fermions, we carefully study the projections to Weyl, Majorana, symplectic and Majorana–Weyl fermions in the dimensions where they exist. Second, we explore theories involving elements of both Type A and Type B theories, which we call Type AB. Their spectrum includes fields of every half-integer spin, and they are expected to be related to the $U\left(N\right)/O\left(N\right)$ singlet sector of the CFT of N free complex/real scalar and fermionic fields. Finally, we explore the Type C theories, which have been conjectured to be dual to the CFTs of p-form gauge fields, where $p=\frac{d}{2}-1$ . In most cases, we find that the free energies at $O\left(N^0\right)$ either vanish or give contributions proportional to the free-energy of a single free field in the conjectured dual CFT. Interpreting these non-vanishing values as shifts of the bulk coupling constant $G_N\sim 1/(N-k)$ , we find the values $k=-1,-1/2,0,1/2,1,2$ . Exceptions to this rule are the Type B and AB theories in odd d; for them, we find a mismatch between the bulk and boundary free energies that has a simple structure, but does not follow from a simple shift of the bulk coupling constant. Full article

We introduce the symplectic structure of information geometry based on Souriau’s Lie group thermodynamics model, with a covariant definition of Gibbs equilibrium via invariances through co-adjoint action of a group on its moment space, defining physical observables like energy, heat, and moment as pure geometrical objects. Using geometric Planck temperature of Souriau model and symplectic cocycle notion, the Fisher metric is identified as a Souriau geometric heat capacity. The Souriau model is based on affine representation of Lie group and Lie algebra that we compare with Koszul works on G/K homogeneous space and bijective correspondence between the set of G-invariant flat connections on G/K and the set of affine representations of the Lie algebra of G. In the framework of Lie group thermodynamics, an Euler-Poincaré equation is elaborated with respect to thermodynamic variables, and a new variational principal for thermodynamics is built through an invariant Poincaré-Cartan-Souriau integral. The Souriau-Fisher metric is linked to KKS (Kostant–Kirillov–Souriau) 2-form that associates a canonical homogeneous symplectic manifold to the co-adjoint orbits. We apply this model in the framework of information geometry for the action of an affine group for exponential families, and provide some illustrations of use cases for multivariate gaussian densities. Information geometry is presented in the context of the seminal work of Fréchet and his Clairaut-Legendre equation. The Souriau model of statistical physics is validated as compatible with the Balian gauge model of thermodynamics. We recall the precursor work of Casalis on affine group invariance for natural exponential families. Full article