Dynamical reduced basis methods for Hamiltonian systems

We consider model order reduction of parameterized Hamiltonian systems describing nondissipative phenomena, like wave-type and transport dominated problems. The development of reduced basis methods for such models is challenged by two main factors: the rich geometric structure encoding the physical and stability properties of the dynamics and its local low-rank nature. To address these aspects, we propose a nonlinear structure-preserving model reduction where the reduced phase space evolves in time. In the spirit of dynamical low-rank approximation, the reduced dynamics is obtained by a symplectic projection of the Hamiltonian vector field onto the tangent space of the approximation manifold at each reduced state. A priori error estimates are established in terms of the projection error of the full model solution onto the reduced manifold. For the temporal discretization of the reduced dynamics we employ splitting techniques. The reduced basis satisfies an evolution equation on the manifold of symplectic and orthogonal rectangular matrices having one dimension equal to the size of the full model. We recast the problem on the tangent space of the matrix manifold and develop intrinsic temporal integrators based on Lie group techniques together with explicit Runge–Kutta (RK) schemes. The resulting methods are shown to converge with the order of the RK integrator and their computational complexity depends only linearly on the dimension of the full model, provided the evaluation of the reduced flow velocity has a comparable cost.

A Zermelo navigation problem with a vortex singularity

Helhmoltz-Kirchhoff equations of motions of vortices of an incompressible fluid in the plane define a dynamics with singularities and this leads to a Zermelo navigation problem describing the ship travel in such a field where the control is the heading angle. Considering one vortex, we define a time minimization problem, geometric frame being the extension of Randers metrics in the punctured plane, with rotational symmetry. Candidates as minimizers are parameterized thanks to the Pontryagin Maximum Principle as extremal solutions of a Hamiltonian vector field. We analyze the time minimal solution to transfer the ship between two points where during the transfer the ship can be either in a strong current region in the vicinity of the vortex or in a weak current region. Analysis is based on a micro-local classification of the extremals using mainly the integrability properties of the dynamics due to the rotational symmetry. The discussion is complex and related to the existence of an isolated extremal (Reeb) circle due to the vortex singularity. Explicit computation of cut points where the extremal curves cease to be optimal is given and the spheres are described in the case where at the initial point the current is weak.

Symplectic formulation of teleparallel gravity

Teleparallel gravity has been formulated in the symplectic-geometrical fashion. For that purpose, the symplectic potential and the pre-symplectic structure on the manifold of all solutions of the field equation of teleparallel gravity have been firstly constructed from the Lagrangian density of the theory. That the obtained pre-symplectic structure is a symplectic structure also has been proved. It has been shown that the symplectic potential obtained from the Lagrangian density up to a factor is equal to the superpotential of teleparallel gravity. The Hamiltonian formulation of teleparallel gravity has been derived. Furthermore, the Poisson bracket between arbitrary two classical observables (i.e. real-valued functional) defined on the symplectic manifold of all solutions of the field equation of teleparallel gravity has constructed. The Hamiltonian vector field associated to an observable has been obtained. The formulation of Noether theorem in the symplectic formulation of teleparallel gravity has been investigated.

Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems

Kahan discretization is applicable to any quadratic vector field and produces a birational map which approximates the shift along the phase flow. For a planar quadratic canonical Hamiltonian vector field, this map is known to be integrable and to preserve a pencil of cubic curves. Generically, the nine base points of this pencil include three points at infinity (corresponding to the asymptotic directions of cubic curves) and six finite points lying on a conic. We show that the Kahan discretization map can be represented in six different ways as a composition of two Manin involutions, corresponding to an infinite base point and to a finite base point. As a consequence, the finite base points can be ordered so that the resulting hexagon has three pairs of parallel sides which pass through the three base points at infinity. Moreover, this geometric condition on the base points turns out to be characteristic: if it is satisfied, then the cubic curves of the corresponding pencil are invariant under the Kahan discretization of a planar quadratic canonical Hamiltonian vector field.

A geometric Hamilton–Jacobi theory on a Nambu–Jacobi manifold

In this paper, we propose a geometric Hamilton–Jacobi (HJ) theory on a Nambu–Jacobi (NJ) manifold. The advantage of a geometric HJ theory is that if a Hamiltonian vector field [Formula: see text] can be projected into a configuration manifold by means of a one-form [Formula: see text], then the integral curves of the projected vector field [Formula: see text] can be transformed into integral curves of the vector field [Formula: see text] provided that [Formula: see text] is a solution of the HJ equation. This procedure allows us to reduce the dynamics to a lower-dimensional manifold in which we integrate the motion. On the other hand, the interest of a NJ structure resides in its role in the description of dynamics in terms of several Hamiltonian functions. It appears in fluid dynamics, for instance. Here, we derive an explicit expression for a geometric HJ equation on a NJ manifold and apply it to the third-order Riccati differential equation as an example.

Bifurcation of Limit Cycles in a 12-Degree Hamiltonian System Under Thirteenth-Order Perturbation

This paper considers the weakened Hilbert’s 16th problem for symmetric planar perturbed polynomial Hamiltonian system. A [Formula: see text]-equivariant planar vector field of degree 12 is deduced to find as many as possible limit cycles and their configuration patterns. By using bifurcation theory of planar dynamical system and the method of detection function, we have obtained that, under the thirteenth-order perturbation, the above [Formula: see text]-equivariant planar perturbed Hamiltonian vector field of 12-degree has at least 117 limit cycles. Moreover, this paper also shows the configuration of compound eyes of the corresponding perturbed systems.

Geometry of physical systems on quantized spaces

We present a mathematical model for physical systems. A large class of functions is built through the functional quantization method and applied to the geometric study of the model. Quantized equations of motion along the Hamiltonian vector field are built up. It is seen that the procedure in higher dimension carries more physical information. The metric tensor appears to induce an electromagnetic field into the system and the dynamical nature of the electromagnetic field in curved space arises naturally. In the end, an explicit formula for the curvature tensor in the quantized space is given.

Geometrical structures on the cotangent bundle

In this paper, we study the geometrical structures on the cotangent bundle using the notions of adapted tangent structure and regular vector fields. We prove that the dynamical covariant derivative on [Formula: see text] fix a nonlinear connection for a given [Formula: see text]-regular vector field. Using the Legendre transformation induced by a regular Hamiltonian, we show that a semi-Hamiltonian vector field on [Formula: see text] corresponds to a semispray on [Formula: see text] if and only if the nonlinear connection on [Formula: see text] is just the canonical nonlinear connection induced by the regular Lagrangian.