scholarly journals Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems

2019 ◽  
Vol 475 (2223) ◽  
pp. 20180761 ◽  
Author(s):  
Matteo Petrera ◽  
Jennifer Smirin ◽  
Yuri B. Suris
Keyword(s):  
Vector Field ◽  
Base Point ◽  
Hamiltonian Vector ◽  
Geometric Condition ◽  
Hamiltonian Vector Field ◽  
Cubic Curves ◽  
Birational Map ◽  
Base Points ◽  
Finite Base ◽  
Pass Through

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.

On the canonical systems of certain algebraic surfaces

1937 ◽  
Vol 33 (3) ◽  
pp. 311-314
Author(s):  
D. Pedoe
Keyword(s):  
Algebraic Surface ◽  
Base Point ◽  
Canonical System ◽  
Algebraic Surfaces ◽  
Simple Point ◽  
Canonical Systems ◽  
Base Points ◽  
Complete Linear System ◽  
Pass Through ◽  
Simple Points

A complete linear system of curves on an algebraic surface may have assigned base points. The canonical system, from its definition, has no assigned base points at simple points of the surface. But we may construct surfaces on which, all the same, the canonical system has “accidental base points” at simple points of the surface. The classical example, due to Castelnuovo, is a quintic surface with two tacnodes. On this surface the canonical system is cut out by the planes passing through the two tacnodes. These planes also pass through the simple point in which the join of the two tacnodes meets the surface again. This point is the accidental base point of the canonical system on the quintic surface.

QUANTUM-DEFORMED GEOMETRY ON PHASE-SPACE

1993 ◽  
Vol 08 (15) ◽  
pp. 1433-1442 ◽  
Author(s):  
E. GOZZI ◽  
M. REUTER
Keyword(s):  
Vector Field ◽  
Phase Space ◽  
Mechanical System ◽  
Time Evolution ◽  
Cotangent Bundle ◽  
Symplectic Manifolds ◽  
Hamiltonian Vector ◽  
Lie Derivative ◽  
Hamiltonian Vector Field ◽  
Quantum Analog

In this paper we extend the standard Moyal formalism to the tangent and cotangent bundle of the phase-space of any Hamiltonian mechanical system. In this manner we build the quantum analog of the classical Hamiltonian vector-field of time evolution and its associated Lie-derivative. We also use this extended Moyal formalism to develop a quantum analog of the Cartan calculus on symplectic manifolds.

scholarly journals A Lie connection between Hamiltonian and Lagrangian optics

1997 ◽  
Vol Vol. 1 ◽  
Author(s):  
Alex J. Dragt
Keyword(s):  
Vector Field ◽  
Lie Algebra ◽  
Vector Fields ◽  
Geometrical Optics ◽  
Hamiltonian Vector ◽  
Hamiltonian Vector Field ◽  
Third Order ◽  
Hamiltonian Vector Fields ◽  
Uniform Medium ◽  
International Audience

International audience It is shown that there is a non-Hamiltonian vector field that provides a Lie algebraic connection between Hamiltonian and Lagrangian optics. With the aid of this connection, geometrical optics can be formulated in such a way that all aberrations are attributed to ray transformations occurring only at lens surfaces. That is, in this formulation there are no aberrations arising from simple transit in a uniform medium. The price to be paid for this formulation is that the Lie algebra of Hamiltonian vector fields must be enlarged to include certain non-Hamiltonian vector fields. It is shown that three such vector fields are required at the level of third-order aberrations, and sufficient machinery is developed to generalize these results to higher order.

Weak Center and Bifurcation of Critical Periods in a Cubic Z2-Equivariant Hamiltonian Vector Field

2015 ◽  
Vol 25 (11) ◽  
pp. 1550143 ◽  
Author(s):  
Yusen Wu ◽  
Wentao Huang ◽  
Yongqiang Suo
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Hamiltonian Vector ◽  
Local Bifurcation ◽  
Hamiltonian Vector Field ◽  
Critical Periods ◽  
Weak Center ◽  
Hamiltonian Vector Fields ◽  
Center Conditions ◽  
Bifurcation Of Critical Periods

This paper focuses on the problems of weak center and local bifurcation of critical periods for a class of cubic Z2-equivariant planar Hamiltonian vector fields. By computing the period constants carefully, one can see that there are three weak centers: (±1, 0) and the origin. The corresponding weak center conditions are also derived. Meanwhile, we address the problem of the coexistence of bifurcation of critical periods that occurred from (±1, 0) and the origin.

scholarly journals Second-order analysis in polynomially perturbed reversible quadratic Hamiltonian systems

2000 ◽  
Vol 20 (6) ◽  
pp. 1671-1686 ◽  
Author(s):  
LUBOMIR GAVRILOV ◽  
ILIYA D. ILIEV
Keyword(s):  
Vector Field ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bound ◽  
Vector Fields ◽  
Hamiltonian Vector ◽  
Hamiltonian Vector Field ◽  
Finite Plane ◽  
Order Analysis ◽  
Hamiltonian Vector Fields

We study degree $n$ polynomial perturbations of quadratic reversible Hamiltonian vector fields with one center and one saddle point. It was recently proved that if the first Poincaré–Pontryagin integral is not identically zero, then the exact upper bound for the number of limit cycles on the finite plane is $n-1$. In the present paper we prove that if the first Poincaré–Pontryagin function is identically zero, but the second is not, then the exact upper bound for the number of limit cycles on the finite plane is $2(n-1)$. In the case when the perturbation is quadratic ($n=2$) we obtain a complete result—there is a neighborhood of the initial Hamiltonian vector field in the space of all quadratic vector fields, in which any vector field has at most two limit cycles.

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