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.

scholarly journals ∞-jets of diffeomorphisms preserving orbits of vector fields

2009 ◽  
Vol 7 (2) ◽  
Author(s):  
Sergiy Maksymenko
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Hamiltonian Vector ◽  
Homogeneous Polynomials ◽  
Local Flow ◽  
Linear Vector ◽  
Identity Component ◽  
Hamiltonian Vector Fields

AbstractLet F be a C ∞ vector field defined near the origin O ∈ ℝn, F(O) = 0, and (Ft) be its local flow. Denote by the set of germs of orbit preserving diffeomorphisms h: ℝn → ℝn at O, and let , (r ≥ 0), be the identity component of with respect to the weak Whitney Wr topology. Then contains a subset consisting of maps of the form Fα(x)(x), where α: ℝn → ℝ runs over the space of all smooth germs at O. It was proved earlier by the author that if F is a linear vector field, then = .In this paper we present a class of examples of vector fields with degenerate singularities at O for which formally coincides with , i.e. on the level of ∞-jets at O.We also establish parameter rigidity of linear vector fields and “reduced” Hamiltonian vector fields of real homogeneous polynomials in two variables.

scholarly journals THE g-AREAS AND THE COMMUTATOR LENGTH

2013 ◽  
Vol 24 (07) ◽  
pp. 1350057 ◽  
Author(s):  
FRANÇOIS LALONDE ◽  
ANDREI TELEMAN
Keyword(s):  
Vector Fields ◽  
Morse Function ◽  
Hamiltonian Vector ◽  
Hamiltonian Vector Field ◽  
Linear Combinations ◽  
Hamiltonian Diffeomorphisms ◽  
Hamiltonian Vector Fields ◽  
Commutator Length ◽  
Lie Brackets ◽  
Hamiltonian Diffeomorphism

The commutator length of a Hamiltonian diffeomorphism f ∈ Ham (M,ω) of a closed symplectic manifold (M,ω) is by definition the minimal k such that f can be written as a product of k commutators in Ham (M,ω). We introduce a new invariant for Hamiltonian diffeomorphisms, called the k+-area, which measures the "distance", in a certain sense, to the subspace [Formula: see text] of all products of k commutators. Therefore, this invariant can be seen as the obstruction to writing a given Hamiltonian diffeomorphism as a product of k commutators. We also consider an infinitesimal version of the commutator problem: what is the obstruction to writing a Hamiltonian vector field as a linear combination of k Lie brackets of Hamiltonian vector fields? A natural problem related to this question is to describe explicitly, for every fixed k, the set of linear combinations of k such Lie brackets. The problem can be obviously reformulated in terms of Hamiltonians and Poisson brackets. For a given Morse function f on a symplectic Riemann surface M (verifying a weak genericity condition) we describe the linear space of commutators of the form {f, g}, with [Formula: see text].

scholarly journals INFINITESIMAL TAKESAKI DUALITY OF HAMILTONIAN VECTOR FIELDS ON A SYMPLECTIC MANIFOLD

2000 ◽  
Vol 12 (12) ◽  
pp. 1669-1688 ◽  
Author(s):  
KATSUNORI KAWAMURA
Keyword(s):  
Lie Algebra ◽  
Symplectic Manifold ◽  
Vector Fields ◽  
Type Theorem ◽  
Hamiltonian Vector ◽  
Crossed Product ◽  
Hamiltonian Vector Fields ◽  
Symplectic Action

For an infinitesimal symplectic action of a Lie algebra [Formula: see text] on a symplectic manifold, we construct an infinitesimal crossed product of Hamiltonian vector fields and [Formula: see text]. We obtain its second crossed product in the case where [Formula: see text] and obtain an infinitesimal version of a Takesaki duality type theorem.

POISSON-NIJENHUIS STRUCTURES AND SATO HIERARCHY

1991 ◽  
Vol 03 (04) ◽  
pp. 403-466 ◽  
Author(s):  
G. MAGNANO ◽  
F. MAGRI
Keyword(s):  
Finite Number ◽  
Lie Algebra ◽  
Vector Fields ◽  
Hamiltonian Structure ◽  
Differential Operators ◽  
Hamiltonian Vector ◽  
Poisson Brackets ◽  
Moody Algebra ◽  
Hamiltonian Vector Fields ◽  
Direct Sum

We show that the direct sum of n copies of a Lie algebra is endowed with a sequence of affine Lie-Poisson brackets, which are pairwise compatible and define a multi-Hamiltonian structure; to this structure one can associate a recursion operator and a Kac-Moody algebra of Hamiltonian vector fields. If the initial Lie algebra is taken to be an associative algebra of differential operators, a suitable family of Hamiltonian vector fields reproduce either the n-th Gel'fand-Dikii hierarchy (for n finite) or Sato's hierarchy (for n = ∞). Within the same framework, it is also possible to recover a class of integro-differential hierarchies involving a finite number of fields, which generalize the Gel'fand-Dikii equations and are equivalent to Sato's hierarchy.

scholarly journals Geometrical structures on the cotangent bundle

2016 ◽  
Vol 13 (05) ◽  
pp. 1650071 ◽  
Author(s):  
Liviu Popescu
Keyword(s):  
Vector Field ◽  
Covariant Derivative ◽  
Cotangent Bundle ◽  
Vector Fields ◽  
Hamiltonian Vector ◽  
Legendre Transformation ◽  
Hamiltonian Vector Field ◽  
Geometrical Structures ◽  
Nonlinear Connection

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.

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