###### On the integrability of the Hamiltonian systems with homogeneous polynomial potentials

1 Introduction In the theory of ordinary differential equations and in particular in the theory of Hamiltonian systems the existence of first integrals is important, because they allow to lower the dimension where the Hamiltonian system is defined. Furthermore, if we know a sufficient number of first integrals, these allow to solve the Hamiltonian system explicitly, and we say that the system is integrable. Almost until the end of the 19th century the major part of mathematicians and physicians believe that the equations of classical mechanics were

###### Analysis of fractional factor system for data transmission in SDN

graph for every independent set I of G , a graph is a fractional independent-set-deletable ( a,b,m )-deleted graph (shortly, fractional ID-( a,b,m )-deleted graph). If a = b = k , then a fractional ID-( a,b,m )-deleted graph is a fractional ID-( k,m )-deleted graph. If m = 0, then a fractional ID-( a,b,m )-deleted graph is just a fractional ID-( a,b )-factor-critical graph. If G has a fractional ( g, f )-factor containing a Hamiltonian cycle, it is said that G includes a Hamiltonian fractional ( g, f )-factor. A graph G is called an ID-Hamiltonian

###### Poisson and symplectic reductions of 4–DOF isotropic oscillators. The van der Waals system as benchmark

1 Introduction The use of computer algebra systems for normal forms computations is considered at present a routine operation. As a general reference see e.g. Sanders et al . [ 36 ] and Meyer et al . [ 32 ]. Nevertheless when we deal with special classes of differential equations, like Poisson or Hamiltonians systems which is our case, it is advisable to employ specific transformations as well as tailored variables for those problems [ 32 ], mostly connected with the symmetries that those systems might possess. More precisely we are interested in

###### Hamilton-connectivity of Interconnection Networks Modeled by a Product of Graphs

the longest path or cycle is required the problem is closely related to well-known hamiltonian problems in graph theory. In the rest of this paper, we will use standard terminology in graphs(see ref.[ 2 ]). It is very difficult to determine that a graph is hamiltonian or not. Readers may refer to [ 4 , 5 , 6 ]. 2 Definitions and Notation We follow [ 2 ] for graph-theoretical terminology and notation not defined here. A graph G = ( V,E ) always means a simple graph(without loops and multiple edges), where V = V ( G ) is the vertex set and E = E ( G

###### Complex variables approach to the short-axis-mode rotation of a rigid body

1 Introduction The rotation of a triaxial rigid body in the absence of external torques is known to be integrable [ 1 , 2 ]. In particular, the canonical transformation to Andoyer variables [ 3 ] reduces the free rigid body rotation to an integrable, one degree of freedom Hamiltonian, which immediately shows the preservation of the total angular momentum and allows for the representation of the possible solutions by contour plots of the reduced Hamiltonian [ 4 ]. However, because the solution to the torque-free motion depends on elliptical integrals and

###### Travelling waves and conservation laws of a (2+1)-dimensional coupling system with Korteweg-de Vries equation

coupling with the Korteweg-de Vries equation, which is associated with non-semisimple matrix Lie algebras. In the references [ 10 ] and [ 11 ], its Lax pair and bi-Hamiltonian formulation were presented respectively. It should be noted that its bi-Hamiltonian structure is the first example of local bi-Hamiltonian structures, which lead to hereditary recursion operators in (2+1)-dimensions. Several methods have been developed to find exact solutions of the NLPDEs. Some of these are the homogeneous balance method [ 12 ], the ansatz method [ 13 ], the inverse scattering

###### The Triaxiality Role in the Spin-Orbit Dynamics of a Rigid Body

Poincaré and Arnold, we split the Hamiltonian into two terms: H = H 0 + H 1 , $$\begin{array}{} \displaystyle {\cal H} = {\cal H}_0 + {\cal H}_1, \end{array}$$ where the intermediary 𝓗 0 defines a non-degenerate and simplified model of the problem at hand, which includes the Kepler and free rigid-body as particular cases and 𝓗 1 is usually dubbed as the perturbation. A special realization of an intermediary occurs for the case in which it is an integrable 1-DOF system. The work of Hill on the Moon motion [ 32 ] is, perhaps, the best known example. The

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A sufficient condition for the existence of a *k*-factor excluding a given *r*-factor

1 Introduction For motivation and background to this work see [ 1 ]. In this paper, we consider only finite and simple graphs. Let G = ( V ( G ) , E ( G )) be a graph, where V ( G ) denotes its vertex set and E ( G ) denotes its edge set. A graph is Hamiltonian if it admits a Hamiltonian cycle. For each x ∊ V ( G ), the neighborhood N G ( x ) of x is the set of vertices of G adjacent to x , and the degree d G ( x ) of x is | N G ( x )|. For S ⊆ V ( G ), we write N G ( S ) = ∪ x∊S N G ( x ). G [ S ] denotes the subgraph of G

###### Centers: their integrability and relations with the divergence

, and we denote it by div( x , y ), as the function div ( x , y ) = ∂ X ∂ x ( x , y ) + ∂ Y ∂ y ( x , y ) . $$\begin{array}{} \displaystyle {\rm div} (x,y) \, = \, \frac{\partial X}{\partial x}(x,y) \, + \, \frac{\partial Y}{\partial y}(x,y). \end{array}$$ System 1 is said to be Hamiltonian if div( x , y ) ≡ 0. In such a case there exists a neighborhood of the origin U and an analytic function H : U ⊆ ℝ 2 → ℝ, called the Hamiltonian, such that X ( x , y ) = − ∂ H ∂ y and Y ( x , y ) = ∂ H ∂ y . $$\begin{array}{} \displaystyle X(x,y) = - \frac

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On the central configurations of the *n*-body problem

–body problem Physica D 238 2009 563 571 10.1016/j.physd.2008.12.014 [26] Llibre, J., Moeckel, R. and Simó, C., Central configurations, periodic orbits and Hamiltonian systems , Advances Courses in Math., CRM Barcelona, Birhauser, 2015. Llibre J. Moeckel R. Simó C. Central configurations, periodic orbits and Hamiltonian systems Advances Courses in Math., CRM Barcelona Birhauser 2015 [27] Long, Y. and Sun, S., Four–Body Central Configurations with some Equal Masses , Arch. Rational Mech. Anal. 162 (2002), 24–44. doi 10.1007/s002050100183 Long Y. Sun S. Four–Body Central