Results 191 to 200 of about 129,282 (240)

Singularities of momentum maps of integrable Hamiltonian systems with two degrees of freedom

Journal of Mathematical Sciences, 1999
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Bolsinov, A. V., Matveev, V. S.
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Lagrangian singularities of invariant tori of hamiltonian systems with two degrees of freedom

Inventiones Mathematicae, 1989
The following main statement is proved, besides another, for the 2-torus \(T^ 2={\mathbb{R}}^ 2/2\pi {\mathbb{Z}}^ 2\). Theorem. Let L be an incompressible torus of class \(C^ 3\) imbedded in the hypersurface \(M=\{x\in T^*X:\) \(H(x)=h\), H being the Hamiltonian function\(\}\) (i.e.
Bialy, M. L., Polterovich, L. V.
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On degenerate resonances in Hamiltonian systems with two degrees of freedom

Chaos, Solitons & Fractals, 2014
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Karabanov, A., Morozov, A. D.
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Integrable Hamiltonian systems with two degrees of freedom associated with holomorphic functions

Theoretical and Mathematical Physics, 2000
We focus on integrable systems with two degrees of freedom that are integrable in the Liouville sense and are obtained as real and imaginary parts of a polynomial (or entire) complex function in two complex variables. We propose definitions of the actions for such systems (which are not of the Arnol'd-Liouville type). We show how to compute the actions
Doss-Bachelet, C., Françoise, Jean-Pierre   +1 more
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Horseshoes in Two-Degree-of-Freedom Hamiltonian Systems with Saddle-Centers

Archive for Rational Mechanics and Analysis, 2000
The author studies two-degree-of-freedom Hamiltonian systems of the form \[ \dot x=JD_x H(x,y),\quad \dot y=JD_y H(x,y),\quad (x,y)\in \mathbb{R}^2\times \mathbb{R}^2 \tag{1} \] where \(H:\mathbb{R}^2\times \mathbb{R}^2\rightarrow \mathbb{R}\) is \(C^{r+1}(r\geq 3) \) and \(J=\left( \begin{smallmatrix} 0&1\\ -1&0\end{smallmatrix}\right).\) It is ...
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A class of Liouville-integrable Hamiltonian systems with two degrees of freedom

Journal of Mathematical Physics, 2000
A class of two-dimensional Liouville-integrable Hamiltonian systems is studied. Separability of the corresponding Hamilton–Jacobi equation for these systems is shown to be equivalent to other fundamental properties of Hamiltonian systems, such as the existence of the Lax and bi-Hamiltonian representations of certain fixed types.
McLenaghan, Raymond G., Smirnov, Roman G.   +1 more
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ORBITAL EQUIVALENCE OF INTEGRABLE HAMILTONIAN SYSTEMS WITH TWO DEGREES OF FREEDOM. A CLASSIFICATION THEOREM. I

Russian Academy of Sciences. Sbornik Mathematics, 1995
This is the second part of a comprehensive study of integrable Hamiltonian systems with 2 degrees of freedom on 3-dimensional constant energy surfaces [the first part was published in ibid. 81, No. 2, 421-465 (1995); transl. from Mat. Sb. 185, No. 4, 27-80 (1994; Zbl 0828.58020)].
Bolsinov, A. V., Fomenko, A. T.
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On stability in Hamiltonian systems with two degrees of freedom

Mathematical Notes, 2014
We consider the stability of the equilibrium position at the origin of coordinates of a Hamiltonian system with two degrees of freedom whose unperturbed part describes oscillators with restoring force of odd order greater than 1. It is proved that if the exponents of the restoring force of the oscillators are not equal, then the equilibrium position is
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Bifurcations in a Hamiltonian system with two degrees of freedom associated with the reversible hyperbolic umbilic

Nonlinear Dynamics, 2021
We deal with Hamiltonian bifurcations associated with the reversible umbilic in two degrees of freedom systems defined by 0:1 resonance, i.e. the unperturbed equilibrium has two purely imaginary eigenvalues and a semisimple double-zero one. The Hamiltonian is written as the sum of integrable Hamiltonian
Xing Zhou, Xuemei Li
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Detailed asymptotics for a convex Hamiltonian system with two degrees of freedom

Journal of Dynamics and Differential Equations, 1993
The authors investigate the simultaneous system \(u''+u(1+u^ 2+v^ 2)=0\), \(v''+v(k+u^ 2+v^ 2)=0\), where \(k>1\). The system has two conserved energies given by \[ E(u,v,u',v')={1\over 2}u^{'2}+{1\over 2}v^{'2}+{1\over 2}u^ 2+{k\over 2}v^ 2+{1\over 4}(u^ 2+v^ 2)^ 2, \] and \[ F(u,v,u',v')=-{(uv'-u'v)^ 2\over 2(k-1)}+v^{'2}+kv^ 2+{1\over 2}v^ 2(u^ 2+v^
Cazenave, Thierry, Haraux, Alain, Weissler, Fred B.   +2 more
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