scholarly journals Integrability of Hamiltonian systems with homogeneous potentials of degrees $\pm 2$. An application of higher order variational equations

2014 ◽  
Vol 34 (11) ◽  
pp. 4589-4615 ◽  
Author(s):  
Andrzej J. Maciejewski ◽  
Guillaume Duval
Keyword(s):  
Hamiltonian Systems ◽  
Higher Order ◽  
Variational Equations ◽  
Homogeneous Potentials

scholarly journals DIFFERENTIAL GALOIS THEORY AND INTEGRABILITY

2009 ◽  
Vol 06 (08) ◽  
pp. 1357-1390 ◽  
Author(s):  
ANDRZEJ J. MACIEJEWSKI ◽  
MARIA PRZYBYLSKA
Keyword(s):  
Galois Group ◽  
Hamiltonian Systems ◽  
Integrable Systems ◽  
Degrees Of Freedom ◽  
Galois Theory ◽  
Differential Galois Theory ◽  
Differential Galois Group ◽  
Variational Equations ◽  
Homogeneous Potentials ◽  
Particular Solution

This paper is an overview of our works that are related to investigations of the integrability of natural Hamiltonian systems with homogeneous potentials and Newton's equations with homogeneous velocity independent forces. The two types of integrability obstructions for these systems are presented. The first, local ones, are related to the analysis of the differential Galois group of variational equations along a non-equilibrium particular solution. The second, global ones, are obtained from the simultaneous analysis of variational equations related to all particular solutions belonging to a certain class. The marriage of these two types of the integrability obstructions enables to realize the classification programme of all integrable homogeneous systems. The main steps of the integrability analysis for systems with two and more degrees of freedom as well as new integrable systems are shown.

scholarly journals Meromorphic integrability of the Hamiltonian systems with homogeneous potentials of degree -4

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jaume Llibre ◽  
Yuzhou Tian
Keyword(s):  
Hamiltonian Systems ◽  
Homogeneous Polynomial ◽  
First Integral ◽  
Liouville Integrability ◽  
Linear Change ◽  
Change Of Variables ◽  
Polynomial First Integral ◽  
Homogeneous Potentials

<p style='text-indent:20px;'>We characterize the meromorphic Liouville integrability of the Hamiltonian systems with Hamiltonian <inline-formula><tex-math id="M2">\begin{document}$ H = \left(p_1^2+p_2^2\right)/2+1/P(q_1, q_2) $\end{document}</tex-math></inline-formula>, being <inline-formula><tex-math id="M3">\begin{document}$ P(q_1, q_2) $\end{document}</tex-math></inline-formula> a homogeneous polynomial of degree <inline-formula><tex-math id="M4">\begin{document}$ 4 $\end{document}</tex-math></inline-formula> of one of the following forms <inline-formula><tex-math id="M5">\begin{document}$ \pm q_1^4 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ 4q_1^3q_2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ \pm 6q_1^2q_2^2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ \pm \left(q_1^2+q_2^2\right)^2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M9">\begin{document}$ \pm q_2^2\left(6q_1^2-q_2^2\right) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ \pm q_2^2\left(6q_1^2+q_2^2\right) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M11">\begin{document}$ q_1^4+6\mu q_1^2q_2^2-q_2^4 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M12">\begin{document}$ -q_1^4+6\mu q_1^2q_2^2+q_2^4 $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M13">\begin{document}$ \mu&gt;-1/3 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M14">\begin{document}$ \mu\neq 1/3 $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M15">\begin{document}$ q_1^4+6\mu q_1^2q_2^2+q_2^4 $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M16">\begin{document}$ \mu \neq \pm 1/3 $\end{document}</tex-math></inline-formula>. We note that any homogeneous polynomial of degree <inline-formula><tex-math id="M17">\begin{document}$ 4 $\end{document}</tex-math></inline-formula> after a linear change of variables and a rescaling can be written as one of the previous polynomials. We remark that for the polynomial <inline-formula><tex-math id="M18">\begin{document}$ q_1^4+6\mu q_1^2q_2^2+q_2^4 $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M19">\begin{document}$ \mu\in\left\{-5/3, -2/3\right\} $\end{document}</tex-math></inline-formula> we only can prove that it has no a polynomial first integral.</p>

scholarly journals Higher Order Hamiltonian Systems with Generalized Legendre Transformation

Mathematics ◽  
2018 ◽  
Vol 6 (9) ◽  
pp. 163
Author(s):  
Dana Smetanová
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Higher Order ◽  
Second Order ◽  
Legendre Transformation ◽  
Lagrange Equations ◽  
Hamilton Equations ◽  
Strongly Regular

The aim of this paper is to report some recent results regarding second order Lagrangians corresponding to 2nd and 3rd order Euler–Lagrange forms. The associated 3rd order Hamiltonian systems are found. The generalized Legendre transformation and geometrical correspondence between solutions of the Hamilton equations and the Euler–Lagrange equations are studied. The theory is illustrated on examples of Hamiltonian systems satisfying the following conditions: (a) the Hamiltonian system is strongly regular and the Legendre transformation exists; (b) the Hamiltonian system is strongly regular and the Legendre transformation does not exist; (c) the Legendre transformation exists and the Hamiltonian system is not regular but satisfies a weaker condition.

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