dorsal/arxiv
View SchemaQuasi-Lagrangian Systems of Newton Equations
| Authors | Stefan Rauch-Wojciechowski, Krzysztof Marciniak, Hans Lundmark |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9909025 |
| URL | https://arxiv.org/abs/solv-int/9909025 |
| DOI | 10.1063/1.533098 |
Abstract
Systems of Newton equations of the form $\ddot{q}=-{1/2}A^{-1}(q)\nabla k$ with an integral of motion quadratic in velocities are studied. These equations generalize the potential case (when A=I, the identity matrix) and they admit a curious quasi-Lagrangian formulation which differs from the standard Lagrange equations by the plus sign between terms. A theory of such quasi-Lagrangian Newton (qLN) systems having two functionally independent integrals of motion is developed with focus on two-dimensional systems. Such systems admit a bi-Hamiltonian formulation and are proved to be completely integrable by embedding into five-dimensional integrable systems. They are characterized by a linear, second-order PDE which we call the fundamental equation. Fundamental equations are classified through linear pencils of matrices associated with qLN systems. The theory is illustrated by two classes of systems: separable potential systems and driven systems. New separation variables for driven systems are found. These variables are based on sets of non-confocal conics. An effective criterion for existence of a qLN formulation of a given system is formulated and applied to dynamical systems of the Henon-Heiles type.
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"abstract": "Systems of Newton equations of the form $\\ddot{q}=-{1/2}A^{-1}(q)\\nabla k$\nwith an integral of motion quadratic in velocities are studied. These equations\ngeneralize the potential case (when A=I, the identity matrix) and they admit a\ncurious quasi-Lagrangian formulation which differs from the standard Lagrange\nequations by the plus sign between terms. A theory of such quasi-Lagrangian\nNewton (qLN) systems having two functionally independent integrals of motion is\ndeveloped with focus on two-dimensional systems. Such systems admit a\nbi-Hamiltonian formulation and are proved to be completely integrable by\nembedding into five-dimensional integrable systems. They are characterized by a\nlinear, second-order PDE which we call the fundamental equation. Fundamental\nequations are classified through linear pencils of matrices associated with qLN\nsystems. The theory is illustrated by two classes of systems: separable\npotential systems and driven systems. New separation variables for driven\nsystems are found. These variables are based on sets of non-confocal conics. An\neffective criterion for existence of a qLN formulation of a given system is\nformulated and applied to dynamical systems of the Henon-Heiles type.",
"arxiv_id": "solv-int/9909025",
"authors": [
"Stefan Rauch-Wojciechowski",
"Krzysztof Marciniak",
"Hans Lundmark"
],
"categories": [
"solv-int",
"nlin.SI"
],
"doi": "10.1063/1.533098",
"title": "Quasi-Lagrangian Systems of Newton Equations",
"url": "https://arxiv.org/abs/solv-int/9909025"
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