Cylindrical type integrable classical systems in a magnetic field

Abstract

We present all second order classical integrable systems of the cylindrical type in a three dimensional Euclidean space ${\mathbb{E}}_3$ with a nontrivial magnetic field. The Hamiltonian and integrals of motion have the form $H=\frac{1}{2}{\left(\vec{p}+\vec{A}(\vec{x})\right)}^2+W(\vec{x})$, $X_1=(p_{\varphi }^A{)}^2+s_1^r(r,\varphi ,Z)p_r^A+s_1^{\varphi }(r,\varphi ,Z)p_{\varphi }^A+s_1^Z(r,\varphi ,Z)p_Z^A+m_1(r,\varphi ,Z)$, $X_2=(p_Z^A{)}^2+s_2^r(r,\varphi ,Z)p_r^A+s_2^{\varphi }(r,\varphi ,Z)p_{\varphi }^A+s_2^Z(r,\varphi ,Z)p_Z^A+m_2(r,\varphi ,Z)$. Infinite families of such systems are found, in general depending on arbitrary functions or parameters. This leaves open the possibility of finding superintegrable systems among the integrable ones (i.e. systems with 1 or 2 additional independent integrals).

arXiv comment26 pages

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Cited by in corpus (9)

• On superintegrability of 3D axially-symmetric non-subgroup-type systems with magnetic fields
• New classes of quadratically integrable systems in magnetic fields: the generalized cylindrical and spherical cases
• Family of nonstandard integrable and superintegrable classical Hamiltonian systems in non-vanishing magnetic fields
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• Integrable systems in magnetic fields: the generalized parabolic cylindrical case
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• Cylindrical first order superintegrability with complex magnetic fields