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Biot-Savart-Laplace dynamical systems. (English) Zbl 0888.34007

Summary: §1 recalls known facts about the magnetic field \(\vec H\) produced by the Biot-Savart-Laplace law for a massive condutor \(\overline D\). §2 proves that generally the part in \(\text{ext }\overline D\) of a magnetic line is a trajectory of a potential dynamical system of order two (a geodesic of Riemann-Jacobi structure) and the part in \(\text{int }\overline D\) is a trajectory of a nonpotential dynamical system of order two (new Lorentz world – force laws) describing new magnetic dynamics. This paragraph presents also some properties of magnetic traps, two significant examples and formulates an open problem. §3 describes magnetic dynamical systems which can be reduced to 2-dimensional Hamiltonian systems. §4 analyses magnetic fields which are symmetric or antisymmetric with respect to some symmetries.

MSC:

34A26 Geometric methods in ordinary differential equations
78A25 Electromagnetic theory (general)
35F10 Initial value problems for linear first-order PDEs
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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