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On the Incorrectness of the Lagrange Formalism as Applied to Magnetic Phenomena

Yuri I. Petrov
Year: 2010
Keywords: freely moving charges, magnetic moment, magnetic energy, Hamiltonian function, Lagrange function, Lorentz? force

A critical analysis reveals the invalidity of the established idea that movement of free charges in a magnetic field could be described with a help of the analytical Lagrange mechanics: the given system is non-conservative and non-holonomic. As the Lorentz force in the equation of charge movement was silently supposed to be an active force, the magnetic energy u0H (u0 is the magnetic moment of orbital electron and H is the magnetic field, and ?dot? denotes scalar product) turns out numerically equal to a fraction of the kinetic energy of the electron in the plane perpendicular to the direction of H. Consequently, a routine introducing into Lagrangian of the term with a vector potential, which is equivalent to u0H, resulting in paradoxical doubling of the corresponding fraction of the kinetic energy. This occurs irrespective of the magnetic field strength. As a result, the applied Lagrange?s formalism and notations of Lagrangian and Hamiltonian lose their meanings. Therefore, expressions obtained with a help of the Lagrange mechanics become inapplicable for deducing the equation of charge motion in a magnetic field or for calculating the magnetic magnitudes. The Lorentz force actually acts upon a moving charge as a passive non-holonomic linkage imposing constraints upon the direction of its velocity, but it is not able to add the kinetic energy to the particle.