Constrained Hamiltonian dynamics for electrons in magnetic field and additional forces besides the Lorentz force acting on electrons

Abstract We consider the forces acting on electrons in magnetic field including the constraints and a condition arising from quantum mechanics. The force is calculated as the electron mass, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>m</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">e</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> , multiplied by the total time-derivative of the velocity field evaluated using the quantum mechanical many-electron wave function. The velocity field includes a term of the Berry connection from the many-body wave function; thereby, quantum mechanical effects are included. It is shown that additional important forces besides the Lorentz force exist; they include the gradient of the electron velocity field kinetic energy, the gradient of the chemical potential, and the ‘force’ for producing topologically protected loop currents. These additional forces are shown to be important in superconductivity, electric current in metallic wires, and charging of capacitors.