The classical particle coupled to external electromagnetic field: symmetries and conserved quantities

Full text: We consider a classical particle minimally coupled to an external electromagnetic field, in both non-relativistic and relativistic regimes. The coupling is constructed via the electromagnetic potential which is assumed to satisfy the classical Maxwell equations. We review Noether's theorem at classical level associating infinitesimal symmetries to conserved quantities. The fundamental space-time symmetries are investigated considering a non-relativistic action, a relativistic action in a particular reference frame and an explicitly Lorentz invariant Lagrangian. We work out in detail the corresponding conserved quantities for each case. The well-known Noether's theorem establishes a connection between continuous infinitesimal symmetries of the action and conserved quantities - given a particular action, for each infinitesimal symmetry there exists an explicit conserved quantity. In particular, a single particle subjected to an external electromagnetic field gives rise to an action which may enjoy space-time symmetries. For the non-relativistic particle, we analyze translations in space and time and spatial rotations, calculating the conserved quantities - linear momentum, energy and angular momentum. The relativistic particle enjoys space-time Lorentz symmetry. Thus we check the six symmetries of the homogeneous Lorentz group, corresponding to three spatial rotations and three boosts, and the four space-time translations extending to the non-homogeneous Lorentz group (Poincare group). We consider two distinct actions describing the relativistic particle minimally coupled to an external electromagnetic field - the first one describes the particle in a particular frame of reference enforcing the relativistic generalization of Newton's second law with the Lorentz force while the second one is obtained from a Lorentz scalar Lagrangian. In all cases the conserved quantities are explicitly calculated via Noether's theorem. (author)