One Classical Particle Subject to Electromagnetic Fields

The equation of motion for a nonrelativistic charged particle with charge cj and mass m subject to time-dependent electromagnetic fields will now be derived within the Lagrangian and also within the Hamiltonian formalism. The 

This is the most prominent and important example of a velocity-dependent potential U. The evaluation of the Euler-Lagrange equations (2.53) together with the relations between electromagnetic fields and potentials in Eq. (2.127) 

It is important to realize that for any system including a vector potential canonical and linear momentum do not coincide. According to Eq. (2.54) the canonical momentum p conjugate to r is given by 

Note that we have introduced the symbol 7t for linear momentum here in order to better distinguish it from canonical momentum. This notational rigor is only needed for the discussion of this section and will thus be dropped elsewhere in the book. In most cases it will become obvious from the context to which kind of momentum we are referring. Obviously, it is rather the linear than the canonical momentum which is gauge invariant. The canonical momentum p satisfies the fundamental Poisson brackets of Eq. (2.81), of course, whereas the components of linear momentum, interpreted as phase space functions tt, = Ki r,p,t), feature nonvanishing but gauge invariant Poisson brackets. 

Taking advantage of the canonical momentum we will now derive the Hamiltonian describing this nonrelativistic particle within electromagnetic fields. After expression of all velocities by canonical momenta p in the Legendre transformation of Eq. (2.71) and basic algebraic manipulations one 

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