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Unformatted text preview: Particle and Field Dynamics Comte Joseph Louis Lagrange (1736  1813) November 9, 2001 Contents 1 Lagrangian and Hamiltonian of a Charged Particle in an External Field 2 1.1 Lagrangian of a Free Particle . . . . . . . . . . . . . . . . . . . . . . 4 1.1.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Lagrangian of a Charged Particle in Fields . . . . . . . . . . . . . . . 6 1.2.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Hamiltonian of a Charged Particle . . . . . . . . . . . . . . . . . . . . 8 1.4 Invariant Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 Lagrangian for the Electromagnetic Field 11 3 Stress Tensors and Conservation Laws 13 3.1 Free Field Lagrangian and Hamiltonian Densities . . . . . . . . . . . 14 3.2 Symmetric Stress Tensor . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.3 Conservation Laws in the Presence of Sources . . . . . . . . . . . . . 19 4 Examples of Relativistic Particle Dynamics 20 1 4.1 Motion in a Constant Uniform Magnetic Induction . . . . . . . . . . 20 4.2 Motion in crossed E and B fields, E < B . . . . . . . . . . . . . . . . 22 4.3 Motion in crossed E and B fields, E > B . . . . . . . . . . . . . . . . 24 4.4 Motion for general uniform E and B . . . . . . . . . . . . . . . . . . . 26 4.5 Motion in slowly spatially varying B ( x ) . . . . . . . . . . . . . . . . . 26 2 In this chapter we shall study the dynamics of particles and fields. For a particle, the relativistically correct equation of motion is d p dt = F (1) where p = m u ; the corresponding equation for the time rate of change of the particles energy is dE dt = F u . (2) The dynamics of the electromagnetic field is given by the Maxwell equations, E ( x , t ) = 4 ( x , t ) B ( x , t ) = 0 E ( x , t ) + 1 c B ( x , t ) t = 0 B ( x , t ) 1 c E ( x , t ) t = 4 c J ( x , t ) . (3) These are tied together by the Lorentz force which gives F in terms of the electro magnetic fields F = q E + 1 c ( u B ) (4) and by the expressions for ( x , t ) and J ( x , t ) in terms of the particles coordinates and velocities ( x , t ) = i q i ( x x i ( t )) J ( x , t ) = i q i u i ( t ) ( x x i ( t )) . (5) In view of the fact that we already know all of this, what further do we want to do? Two things: (1) Formulate appropriate covariant Lagrangians and Hamiltonians from which covariant dynamical equations can be derived; and (2) applications. 1 Lagrangian and Hamiltonian of a Charged Par ticle in an External Field We want to devise a Lagrangian for a charged particle in the presence of given applied fields which are treated as parameters and not as dynamic variables. This Lagrangian 3 is to yield the equations of motion Eqs. (1) and (2) with F given by the Lorentz force....
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 Fall '11
 Electrodynamics
 Charge

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