By Martin Berz
Advances in Imaging & Electron Physics merges long-running serials-Advances in Electronics & Electron Physics and Advances in Optical & Electron Microscopy. The sequence gains prolonged articles at the physics of electron units (especially semiconductor devices), particle optics at low and high energies, microlithography, picture technology and electronic photograph processing, electromagnetic wave propagation, electron microscopy, and the computing tools utilized in a majority of these domain names.
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Additional info for Modern Map Methods in Particle Beam Physics
We note that using Eqs. 44), we have <,(t1,2,a) = <1,2 for 1 5 j 5 n , and thus all the curves in the family go through the same starting and ending points, as demanded by the requirement of the minimization problem. Because < j minimizes the action integral, when aqj . 7, is substituted into the integral, a minimum is assumed at a! 48) Thus, we obtain L ( < j ( r , a ! ) , i j j ( r , a ) , r ) d=r[ (-aqja L - A S) dr a g j This condition must be true for every choice of q, satisfying can only be the case if indeed q j dr.
This is indeed frequently possible, but it is difficult to classify those systems that are Hamiltonian up to coordinate transformations or to establish a general mechanism to find a suitable transformation. However, a very important special case of such transformations will be discussed in the next section. 2,the question of the existence of a Hamiltonian to a given motion is somewhat restrictive, and in general it is not clear whether there may be certain changes of variables that will lead to a Hamiltonian system.
It is somewhat illuminating, however, that in the new Hamiltonian the conjugate of the time can be chosen as the old Hamiltonian. The transformation to autonomous Hamiltonian systems is performed frequently in practice, and the 2(n 1) dimensional space ( i ,s, fi, p s ) is referred to as the extended phase space. In conclusion, we show that despite the unusual intermixed form characteristic of Hamiltonian dynamics, there is a rather straightforward and intuitive geometric interpretation of Hamiltonian motion.
Modern Map Methods in Particle Beam Physics by Martin Berz