Resumo

WOLF, Kurt Bernardo. Por qué y cómo exponenciamos matrices hamiltonianas. Rev. mex. fis. [online]. 2003, vol.49, n.5, pp.465-476. ISSN 0035-001X.

The trajectories of mass points in the classical mechanics of oscillators, and light rays in geometric paraxial optics, are obtained exponentiating matrices. Hamiltonian matrices represent and classify through equivalence the possible dynamics of linear systems. In one-dimensional mechanics and plane waveguides, the possible systems are harmonic, repulsive, or free; this is well known and only requires 2 X matrices with 3 independent parameters. Here we address the problem of mechanical systems in two dimensions, which coincides with that of waveguides in three dimensions, where 4 X 4 matrices are required, with 10 parameters. Knowing the eigenvalue structure, we reduce the exponential of a hamiltonian matrix to the sum of its first four powers, with coefficients that we compute analytically, and resolve the degeneracy which is present in the eigenvalue plane. We comment on the linear wave systems where these results are applied.

Palavras-chave : Hamiltonian systems; canonical transformations; symplectic groups and algebras; matrix exponentiation; equivalence orbits.

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