International Journal of Bifurcation and Chaos, Vol. 22, No. 6 (2012) 1250135 (5 pages) c  World Scientific Publishing Company DOI: 10.1142/S0218127412501350 DIRECT HAMILTONIZATION — GENERALIZATION OF AN ALTERNATIVE HAMILTONIZATION MARIA LEWTCHUK ESPINDOLA Dpto. Matem., CCEN, Universidade Federal da Para´ ıba, Campus I, Cidade Universit´ aria, Jo˜ ao Pessoa, 58051-970, PB, Brazil mariia@mat.ufpb.br Received February 28, 2011 A new procedure named direct Hamiltonization presents an alternative foundation to Analytical Mechanics, since in this formalism the Hamiltonian function can be obtained for all mechanical systems. The principal change proposed in this procedure is that the conjugate momenta cannot be defined a priori, but are established as a consequence of a canonical description of the mechanical system. The direct Hamiltonization is a generalization of the alternative one, where the usual Hamiltonization and momenta are recovered whenever they exist. Also this procedure assures the existence of a Hamiltonian function without any constraints for any mechanical system, therefore the usual quantization is always allowed. This procedure can be applied to non-Lagrangian, Nambu, nonholonomic and dynamical systems since there are no restrictions in this formalism as, for example, the number of equations of motion. Keywords : Direct Hamiltonization; alternative Hamiltonization; analytical mechanics founda- tions; Hamiltonian mechanics. 1. Introduction A generalized approach for the foundation of Ana- lytical Mechanics is obtained by an alternative Hamiltonization procedure. This procedure is devel- oped defining the Hamiltonian function as a solution of the partial differential equation obtained by the usual definition of the Hamiltonian func- tion from the Lagrangian one and the first set of Hamilton equations of motion. The momenta is obtained from the second set of Hamilton equations of motion furnishing always a canonical description of the system. The usual Hamiltonization and the usual definition of the momentum, when they exist, are recovered as a particular case by the envelope solution of the above equation. The procedure of alternative Hamiltoniza- tion for mechanical systems has been developed and applied to singular and nonholonomic sys- tems in previous papers [Espindola et al., 1986a, 1986b, 1986c; Espindola, 1993; Espindola et al., 1987a]. In a similar manner the procedure is extended for field theory [Espindola et al., 1987b], recover- ing the usual Hamiltonian density as a particular case. The direct Hamiltonization for mechanical sys- tems [Espindola, 1996] generalizes the alternative one. This procedure can be applied to non- Lagrangian, Nambu [Espindola, 2008], nonholo- nomic [Espindola et al., 1987a] and dynamical systems since there are no restrictions in the for- malism as, for example, the number of equations of motion. In Sec. 2, the two-fold or alternative Hamil- tonization procedure is abridged, in Sec. 3 the direct Hamiltonization procedure is described. The final remarks and a lot of possibilities of future exten- sion and applications are given in Sec. 4. 1250135-1