2013/08/31 by Steven Duplij · 2 citations
Mathematics · Physics and Astronomy · #Algebraic number #Antisymmetric relation #Degenerate energy levels #First class constraint #Hamiltonian (control theory) #Hamiltonian mechanics #Homotopy and Cohomology in Algebraic Topology #Noncommutative and Quantum Gravity Theories #Nonlinear Waves and Solitons #Partial differential equation #Phase space #Poisson bracket #gr-qc #hep-th #math-ph #math.MP #msc:37K05 #msc:70H05 #msc:70H20 #msc:70H45 #msc:70S05 #quant-ph
paper · pdf · doi:10.1142/s0219887815500012
published in International Journal of Geometric Methods in Modern Physics 12(01), 1550001 (World Scientific) · 17 pages, ws-ijgmmp.cls
openalex publication_date 2014/08/22 · arxiv created 2014/09/07 · arxiv updated 2014/09/09 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
A formulation of singular classical theories (determined by degenerate Lagrangians) without constraints is presented. A partial Hamiltonian formalism in the phase space having an initially arbitrary number of momenta (which can be smaller than the number of velocities) is proposed. The equations of motion become first-order differential equations, and they coincide with those of multi-time dynamics, if a certain condition is imposed. In a singular theory, this condition is fulfilled in the case of the coincidence of the number of generalized momenta with the rank of the Hessian matrix. The noncanonical generalized velocities satisfy a system of linear algebraic equations, which allows an appropriate classification of singular theories (gauge and nongauge). A new antisymmetric bracket (similar to the Poisson bracket) is introduced, which describes the time evolution of physical quantities in a singular theory. The origin of constraints is shown to be a consequence of the (unneeded in our formulation) extension of the phase space, when the new bracket transforms into the Dirac bracket. Quantization is briefly discussed.