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The set of lagrange and routh formulations for non‐linear networks
Author(s) -
Shragowitz Eugene,
Gerlovin Emmanuel
Publication year - 1988
Publication title -
international journal of circuit theory and applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.364
H-Index - 52
eISSN - 1097-007X
pISSN - 0098-9886
DOI - 10.1002/cta.4490160203
Subject(s) - mathematics , constraint algorithm , lagrange multiplier , duality (order theory) , uniqueness , mathematical analysis , discrete mathematics , mathematical optimization
Formulations of systems of Lagrange and Routh equations for arbitrary non‐linear electrical circuits are given. the use of Routh equations for this purpose is new. It is proved that these formulations are equivalent to the complete system of Kirchhoff equations (instead of only a part of it as in prior works). the vector of generalized coordinates for the system of Lagrange equations consists of four subvectors (loop charges for fundamental loops, cut‐set fluxes for fundamental cut‐sets, branch fluxes for voltage and flux controlled elements and branch charges for current and charge controlled elements). For the defined set of Lagrange formulations, the uniqueness of a parametric representation is proved. the structure of the Lagrange (Hamilton, Routh) formulation set is then studied and it is proved that this set is an Abelian group. A duality of Lagrange triples for electrically and topologically dual circuits is established and it is proved that this relation between the sets of Lagrange triples is an isomorphism. It is also shown that the Brayton‐Moser equations and the anti‐Lagrangian equations similar to those of M. Milić and L. Novak represent partial cases of the formulated set.