Lagrangian for Circuits with Higher-Order Elements

The necessary and sufficient conditions of the validity of Hamilton’s variational principle for circuits consisting of (α,β) elements from Chua’s periodical table are derived. It is shown that the principle holds if and only if all the circuit elements lie on the so-called Σ-diagonal with a constant...

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Bibliographic Details
Published in:Entropy (Basel, Switzerland) Switzerland), 2019-11, Vol.21 (11), p.1059
Main Authors: Biolek, Zdenek, Biolek, Dalibor, Biolkova, Viera
Format: Article
Language:English
Subjects:
chua’s table
Circuits
Computer engineering
Control theory
Electrical engineering
Equations of motion
euler–lagrange equation
hamilton’s variational principle
higher-order element
Independent variables
lagrangian
Mathematical analysis
memristor
Velocity
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Summary:The necessary and sufficient conditions of the validity of Hamilton’s variational principle for circuits consisting of (α,β) elements from Chua’s periodical table are derived. It is shown that the principle holds if and only if all the circuit elements lie on the so-called Σ-diagonal with a constant sum of the indices α and β. In this case, the Lagrangian is the sum of the state functions of the elements of the L or +R types minus the sum of the state functions of the elements of the C or −R types. The equations of motion generated by this Lagrangian are always of even-order. If all the elements are linear, the equations of motion contain only even-order derivatives of the independent variable. Conclusions are illustrated on an example of the synthesis of the Pais–Uhlenbeck oscillator via the elements from Chua’s table.
ISSN:1099-4300
1099-4300
DOI:10.3390/e21111059