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Open AccessArticle

Lagrangian for Circuits with Higher-Order Elements

1
Department of Microelectronics, Brno University of Technology, 616 00 Brno, Czech Republic
2
Department of Electrical Engineering, University of Defence, 662 10 Brno, Czech Republic
3
Department of Radio Electronics, Brno University of Technology, 616 00 Brno, Czech Republic
*
Author to whom correspondence should be addressed.
Entropy 2019, 21(11), 1059; https://doi.org/10.3390/e21111059
Received: 20 September 2019 / Revised: 26 October 2019 / Accepted: 26 October 2019 / Published: 29 October 2019
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. View Full-Text
Keywords: Hamilton’s variational principle; higher-order element; memristor; Lagrangian; Chua’s table; Euler–Lagrange equation Hamilton’s variational principle; higher-order element; memristor; Lagrangian; Chua’s table; Euler–Lagrange equation
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Biolek, Z.; Biolek, D.; Biolkova, V. Lagrangian for Circuits with Higher-Order Elements. Entropy 2019, 21, 1059.

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