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Relating Lagrangian and Hamiltonian descriptions of electrical circuits

Project members: J. Clemente Gallardo, J.M.A. Scherpen, D. Jeltsema

In the last years, an evident interest for the Lagrangian and Hamiltonian description of electrical circuits has arisen in the literature. A recent Lagrangian description [29,30] leads to a successful picture of RLC circuit dynamics and provides a step-by-step construction for the description of the components, the definition of the Lagrangian, and the corresponding Euler-Lagrange dynamics. Kirchhoff's current law defines a set of holonomic constraints for the corresponding Lagrangian system, while the corresponding voltage law defines the Euler-Lagrange equations for the system. Regarding the Hamiltonian description of the dynamics of electrical circuits, a recent and successful approach is based on the concept of Dirac structures and port-controlled Hamiltonian systems. This approach also provides a suitable description of the dynamics of the system.

It seems quite natural to compare both approaches and to try to relate the solutions of both methods for electrical circuits. Since dissipative elements and sources can be viewed as external elements, we only consider electrical LC circuits here. The formulation of both frameworks is done in $R^{n}$ and hence the canonical procedure would suggest to use the Legendre transform to go from dynamics given by the Lagrangian formalism into dynamics given by the Hamiltonian formalism, and vice versa. The problem in this case is that the Lagrangian formalism proposed in [29,30] yields a singular Lagrangian description, which makes the Legendre transform ill-defined and thus no straightforward Hamiltonian formulation can be related. We complement the original Lagrangian picture proposed in [29,30] with a procedure that transforms the singular Lagrangian system into a regular Lagrangian system. Then the Lagrangian system can be related with a Hamiltonian system by using a well defined Legendre transform. The main new ingredient of the approach is the use of Lie algebroids in the description. A Lie algebroid is a geometrical object which generalizes the concept of tangent bundles (which is the natural framework of usual Lagrangian mechanics) such that a Lagrangian formulation on them is still possible. Essentially, we just need one of the simplest examples of the Lie algebroid, namely an integrable subbundle of a tangent bundle, which in the case of electrical LC circuits is even a vector space. For the case of networks without switches, this approach is equivalent to use the integrated version of Kirchhoff current law. This implies the use of the condition of charge conservation, to define a regular Lagrangian description by using only the inductances of the system.

The future research includes the extension of the new framework to more general circuits, including switched networks, and to merge this approach with the extension of Brayton-Moser equations.

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