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 stepbystep
construction for the description of the components, the definition of
the Lagrangian, and the corresponding EulerLagrange dynamics.
Kirchhoff's current law defines a set of holonomic constraints for the
corresponding Lagrangian system, while the corresponding voltage law
defines the EulerLagrange 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 portcontrolled 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 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 illdefined
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 BraytonMoser equations.
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