Variational analysis of electrical networks

Jones, Dominic; Evans, F.J.

doi:10.1016/0016-0032(73)90249-4

Cited by 10 publications

(3 citation statements)

References 5 publications

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“…Also, the formulation (4.68) of an implicit Hamiltonian system satisfying the closedness condition suggests a connection with variational principles via the first-order condition of Pontryagin's maximum principle. In the case of electrical circuits, where the interconnections are defined by Kirchhoff 's laws and the closedness conditions are trivially satisfied (see Example 3.1), some important work concerning a variational formulation of Kirchhoff 's laws and the resulting variational characterization of the overall circuit has been done (see, e.g., [JE,M1]), and it seems of interest to extend these ideas to the general situation considered in Proposition 5.1.…”

Section: Closedness Of Generalizedmentioning

confidence: 99%

On Representations and Integrability of Mathematical Structures in Energy-Conserving Physical Systems

Dalsmo¹,

Schaft²

1998

SIAM J. Control Optim.

242

324

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In the present paper we elaborate on the underlying Hamiltonian structure of interconnected energy-conserving physical systems. It is shown that a power-conserving interconnection of port-controlled generalized Hamiltonian systems leads to an implicit generalized Hamiltonian system, and a power-conserving partial interconnection to an implicit port-controlled Hamiltonian system. The crucial concept is the notion of a (generalized) Dirac structure, defined on the space of energy-variables or on the product of the space of energy-variables and the space of flow-variables in the port-controlled case. Three natural representations of generalized Dirac structures are treated. Necessary and sufficient conditions for closedness (or integrability) of Dirac structures in all three representations are obtained. The theory is applied to implicit port-controlled generalized Hamiltonian systems, and it is shown that the closedness condition for the Dirac structure leads to strong conditions on the input vector fields.

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Section: Closedness Of Generalizedmentioning

confidence: 99%

On Representations and Integrability of Mathematical Structures in Energy-Conserving Physical Systems

Dalsmo¹,

Schaft²

1998

SIAM J. Control Optim.

242

324

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“…Even the generally basic variational approach is acquired by circuit theory (e.g. [60]) from physics.…”

Section: E6 Two Different Methodologiesmentioning

confidence: 99%

Nonlinear systems: between a law and a definition

Gluskin

1997

Rep. Prog. Phys.

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A discussion relevant to the logical foundation of nonlinear dynamic systems theory and statistical theory is presented. The initial point of view is associated with the question of whether or not an equation can represent a physical law. It is argued that the answer depends on which of the associated concepts/processes-that of direct physical measurement or that of counting-is more important in the theory of the relevant system. Without the inclusion of the concept of counting in the set of the most basic physical concepts, consideration of the nonlinear dynamic (or statistical) systems which may have an 'iterative mathematical nature' is seen to be logically unsatisfactory. The ergodic hypothesis is considered, from the points of view of direct physical measurement and counting. It is shown that for the foundation of statistical theory we must consider 'measurement in principle' even when we cannot, by reason of the complexity of the system, actually perform this measurement. The discussion inevitably involves many concepts, which reflects the essence of the very complicated situation under discussion. This gives the work, which is intended, first of all, for a reader with a physics background, a logical-critical inclination. The latter is justified also by the opinion here that the solution to the fundamental problems of 'nonlinear physics' cannot come from the existing physics. It is argued that the topic of intuitionistic logic has to be considered for axiomization of the nonlinear dynamic theory. Though the work is not a review of the results of the existing dynamic or statistical theories, the discussion of the logical problems in the foundation of these theories is relatively much more complete than it usually is in such reviews. The classical results of analytical mechanics are finally considered, including some nontrivial applications.

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“…Its total energy is given by 54with p,, p1 denoting the momenta of mass IJI, and w?, respectively, and s,' the distance between them and k denoting the spring constant. The Poisson structure is again given by (23). resulting in the equations of motion (compare with (22)) : Notice that r is the total angular momentum, which indeed is a conserved quantity.…”

Section: Let M Be An M-dimensionalmentioning

confidence: 99%

An intrinsic hamiltonian formulation of network dynamics: non-standard poisson structures and gyrators

Maschke

Schaft

Breedveld

1992

Journal of the Franklin Institute

148

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The aim of this paper is to provide an intrinsic Hamiltonian jormulation of the equations of motion ofnetwork models of non-resistive physical systems. A recently developed extension of the classical Hamiltonian equations of motion considers systems with state space given by Poisson manifolds endowed with degenerate Poisson structures, examples of which naturally appear in the reduction of' systems with symmetry. The link with network representations of non-resistive physical systems is established using the generalized bond graph formalism which has the essential feature of symmetrizing all the energetic network elements into a single class and introducing a coupling unit gyrator. The relation between the Hamiltonian formalism and network dynamics is then investtgated throqh the representation of the invariants of the system, either captured in the degeneracy of the Poisson structure or in the topological constraints at the ports of the gyrative type network structure. This provides a Hamiltonian formulation of dimension equal to the order of the physical system, in particular, for odd dimensional systems. A striking example is the direct Hamiltonian formulation of electrical LC networks.

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Variational analysis of electrical networks

Cited by 10 publications

References 5 publications

On Representations and Integrability of Mathematical Structures in Energy-Conserving Physical Systems

On Representations and Integrability of Mathematical Structures in Energy-Conserving Physical Systems

Nonlinear systems: between a law and a definition

An intrinsic hamiltonian formulation of network dynamics: non-standard poisson structures and gyrators

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