# On variational principles for dissipative networks and systems

1976

## Publisher

Virginia Polytechnic Institute and State University

## Abstract

Dynamical equations for dissipative systems are in general not derivable from a Lagrangian, but variational principles can nevertheless be formulated using auxiliary variables or a multiplying factor. Both methods are examined, generalized, and applied to some dissipative systems.

Composite variational principles employ auxiliary variables in addition to the original ones, and achieve stationarity of an action functional in the enlarged space of both sets of variables. The original version of the method, known as the "image" method, was based on a physical interpretation, which is shown to be erroneous. The method is given, instead, a mathematical justification.

A composite variational formalism is developed for linear time-varying systems, and applied to linear reciprocal networks with bilateral elements. Lagrangians and Hamiltonians are formulated for various second-order network models: loop, nodal, node-pair, etc.

Variational formulation for single-branch network models is recognized as an inverse variational problem with constraints. A system admits a variational principle if the constraints are ideal. The network constraints are demonstrated to be ideal, and Lagrangian multipliers are interpreted as node voltages in the impedance models, or loop currents in the admittance models.

In the time-varying case the image network contains negative resistors - R, accompanied by the series capacitors of elastance S = -R (in the impedance models), or the parallel inductors of inverse inductance Γ = -G (in the admittance models).

The network state model presents an obstacle to Hamiltonian formulation because of generally unequal number of tree-capacitors and link-inductors. This problem appears in conservative as well as in dissipative networks. The solution for the time-invariant networks is found by considering a "complementary" or an "adjoint" network together with the given network. The "complementary" and the "adjoint" networks differ from the "image" network only in the sign convention employed, and the choice of generalized coordinates and momenta. Combining the state model for the original network with the "co-state" model for the "complementary" or the "adjoint" network, Hamiltonian models are formulated for the combined network. Networks with all-capacitor loops,and all-inductor cut-sets, require an additional transformation of variables.

Hamilton's principle for the combined state model has the form of "canonical integral", in which the momenta are independent variables, not related to generalized velocities.

The multiplier method is applied to a linear harmonic oscillator. The time-varying Hamiltonian model is reduced by a canonical transformation to a time-invariant model. Dissipation now enters Lagrangian and Hamiltonian equations through a modified restoring constant. For a coupled linear system a transformation of variables is applied in order to eliminate dissipative terms; variational principle exists in four special cases.

The multiplier method is extended to a class of nonlinear time- invariant circuits. An important special case contains linear and quadratic dissipative terms.

The multiplier method is applied to the lossy transmission line with constant parameters, and a Hamiltonian principle is derived for it. Hamilton's formalism is improved by a canonical transformation. The relation of Hamiltonian to energy is investigated.