Figure 1.
We consider the simple introductory problem of a mass, m, on a spring, with stiffness constant of k. The kinetic energy stored by the inertia of the mass is denoted by . The elastic potential energy stored in the spring is denoted by
. The independent coordinate is denoted by the position, x. The Lagrangian function is traditionally written as
, which can be written explicitly as
.
Table 1.
A homomorphic mapping due to Karnopp et al.
Figure 2.
An LC electromagnetic harmonic circuit.
We consider a capacitor, C, in parallel with an inductor, L. We consider the idealised case where there is no dissipative loss, or resistance R. The Lagrangian function can be written as , where the magnetic energy stored by the field of the inductor, denoted by
, and the electrical potential energy stored in the capacitor is denoted by
, and q = Cv is the coordinate, which we interpret as the electrical charge that is transferred through the circuit.
Table 2.
Table of Lagrangian terms, in terms of current.
Table 3.
Table of Lagrangian terms, in terms of voltage, v.
Table 4.
Table of mechanical Lagrangian terms.
Figure 3.
A damped mechanical harmonic oscillator.
By introducing non-conservative, or dissipative, elements into a system we need to generalise our concept of potential. The Lagrangian for this system can be written as . The additional term differs from the terms for the un-damped system in two key ways: the term is complex has an imaginary phase, of +j, and there is a fractional derivative of the coordinate, x(1/2).
Figure 4.
The voltage source, vS places a constraint on the voltages across the two resistors, v1 and v2. Kirchhoff's voltage law implies that Ψ = v1 + v2 − vS = 0. We can regard the function Ψ as a function of constraint. Kirchhoff used a function that is equivalent to the dissipated power as a Lagrangian function. In modern notation we can write . In order to ensure compatibility with other existing Lagrangian functions, we need to apply a transformation to obtain a new Lagrangian function of
. This Lagrangian function has been multiplied by a scalar of −j/2 and the order of differentiation has been reduced from v(0) to v(−1/2), which is equivalent to a half-integration of the Lagrangian function, or a full integration of the resulting Euler Lagrange equation.
Figure 5.
A parallel RLC circuit, with source.
We can use the established rules to write the Lagrangian function for the circuit as: . We also use Kirchhoff's current law to impose a constraint function of Ψ = iR + iL + iC − iS = 0, and we use the principle of the Lagrange multiplier to obtain an ordinary differential equation to describe the dynamical behaviour of v(t).
Figure 6.
An electrical ladder-filter circuit.
In this circuit, we make a careful choice of generalised coordinates, which allows us to avoid the explicit use of functions of constraint. We can use the established rules to write down the Lagrangian function for the circuit as shown in Equation 30. We apply the rules for the Euler-Lagrange equation to this Lagrangian function to obtain a pair of ordinary differential equations that describe the dynamics of this circuit.
Figure 7.
Physical layout of the D'Arsonval galvanometer.
We model the essential features of the D'Arsonval meter as: the rotational moment of inertial of the coil J, the torsional spring constant, κ, the torsional damping constant, χ and the maximum magnetic-flux linked by the coil, Φ0. We use a linear model for the stored energy in the coil, . The Lagrangian function can be written in terms of these fundamental parameters. (Adapted from the Wikimedia commons.)
Figure 8.
An equivalent electro-mechanical circuit for a D'Arsonval galvanometer.
The Lagrangian function for this electromechanical system is written in Equation 37. The current, i, comes from an ideal current source, so it is essentially a constraint, rather than an independent coordinate. The last term in this Lagrangian function determines the coupling between the electrical and mechanical aspects of this system.