Euler-Lagrange equation with a damping term

Question

We know that there is a function VariationalMethods`EulerEquations readily handling the Euler-Lagrange equation of the "standard" form

$$ \frac{\partial L}{\partial q} - \frac{\mathrm d}{\mathrm d t}\left(\frac{\partial L}{\partial \dot q}\right) = 0. $$

But today, it suddenly occurs to me that problems [e.g., see Eqs. (5) & (6) in this note] with a damping term are not included, are they?

My question is, is there any hidden magic in VariationalMethods`EulerEquations that makes it still capable of dealing with equations like

$$ \frac{\partial L}{\partial q} - \frac{\mathrm d}{\mathrm d t}\left(\frac{\partial L}{\partial \dot q}\right) - \frac{\partial F}{\partial \dot q} = 0, \qquad F = \frac{b}{2}\dot q^2, $$

or one has to return to manual establishment of equations of motion using functions like D or VariationalMethods`VariationalD?

Answer 1

Yes, it is possible. First of all I recommend you to read these two papers:

1. J. Guerrero, F. F. López-Ruiz, V. Aldaya, and F. Cossio A round trip from Caldirola to Bateman systems, J. Phys.: Conf. Ser. 284, 012062 (2011)
2. H. Dekker, Classical and quantum mechanics of the damped harmonic oscillator, Phys. Rep. 80, 1-110 (1981)

There are basically two ways to deal with dissipation (damping) in the Lagrangian formalism of classical mechanics:

1. The Bateman method,
2. The Caldirola-Kanai method.

They can also be used in order to quantize the classical equations of motion. As an example, consider a harmonic oscillator with frequency $ \Omega $ and damping $ \lambda $, where we set the mass to be equal to 1.

Needs["VariationalMethods`"]

BatemanL = x'[t] y'[t] - Ω^2 x[t] y[t] + λ (x[t] y'[t] - x'[t] y[t]);

EulerEquations[BatemanL, {x[t], y[t]}, t]
(*{-Ω^2 y[t] + 2 λ y′[t] - y′′[t] == 0, -Ω^2 x[t] - 2 λ x′[t] - x′′[t] == 0*)

CaldirolaKanaiL = Exp[λ t] (1/2 x'[t]^2 - 1/2 Ω^2 x[t]^2)

EulerEquations[CaldirolaKanaiL, x[t], t]
(*-E^(t λ) (Ω^2 x[t] + λ x′[t] + x′′[t]) == 0*)

The idea of the Bateman's method is to introduce an auxiliary coupled system $ y(t) $ that evolves in negative time-direction (notice a sign difference in the equation of motion for $ y(t) $ as compared to $ x(t) $), so that in total the energy is conserved.

The idea of Caldirola-Kanai is in introducing the time-dependence into the Lagrangian explicitly. It is worth noting that these two methods can be generalized to more complicated scenarios.