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I am working in a non-conservative system. Would it make a difference if I

  1. Formulate the Lagrange Equation with an additional term on the right hand side of the equation to account for the Rayleigh Dissipation Function, and derive the equation of motion $$ \frac{d}{dt}\left( \frac{\partial L}{\partial \dot \theta_k} \right) - \frac{\partial L}{\partial \theta_k} = - \frac{\partial R}{\partial \dot \theta_k} $$ versus

  2. treating the system as conservative and formulate the standard Euler-Lagrange Equation =0, derive the equation of motion, then add in a dissipation term to the equation of motion.

Qmechanic
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Ee Kin Chan
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1 Answers1

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I expect a difference.

The relation between energies as expressed by the Euler-Lagrange equation has assured validity only when all the potentials involved are potentials of conservative forces.

I assume that when you write: 'derive the equation of motion' you mean that you execute the differentiations specified by the Euler-Lagrange equation

Once those differentiations have been executed the result is in terms of generalized force and generalized acceleration. If you are going to add a frictional term, that is the expression to add it to.

(The word 'dissipation' is associated with the concept of energy. The equation in terms of (generalized) force and (generalized) acceleration is not in terms of energy, so in that context it seems better to me to use the concept of friction rather than the word 'dissipation'.)


I guess it may be tempting to port the frictional term back to the form of the Euler-Lagrange equation. I recommend against that, if only for this purpose: the validity of the Euler-Lagrange equation is assured only when all forces involved are conservative forces. If you limit yourself to use the Euler-Lagrange equation with conservative force potentials only then you have that guaranteed validity.

If you do add a term corresponding to a non-conservative force then a valid result is still possible, but if it happens to yield a valid result it's a fluke.

Cleonis
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