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Consider $\vec{F}$, as the total force applied on the system, $U$ the potential energy of the existent field, and $\vec{Q}$ a non-conservative force.

We have that:

$$F_k=-\frac{\partial U}{\partial q_k}+Q_k.$$

The Lagrange's equations for this type of system are:

$$\frac{d}{dt}\frac{\partial L}{\partial \dot{q_k}}-\frac{\partial L}{\partial q_k}=Q_k.$$

I know the deduction of Lagrange's equation of a conservative system, but not for this one.

I want to deduce those equations using Hamilton's Principle. Could you show me their derivation, or advise me some links of interest?

Élio Pereira
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  • Comment to the post (v1): The general derivation of Lagrange eqs. from Newton's laws is e.g. explained in chapter 1 of Goldstein's Classical Mechanics. The case where all the generalized forces has generalized (possibly velocity-dependent) potentials is discussed in e.g this Phys.SE post. – Qmechanic Nov 14 '15 at 15:02
  • I forgot to mention one thing: I pretend to deduce those equations using Hamilton's Principle. – Élio Pereira Nov 14 '15 at 15:18
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    Comment to the post (v3): Note that it is not possible to apply the stationary action principle (aka. Hamilton's principle) unless all generalized forces have generalized potentials. Related: http://physics.stackexchange.com/q/20298/2451 and links therein. – Qmechanic Nov 14 '15 at 15:20
  • Couldn't we use a Lagrangian $L=T-U$ where, $U$ is the field potential, and $\tau=\int_{\vec{q}{k1}}^{\vec{q}{k2}}(\vec{Q_k}.d\vec{q})$, where $\tau$ is the work done by the non-conservative force? – Élio Pereira Nov 14 '15 at 15:25
  • How is $\tau$ supposed to enter Hamilton's principle? – Qmechanic Nov 23 '15 at 15:26
  • @Qmechanic I was thinking about something like $L=T-U-\tau$. But I know that it is wrong. Forget what I said, it didn't make sense. – Élio Pereira Nov 23 '15 at 15:31

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