My first instinct would be to use the force

$$\vec{F} =- \alpha \vec{v}$$

and therefore

$$V(\vec{r}) = \alpha \int_C \vec{v}\cdot d\vec{s} = \alpha \int_C \vec{v}\cdot \vec{v} dt = \alpha \int_C g_{ab} \frac{dx^a}{dt} \frac{dx^b}{dt} dt$$

(where $C$ is the projectile's path) and go from there. Would this work?


Since velocity dependent forces are not conservative, you cannot write them as the gradient of a potential function - ie. you cannot write a potential function as the integral of your non-conservative (dissipative) force. However, you can introduce a dissipative term into the equations and I believe it will work out from there. See, for example, http://www.phys.uri.edu/~gerhard/PHY520/mln9.pdf

or other google links for lagrangian with friction


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