# Lagrangian formalism application on a particle falling system with air resistance

I have this problem, with a first-step resolution:

Obtain the equation of motion for a particle falling vertically under the influence of gravity when frictional forces obtainable from a dissipation function $$\frac12kv^2$$ are present. Integrate the equation to obtain the velocity as a function of time and show that the maximum possible velocity for a fall from rest is $$v+mg/k$$.

Work in one dimension, and use the most simple Lagrangian possible: $$L = \frac 12 m \dot z^2 - mgz$$ With dissipation function: $$F=\frac 12 k \dot z^2$$ The lagrangian formulation is now: $$\frac{d}{dt} \frac{\partial L}{\partial \dot z} - \frac{\partial L}{\partial z} + \frac{\partial F}{\partial \dot z} = 0$$

So, I just don't know why they put the term $$\frac{\partial F}{\partial \dot{z}}$$ in Lagrange's equations. Why? I know that the Rayleigh dissipation function isn't a conservative force, but I don't know why the partial derivation. For holonomic constraints we need to partially derivate the function of constraint $$f=0$$ in order to $$q$$, the generalized coordinate: $$\frac{\partial f}{\partial{q}}$$. And we introduce it in Lagrange's equation multiplied by Lagrange's multiplier $$\lambda$$, on RHS.

But we have here some kind of constraint with a velocity $$\dot{z}$$ dependence. That's why we need to put the term $$\frac{\partial F}{\partial \dot{z}}$$ in Lagrange's equations? But the term isn't null and they didn't had the Lagrange multiplier, so is it true that it isn't relationated to the constraints formalism?

• – Qmechanic Oct 31 '19 at 13:55

The dissipation function is a bit like a potential. If your particle were only influenced by the gravitational force, you would say that $L=T-U$ where $U$ is the gravitational potential and the Euler-Lagrange equations would be the normal ones. But as you have a friction force, the Euler-Lagrange equations become: $$\frac{d}{dt}\frac{\partial L}{\partial \dot{z}} - \frac{\partial L}{\partial z}=Q_j$$
where $Q_j$ is the generalized force and can be shown by its definition from the D'Alembert's principle that $Q_j=-\frac{ \partial F}{ \partial \dot{z}}$ in this particular case. Indeed, if you have a dissipation function, that's always the form of your equations I think.