# How to use Euler-Lagrange equation to model a linear particle?

Suppose I have a particle with mass $$m$$ situated on a frictionless line which we can model using $$\mathbb{R}$$. Suppose that we supply/push the particle $$m$$ with a force $$f(t)$$. Let the position of this mass be denoted as $$p$$. The particle moves horizontally along the line.

I wish to derive the equation $$m \ddot p(t) = f(t)$$ using Lagrangian mechanics.

To do this, I need to identify the kinetic and potential energy, and then substitute the Lagrangian into the Euler-Lagrange equation, which are from these slides on page 3: https://publish.illinois.edu/ece470-intro-robotics/files/2021/10/ECE470FA21Lec16.pdf

$$f= \dfrac{d}{dt} \dfrac{\partial L}{\partial \dot q}- \dfrac{\partial L}{\partial q}$$ where $$f$$ is the external force, $$L$$ is the Lagrangian, and $$q$$ is the generalized variable.

The kinetic energy is simple: $$T = \dfrac{1}{2}m \dot p^2$$. The external force is $$f(t)$$, which appears at the left-hand-side of my equation. However, I cannot figure out what is the potential energy $$P$$ in order to form $$L$$.

• The particle moves horizontally along the line. Jan 26 at 16:23

well it depends on whether that force is conservative or not. However, in both cases the result is the same. I will change your notation and use $$q$$ as coordinate.
a) Conservative force: In this case $$f := -\partial P(q) / \partial q$$. Thus applying the Euler-Lagrange equation to $$L=T-P$$, you obtain the result.
b) Non-conservative force: In this case $$f := d(\partial L / \partial \dot{q}) /d{t} - \partial L / \partial q$$ and $$P:=0$$ which implies $$L=T$$.
Note that the only different thing is how you interpret the origin of $$f$$ but the computations are the same. In general, a generalized force $$f(q,\dot{q},t)$$ can be written as $$f(q,\dot{q},t) := d(\partial L / \partial \dot{q}) /d{t} - \partial L / \partial q,$$ which of course, includes conservative forces.
• Conservative forces are those who preserve energy and non-conservative allow for things like dissipation. An example of conservative force is the elastic force caused by a spring on a punctual mass. In this case $P(q)=q^2/2$ and the force is just $f=-q$. A standard example of non-conservative force is that produced by viscous friction. This force can be modelled as $f=-\dot{q}$ and cannot be derived from a potential $P(q)$. More generally, conservative forces can be derived from a generalized potential $P(q,\dot{q},t)$ applying $f=d(\partial P /\partial \dot{q}) /dt- \partial P / \partial q$. Jan 26 at 16:55