Lagrangian Involving a Integral
This is my first post here, so do let me know if I am asking anything or in any manner unproperly!
I am trying to construct a Lagrangian for a dynamic Euler-Bernoulli beam (with one end cramped) with a large deflection. The linearize case is well recorded by Wikipedia: https://en.wikipedia.org/wiki/Euler%E2%80%93Bernoulli_beam_theory#Dynamic_beam_equation
However, with large deflection, I need to use the arclength together with the local deflection angle to represent the status of the beam, so the cartesian coordinates of the beam located $s$ from the cramped end can be represented as:
$x(s,t)=\int_0^scos(\theta(s',t))ds,\space y(s,t)=\int_0^ssin(\theta(s',t))ds \space \space \space [1]$
Thus, at a specific location $s$, with linear density $\mu$ and external load $q$ (which is perpendicular to the beam locally), the Lagrangian can be written as:
$\mathcal{L}=\frac{1}{2}\mu \dot{x}^2+\frac{1}{2}\mu \dot{y}^2-\frac{1}{2}EI(\frac{\partial\theta(s,t)}{\partial s})^2+q(s,t)x(s,t)sin(\theta(s,t)) +q(s,t)y(s,t)cos(\theta(s,t))\space \space \space [2]$
Here the term $\frac{1}{2}EI(\frac{\partial\theta(s,t)}{\partial s})^2$ is associated with the energy stored in the bending moment. From here if we plug in $[1]$ to $[2]$, we obtain a Lagrangian containing an integral:
$\mathcal{L}=\frac{1}{2}\mu(\int_0^s sin(\theta(s',t))\dot{\theta}(s',t)ds)^2+\frac{1}{2}\mu(\int_0^s cos(\theta(s',t))\dot{\theta}(s',t)ds)^2-\frac{1}{2}EI(\frac{\partial\theta(s,t)}{\partial s})^2+q(s,t)sin(\theta(s,t))\int_0^scos(\theta(s',t))ds+q(s,t)cos(\theta(s,t))\int_0^s sin(\theta(s',t))ds$
Obviously, this Lagrangian contains some integrals that we don't see in those simpler cases we find in the textbook. Does anyone have an idea on how to deal with this sort of Lagrangian? A reference to a similar problem would definitely help as well.
Many thanks! 101