Lagrange Densities: Intuitive Understanding

[Higgsono]

I'm a little confused when we transition from the Lagrangian of a point particle to consider Lagrangian of objects with dimension bigger then zero. For instance, the Lagrangian of a string is the sum of the Lagrangians for each infinitesimal part of the object. which means we are integrating over a Lagrange density to get the full Lagrangian. Intuitively I'm not sure why this works. But I think that it's analogous to the situation when we sum up all the forces on the object to determine the total force on the object.

In the case of solving Newton's equation of motion, the superposition principle for the forces seems intuitive, but it is not obvious that it carries over so that the superposition principle applies to the action as well.

 

[Orodruin]

The Lagrangian is kinetic - potential energy. You are just finding those quantities for a string. It should be quite clear that the kinetic energy is the sum of the kinetic energies of each part of the string and that the potential of the string is $S(\ell-\ell_0)$, where $S$ is the tension in the string.