When discussing strings, it is useful to talk about tension. Consider a small piece of string from to , with

In the case of transverse oscillations, the longitudinal force is negligible and so it is only neccessary to compute the net vertical force. If we let be the tension, we find that for a small amount of vertical force

Since the length of the string is and if we denote the mass per unit length by , the total mass is . According to Newton’s second law,

This is a just wave equation describing a wave with transverse velocity .

To compute the action for a nonrelativistic string, we must find the Lagrangian, . In this case, the kinetic energy, is equal to the sum of the kinetic energies of all the infinitesimal pieces of the string:

The potential energy, can be thought of as the spring potential when a piece of string with ends at and is stretched by in the x-direction and in the y-direction. The change in length can be computed as

so our potential becomes

Our Lagrangian is therefore

and so our action becomes

By varying by and using the Euler-Lagrange equation,

we are able to get the equation of motion:

where

Close examination reveals this to be the wave equation.

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Sean, do you want to comment on the identity of and . If memory serves these are momentum densities, right?

That’s correct, these are the conjugate momenta. I suppose the notation should actually read

where

I’d like to know more about those in general. In the case of a system with finite degrees of freedom we can use the Lagrangian to create a map (Legendre transform) between the tangent bundle and cotangent bundle. I’ve been interested in what happens if there are an infinite number of freedoms. I’d really like to know the connection between classical differential geometry and the Hilbert spaces where such functions (with infinite number of freedoms) live.

and this is where i get lost having no differential geometry or functional analysis background. There’s obviously more behind the scenes.