From Coordinates to Fields: the Lagrangian Density
A field has a value at every point, so the sum over coordinates becomes an integral of a Lagrangian density and the equations of motion become partial differential equations. The vibrating string is the case where the step can be checked against an answer already known.
What a learner can do afterwards
- Writes a Lagrangian density for a continuous system and derives its field equation
- Explains what replaces the index labelling a coordinate when the system becomes a field
- Recovers the wave equation from a field Lagrangian
- Finds arrival time of a spherical wave front from distance divided by wave speed
1 · Read
You used the action to find what a symmetry conserves. That stretches to a field, which assigns a value to every point in space, so there are infinitely many parts. The label that numbered coordinates one after another is replaced by the position itself: where you summed over coordinates, you now integrate a Lagrangian density over space. The equations of motion turn into partial differential equations in space and time, because the field can bend from point to point and change with time. In three dimensions a point push spreads as a sphere at uniform speed c, so time is distance over speed.
The stretched string checks against an answer you know. Its density leads to the wave equation, and d'Alembert's solution says u of x and t equals f of x minus c t plus g of x plus c t. That means any motion is two shapes sliding opposite at speed c, added. Released from rest with shape f, it is one half of f of x minus c t plus f of x plus c t: with c equals 3 and f of x equals x squared, u of 0 and 1 is half of 9 plus 9, which is 9.
A string of length L fixed at both ends only admits sine shapes that vanish at the ends. Those are the functions sin of n pi x over L with whole numbers n, since each equals zero at both x equals 0 and x equals L. To reuse the infinite string method, extend the shape outside as an odd, periodic function and then apply d'Alembert's formula.
Whenever you derive a field equation, test it on the one case whose answer you trust. For the string that means recovering the wave equation and its traveling shapes. If your derivation gives standing still shapes that never travel, the density or the variation step went wrong somewhere.
Replace the coordinate list with a field, integrate the density, and check the string gives traveling waves.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.