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Dfault

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*The Lazy Universe: An Introduction to the Principle of Least Action,*and on page 72 it says:

Finally, we can free ourselves from thinking of ‘motions’ as just translations or rotations, and consider also changes in capacitance, surface tension, magnetic field, phase of a wave, strain in a beam, pressure within a fluid, and so on. In fact, any variable that can be quantified, is expressible as a function, and characterizes the physical system, can serve as a coordinate of that system.

If I understand it right, she’s saying that in our Euler-Lagrange equation ## \frac {\partial L} {\partial q} - \frac {d} {dt} \frac {\partial L} {\partial \dot q} = 0## , q(t) doesn’t have to be a

*position*coordinate of an object at all: it can represent

*any*physically-measurable time-varying quantity of the system (provided we can write it as a function). I’m realizing that I would have a hard time explaining to someone else

*why*that’s true, which is making me wonder whether I really understand it.

I’ve worked through where Hamilton’s Principle comes from, so I suppose I understand that nature will try to minimize the time-integral of the kinetic energy minus the potential for some given physical system; and I’ve worked through where the Euler-Lagrange Equation comes from, how it can be used to find the solution to a minimization problem for an integral like the one in Hamilton’s Principle, and how it’s pretty flexible with regards to which set of coordinates you use, since as long as I can write

*my*coordinates as some function of

*your*coordinates (and possibly also time), my choice of coordinates will

*also*satisfy a set of Euler-Lagrange equations. Good so far.

I’ve also seen for myself that if our choice for q(t) is, for example, the

*charge in a wire*instead of some spatial coordinate

*,*we can throw the Euler-Lagrange equation at it and get the correct solution to various circuits problems (Claude Gignoux’s

*Solved Problems in Lagrangian and Hamiltonian Mechanics*). I’ve even seen how you can

*mix and match*various q(t)’s so that

*one*q(t) represents a position coordinate, while

*another*q(t) represents the charge in a wire, for the

*same*physical system (like a circuit with a capacitor whose top plate is attached to an oscillating spring; see Dare Wells’

*Schaum’s Outline of Lagrangian Dynamics*, for example). So I can see that you

*do*get the right answers when q(t) represents something other than a position coordinate, but I can’t understand

*why*that’s the case. Is there some general

*proof*of this? How did people know to apply the Euler-Lagrange equation to systems where q(t) represented something

*other than*a spatial coordinate?