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Dfault
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Hello all, so I’ve been reading Jennifer Coopersmith’s The Lazy Universe: An Introduction to the Principle of Least Action, and on page 72 it says:
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?
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?
