Why can q(t) be something other than a spatial coordinate in the Euler-Lagrange equation?

  • #1
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:

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?
 

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  • #2
anuttarasammyak
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In Feynman Lectures on Physics 19 https://www.feynmanlectures.caltech.edu/II_19.html he started saying
“Now I want to talk about other minimum principles in physics. There are many very interesting ones.
to show non mechanical applications. It may help you.
 
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  • #3
Dfault
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Yeah, Feynman has some good examples that show that Hamilton's Principle does apply to some problems in E&M (and implicitly uses the Euler-Lagrange equation to solve it by introducing a variation, doing the difference between the varied answer and the the minimum answer, and setting the first-order portion of the difference to zero), but I haven't seen anything where he explains why it applies to E&M. You do get Lorentz forces and Poisson's equation out of it, so I can see that it does work, but how did anybody ever think to try that? That's the part that's kind of mystifying to me 😂
 
  • #4
robphy
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In Lagrangian mechanics, one talks about
generalized displacements (generalized configuration variables), generalized velocities, generalized momenta
and generalized forces.

Think (Work)Energy-per-unit-configuration-variable with units
... the rate at which work is done when one performs a particular "displacement in the configuration" of the system...

Some "generalized forces" have units:
force: Joules/meter = Newton
torque: Joules/radian =meter-Newton [as in angular displacement]
voltage: Joules/Coulomb=Volts [ as in displacing charges in a capacitor from one plate to the other]
pressure: Joules/meter^3 = Pascal [as in displacing volume]
"gravitational voltage": Joules/kg = (m/s)^2 = (m/s^2)* m ["gh" in "mgh"] [displacing mass]
surface tension : Joules/m^2 = (Newton/m)
temperature : Joules/(units of Boltzmann) = Joules/( m^2 kg s^-2 K^-1 ) = Kelvin

These
http://www.discretephysics.org/Diagrams/THE3-9.pdf (thermodynamics)
http://www.discretephysics.org/Diagrams/ANd6-4.pdf (analogies)
are from Enzo Tonti's website: http://www.discretephysics.org/en/classification-diagrams/

Possibly enlightening:
http://www.discretephysics.org/en/classification-diagrams/

His notions of classification are more subtle than what I gave above

 
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  • #5
Dfault
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Interesting! I just found prof. Tonti's book and am reading through it now. It seems like the methods he's looking at apply to a whole bunch of different branches/subjects; I'm surprised the Youtube videos don't have more views than they do. How did you find these, anyway?
 
  • #6
robphy
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