Exploring the Flexibility of Coordinates in Euler-Lagrange Equations

In summary, according to The Lazy Universe, Jennifer Coopersmith says that any variable that can be quantified, can serve as a coordinate of a system. This is useful because it means that Hamilton's Principle can be applied to problems that would otherwise be difficult to solve. Additionally, Enzo Tonti's book provides a more detailed classification of variables that are related to forces.
  • #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
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
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
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
robphy said:
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?
 
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Related to Exploring the Flexibility of Coordinates in Euler-Lagrange Equations

1. What are Euler-Lagrange equations?

Euler-Lagrange equations are a set of equations used in classical mechanics to describe the motion of a system. They are based on the principle of least action, which states that the path taken by a system between two points in time is the one that minimizes the action, a mathematical quantity that represents the energy of the system.

2. How do Euler-Lagrange equations relate to coordinates?

Euler-Lagrange equations involve the use of coordinates to describe the position and motion of a system. These coordinates can be any set of variables that define the state of the system, such as position, velocity, and time. The equations take into account the changes in these coordinates over time to determine the path of the system.

3. What is the flexibility of coordinates in Euler-Lagrange equations?

The flexibility of coordinates in Euler-Lagrange equations refers to the fact that the equations can be solved using different sets of coordinates. This allows for a more general and flexible approach to solving problems in classical mechanics, as different coordinate systems may be more suitable for different types of systems.

4. How can the flexibility of coordinates be explored in Euler-Lagrange equations?

The flexibility of coordinates in Euler-Lagrange equations can be explored by using different coordinate systems to solve problems. This can involve transforming the equations into different coordinate systems, such as polar coordinates or spherical coordinates, and seeing how the solutions differ.

5. What are some applications of exploring the flexibility of coordinates in Euler-Lagrange equations?

The flexibility of coordinates in Euler-Lagrange equations has many practical applications in physics and engineering. It can be used to solve problems in mechanics, electromagnetics, and quantum mechanics, among others. It also allows for a deeper understanding of the underlying principles and symmetries in physical systems.

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