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Old Aug 19th 2017, 02:41 PM   #1
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Is "Action" a contrived concept to make the Lagrangian work?

See (attached) how the Lagrangian really was only formulated for T (kinetic energy) but by assuming conservative forces only they were able to do some hocus pocus and invent some terms (namely the time derivative of the potential) which must be zero for conservative forces so that they could make the equation "look" more symmetrical and replace the T with an L.

My question is is this not a bit disingenuous? Is "Action" a real thing or just a mathematical contrivance which only works for conservative forces (and not magnetism)?
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Old Aug 20th 2017, 08:54 AM   #2
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Originally Posted by kiwiheretic View Post
See (attached) how the Lagrangian really was only formulated for T (kinetic energy) but by assuming conservative forces only they were able to do some hocus pocus and invent some terms (namely the time derivative of the potential) which must be zero for conservative forces so that they could make the equation "look" more symmetrical and replace the T with an L.
Where are you getting this "time derivative of the potential" from? It's not in the page you posted. And its not hocus pocus in anyway and the generalized force doesn't have to always be zero since not all forces are conservative. The potential is introduced only in situations when the force is conservative, i.e. is the gradient of a potential function.


When the force is magnetic then one uses what's known as a velocity dependent potential, U, in the Lagrangian which is a more general potential. The Lagrangian is the expressed by L = T - U r(rather than L = T - V). In such cases Lagrange's equations still work.

Originally Posted by kiwiheretic View Post
My question is is this not a bit disingenuous?
Absolutely not.

Originally Posted by kiwiheretic View Post
Is "Action" a real thing or just a mathematical contrivance which only works for conservative forces (and not magnetism)?
I don't know what you mean by "real thing." Please clarify.

Action is an integral. Where does your question come from? It doesn't appear to be connected to the page you posted.

I recommend reading more about the concept of action. See
https://en.wikipedia.org/wiki/Action_(physics)

I recommend finding Classical Mechanics - 3rd Ed by Goldstein, Safko and Poole on this subject. You can download it from here: http://booksc.org/book/2059463/88e59f

That might be an earlier edition though. But it doesn't matter since the relevant chapter will be the same.

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Old Aug 20th 2017, 11:43 AM   #3
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Originally Posted by Pmb View Post
Where are you getting this "time derivative of the potential" from? It's not in the page you posted. And its not hocus pocus in anyway and the generalized force doesn't have to always be zero since not all forces are conservative. The potential is introduced only in situations when the force is conservative, i.e. is the gradient of a potential function.
I think it was this step, which may not have been made explicit in attachment, that I saw somewhere and I will see if I can reproduce:

$\displaystyle \frac{\partial T}{\partial \dot{q}_a} \; \textrm{became} \; \frac{\partial (T-V)}{\partial \dot{q}_a} \: \textrm{only because} \; \frac{\partial V}{\partial \dot{q}_a}=0$

The part I struggle with is that this step is only valid if potential is not related to generalised velocity which is not true for magnetism but, as you point out, they define the Lagrangian differently in such cases. However arbitrary choices for the Lagrangian make this makes it appear more mathematical than physical in its widest application. Or is there still well defined criteria about what the Lagrangian can or cannot be?
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Old Aug 20th 2017, 12:36 PM   #4
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Mathematical Physics has moved in in the centurie(s) since Lagrange and Hamilton.

The Lagrangian has been greatly generalised and Hamilton's formulation is just one possibility, although it is still an integral as PMB notes.
Note also that Goldstein is still the gold standard for hamiltonian mechanics.
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Old Aug 20th 2017, 12:54 PM   #5
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Even in @studiot's attachment it still say L = T - V only for conservative forces. I am still left wondering whether there is any rationale to constructing the Lagrangian for non-conservative forces other than trying to find one which is in keeping with the spirit of generalised forces and there seems like it would be a non-trivial task for non-conservative forces.
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Old Aug 20th 2017, 01:15 PM   #6
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Originally Posted by kiwiheretic View Post
Even in @studiot's attachment it still say L = T - V only for conservative forces.
When the problem at hand contains only conservative forces then the Lagrangian is L = T - V. When there are non-conservative forces as well as conservative forces then V refers to the potential for those conservative forces and Lagrange's equations are then expressed in terms of both the L = T = V and the non-conservative forces.

For example: it frequently happens that frictional forces are proportional to the velocity of a particle. In Goldstein's text that's what drag forces are referred to. In those cases the frictional forces may be derived in terms of a function F known as Rayleigh's dissipation function. In that case Lagrange's equations can be expressed in terms of both L and F.


Originally Posted by kiwiheretic View Post
I am still left wondering whether there is any rationale to constructing the Lagrangian for non-conservative forces other than trying to find one which is in keeping with the spirit of generalised forces and there seems like it would be a non-trivial task for non-conservative forces.
The benefit of Lagrange's equations is that there are times when it makes problem solving easier, i.e. more tractable. In other case there are no other ways to solve a problem. E.g. in quantum mechanics a system is represented by the Hamiltonian which is derived from the Lagrangian. In the case of a charged particle moving in a magnetic field that's the way Schrodinger's equations are formed.
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Old Aug 20th 2017, 06:00 PM   #7
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Have you considered buying copy of Schaum's Outline of Lagrangian Dynamics? It's a great book and very inexpensive. I bought myself one years back but for the live of me I can't find it.

Also, one of the most well-known and respected books in physics is The Variational Principles of Mechanics by Cornelius Lanczos.

A book that came out in 2009 is Emmy Noether's Wonderful Theorem by Dwight E. Neuenschwander. Great read which also gives a solid understanding of the history of the subject. Noether's theorem is about constants of motion in discrete systems and systems having continuous media.

Notice how that if the Lagrangian is not an explicit function of a particular variable then the time derivative of quantity becomes zero and thus that quantity does not vary with time, i.e. is conserved. Cool, huh?

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