WHY does the lagrangian method of classical mechanics work?

Darien

Platinum Member
Feb 27, 2002
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L = T - V, and after plugging it in to

((d/dt)(d/dv) - (d/dx))L = 0

you generate the equations of motion.

I've been using this all semester, and I never knew "why" it works. I just knew it works, and worked damn well :beer::D



What is it about knowing kinetic and potential energy that leads one to believe you can understand motion?
 

LurchFrinky

Senior member
Nov 12, 2003
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I'm having a tough time following you.

You've just gone through a semester of Lagrangian equations, but you can't understand the relationship between kinetic energy and motion?

The Lagrangian equation is just stating that the sum of the total energy of the object doesn't change. We often call this phenomenom "conservation of energy." The concept is simple enough, but the mathematics can become daunting in higher DOF systems. If you feel like it, you can add in additional terms to account for various sources of friction or anything else which might affect the total energy of the system.

LF
 

ZeroNine8

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Oct 16, 2003
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Basically you start off writing an equation for the total energy of the system, kinetic and potential, and saying that the change in this total energy is equal to the work done on (or by) the system. By taking the partial derivatives of the equations in the prescribed method, you end up with a model of system behavior under changes (differentiation yields change in one coordinate w.r.t. another) in various system coordinates. Eventually, this gets into generalized coordinates and other methods that are extremely powerful modeling tools of dynamic systems.
 

Darien

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Feb 27, 2002
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that's HOW it works.



I want to know WHY understanding energy gives such an insight, so the answer i'm looking for is probably more philosophical than mathematical...
 
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rjain

Golden Member
May 1, 2003
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You're asking us why energy is conserved...?

Because God said so.

Why is gravity attractive? Why are there two types of electric charge?
 

ZeroNine8

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Oct 16, 2003
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If you want a why beyond the law of conservation of energy, check with your favorite local supreme being.
 

sparkyclarky

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May 3, 2002
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Originally posted by: OrganizedChaos
because obviously god spontaneously created himself

Coming from somebody in highly technical, I'm sure you can grasp the concept of the infinite. Our best theory for the universe is incredibly vague at the moment. We simply don't have any observable evidence from before 300,000 years after the Big Bang. Supposedly this massive explosion happens, and matter is essentially created from nothing. I personally find it much easier to accept that the universe was created by a supreme being which has existed eternally (and therefore out of our concept of time) and does not contain the terms of its own negation within it. But anyways, I think a much better means of explanation is found within the first 17 verses of John's Gospel. Absolutely fascinating reading, especially when read in its original Greek. It references a "Big Bang" sort of event. Oh, and in case anyone is wondering - no, I'm not trolling. I just find this text a fascinating example of antiquity, and especially profound given that the writer is hardly one of the intellectual elites of his day....
 

sparkyclarky

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May 3, 2002
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Originally posted by: OrganizedChaos
because obviously god spontaneously created himself

BTW, I think you're name is great, especially as to how it relates to Greek mythology and the concept of creation.
 

drag

Elite Member
Jul 4, 2002
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Originally posted by: sparkyclarky
Originally posted by: OrganizedChaos
because obviously god spontaneously created himself

BTW, I think you're name is great, especially as to how it relates to Greek mythology and the concept of creation.

What is that like the old story were the universe was created by accident likened a volcano exploding and flinging stone about to create a perfect habitable city?
 

sparkyclarky

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May 3, 2002
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Originally posted by: drag
Originally posted by: sparkyclarky
Originally posted by: OrganizedChaos
because obviously god spontaneously created himself

BTW, I think you're name is great, especially as to how it relates to Greek mythology and the concept of creation.

What is that like the old story were the universe was created by accident likened a volcano exploding and flinging stone about to create a perfect habitable city?

Umm, no. Chaos is a Greek god. He's essentially the great void (more of a personification of a concept than most of the other gods) and out of Chaos comes creation. It's more complicated than that, and if you really are interested I'd suggest picking up a book/textbook on mythology.
 

Richard David Feinman

Junior Member
Jan 8, 2017
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that's HOW it works.



I want to know WHY understanding energy gives such an insight, so the answer i'm looking for is probably more philosophical than mathematical...

Don't quite understand the philosophy yet but if you have a function of kinetic and potential energy that is appropriate and if you take appropriate derivatives, you will have shown that any change in kinetic energy will be balanced by changes in the potential energy, that is, the derivatives of L show conservation of energy. How you know that it is appropriate is probably from working backwards from what you do in Newtonian mechanics. Best I can do.

Somehow, I doubt that Darien is still looking for an answer after 13 years.
admin allisolm
 
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Richard David Feinman

Junior Member
Jan 8, 2017
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It comes from quantum mechanics. I don't understand it but I think Dirac said all paths are possible but there is a most probable one. That is the best I can do but that is the deep philosophical answer. If you ask "Why?" after that you ar back to God which for us is a metaphor for "that's how the universe is."