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I'll be generous and say it might be reasonable to assume that nature would tend to minimize, or maybe even maximize, the integral over time of $T-V$. Okay, fine. You write down the action functional, require that it be a minimum (or maximum), and arrive at the Euler-Lagrange equations. Great. But now you want these Euler-Lagrange equations to not just be derivable from the Principle of Least Action, but you want it to be equivalent to the Principle of Least Action. After thinking about it for awhile, you realize that this implies that the Principle of Least Action isn't really the Principle of Least Action at all: it's the "Principle of Stationary Action". Maybe this is just me, but as generous as I may be, I will not grant you that it is "natural" to assume that nature tends to choose the path that is stationary point of the action functional. Not to mention, it isn't even obvious that there is such a path, or if there is one, that it is unique.

But the problems don't stop there. Even if you grant the "Principle of Stationary Action" as fundamentally and universally true, you realize that not all the equations of motions that you would like to have are derivable from this if you restrict yourself to a Lagrangian of the form $T-V$. As far as I can tell, from here it's a matter of playing around until you get a Lagrangian that produces the equations of motion you want.

From my (perhaps naive point of view), there is nothing at all particularly natural (although I will admit, it is quite useful) about the formulation of classical mechanics this way. Of course, this wouldn't be such a big deal if these classical ideas stayed with the classical physics, but these ideas are absolutely fundamental to how we think about things as modern as quantum field theory.

Could someone please convince me that there is something natural about the choice of the Lagrangian formulation of classical mechanics (I don't mean in comparison with the Hamiltonian formulation; I mean period), and in fact, that it is so natural that we would not even dare abandon these ideas?

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possible duplicate of Hamilton's Principle –  Mark Eichenlaub Oct 19 '11 at 4:16
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Lagrange's equation was originally discovered without the Principle of Least action, and can be derived directly from the Newtonian formulation of mechanics. Goldstein does it that way (and has a discussion of the history of stationary principles in classical physics). –  dmckee Oct 19 '11 at 15:36
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While I can see how this could be considered a duplicate of the other question, it's well written and its getting a good response so I don't feel particularly compelled to close it at this point. –  David Z Oct 19 '11 at 22:12
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A fresh article trying to answer such questions: motls.blogspot.com/2011/10/… –  Luboš Motl Oct 22 '11 at 7:30

7 Answers 7

The functional integral in quantum mechanics includes a summation over all possible paths in the configuration space a quantum system might take . These paths are weighted by an exponential imaginary function whose phase is the action .Using the method of steepest descent , one can pass to the classical limit which shows that the Euler-lagrange equations should hold for the classical path .

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The intuition for the Lagrangian principle comes specific applications of Newton's laws, especially reversible systems with constraints, like nonspherical particles rolling along complicated surfaces. Newton's formulation of Newton's laws was not the end of the story, because there was more structure in the solutions of these types of problems than that which Newton made obvious.

One thing left unsaid by Newton is conservation of energy. Elastic processes are more fundamental than inelastic ones. But energy conservation is only part of the story. Suppose you have a bunch of masses connected by springs, and one of them is attached to a double-pendulum. You could theoretically have energy conservation in such a system by having all the energy leak out of the masses on the springs and go into the double pendulum. Perhaps every frictionless motion of the springs eventually settles all the energy into a single mode.

Your intuition is probably rebelling, telling you "that's infinitely unlikely! How could the pendulum move around and not set the springs vibrating!" But there is nothing in Newton's laws by themselves, even with the principle of conservation of energy, that prevents this sort of concentration of energy. But the solutions do not exhibit such phenomena, and there must be a reason why.

This intuition tells you that a perfect frictionless mechanical system is more than energy-conserving, it must conserve some notion of "motion-volume", so that if you alter the initial state by a certain amount, the final state should alter the same way. It can't concentrate all motion into one mode. This principle is the principle of conservation of phase-space volume, or the conservation of information. If all the motion got concentrated into one mode, the information about where everything was would have to get absurdly compressed into a tiny region of the phase space, the space of all possible motions.

The conservation of information is just about as fundamental as Newton's laws of motion--- it is revealing new facts about nature which are essential for the description of statistical and quantum systems. But it is nowhere to be found in Newton's formulation, because it does not follow from Newton's laws alone, even with the principle of conservation of energy added.

So you need to understand what type of law will give a law of conservation of information. There are two paths to go down, and both lead to the same structure, but from two different points of view, local in time and global in time.

One path is Hamiltonian: you consider formulating the law of motion as a set of symplectic equations for the position and momentum. This formulation clearly separates between reversible and irreversible dynamics, because it only works for reversible. It also explains the fundamental mathematical structure behind reversible classical mechanics, the symplectic geometry. The volume of symplectic geometry gives the precise law of information conservation, and further, the geometrical structure of systems with multiperiodic solutions, the integrable systems, is made clear.

But this point of view is centered on a time-slicing--- it describes things going from one instant of time to another. This is not playing very nice with relativity. So you also want to think about the solution globally, and consider the space of all solutions as the phase space. The initial position and velocities are good coordinates, and intuitive ones, because they determine the future. But if you want a global picture, you want coordinates which are symmetric between the final and initial state, since the dynamics are reversible. An explicit revesible description should treat the initial time and final time symmetrically. So you can use the initial positions and final positions, which also, generically, away from certain bad choices, determine the motion.

For these types of coordinates on phase space, you give the dynamical law as a condition on the trajectory between the intial and final positions. The condition should not be stated as a differential equation, because such a description is unnatural for boundary conditions of this sort. But when you have an action principle, you determine the trajectory by extremizing the action between the end points, you automatically have a notion of phase space volume, which is intuitive--- the phase space volume is defined by the change in the action of extremal trajectories with respect to changes in the initial velocities. This volume is the same as for the changes of the extremal trajectories with respect to changes in the final velocities. This is a straightforward consequence of the equivalence of Lagrangian and Hamiltonian formulation.

The full justification for both principles comes only with quantum mechanics. There you learn that the least action principle is a geometric optics Fermat principle for matter waves, and it is saying that the trajectories are perpendicular to constant-phase lines. But historically, the Lagrangian formulation was recognized to be more fundamental a century before Hamilton conjectured that classical mechanics was a wave mechanics, and this was many decades before Schrodinger. Still, with our modern point of view, it does not hurt to learn the quantum version of these formulations first, and it certainly provides a more solid motivation than the heuristic considerations I gave above.

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What do you mean by conservation of information? –  Arafat Nov 18 '13 at 14:06

The main point of Lagrangian formulation of classical mechanics was to get rid of the constraint relations completely so that one does not have to bother about them while calculating anything (see this answer of mine. Remember all the valuable symmetries of the physical situation is automatically inbuilt in this formulation of mechanics.

Now after having such a powerful technique at our disposal, it is natural to ask the simple question. Can the classical theory of electromagnetism i.e. Maxwell's equations, be expressed as Euler-Lagrange equations by suitably defining a Lagrangian of the electromagnetic field, so that we may readily get all those beautiful results of the structure of this formulation (for example avoid annoying field constraints)? The answer is 'yes', provided we suitably define a Lagrangian.

In fact, it is observed that any general classical field equations can be expressed by Euler Lagrange equation of motion and in each case you need to define a Lagrangian (you get a corresponding "Action") and get everything for free.

This Lagrangian approach is so powerful that even quantum field theories exploit them fully and almost all modern theories of physics exploit them in some way.

However, it is not at all clear whether this approach is completely general. Can't it be the case for a future theory that the equations of the theory can't be expressed in Lagrangian formulation? I asked this question here. See some good answers.

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OP wrote:

As far as I can tell, from here it's a matter of playing around until you get a Lagrangian that produces the equations of motion you want.

Too often, as a student, one is only shown how to derive Newton's 2nd law from Euler-Lagrange equations by postulating some particular Lagrangian $L$. If one believes that Newton's laws are more natural (within the context of non-relativistic classical mechanics), then perhaps it would be more satisfying to see a derivation in the other direction, i.e. to see Euler-Lagrange equations derived from Newton's laws. This is e.g. done in the first chapter of Herbert Goldstein, Classical Mechanics. (An important element in this derivation is to show that a large class of constraint forces do no virtual work, leading to D'Alembert's principle.)

Throwing a wrench into the works, let me finally mention that there exist equations of motion that have no action principle.

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The derivation of Lagrange equations from Newton's law's is discussed in more detail in this Phys.SE post. –  Qmechanic Sep 22 '13 at 14:12

The Lagrangian is just a (special, functional kind of) anti-derivative of an equation of motion. In simple cases, the equation of motion of classical particles is a function $F(x,\dot x,\ddot x,...)$ of the positions, velocities, accelerations, etc. of the particles, for which the equation $F(x,\dot x,\ddot x,...)=0$ determines the trajectory. When a function $F(...)$ admits a Lagrangian anti-derivative, the maxima and minima of the Lagrangian anti-derivative determine where to find the zeros of the function $F(x,\dot x,\ddot x,...)$, the derivative of the Lagrangian.

There is not necessarily anything fundamental or natural about a Lagrangian. The Lagrangian has many formal properties that often make it extremely useful, and quite often make it rather simple or beautiful. We like beautiful things. Appreciating beauty is a tricky thing, to some extent a matter of experience, to some extent a matter of just seeing it. Mark and Anna are both pretty good guides.

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Could someone please convince me that there is something natural about the choice of the Lagrangian formulation...

If I ask a high school physics student, "I am swinging a ball on a string around my head in a circle. The string is cut. Which way does the ball go?", they will probably tell me that the ball goes straight out - along the direction the string was pointing when it was cut. This is not right; the ball actually goes along a tangent to the circle, not a radius. But the beginning student will probably think this is not natural. How do they lose this instinct? Probably not by one super-awesome explanation. Instead, it's by analyzing more problems, seeing the principles applied in new situations, learning to apply those principles themselves, and gradually, over the course of months or years, building what an undergraduate student considers to be common intuition.

So my guess is no, no one can convince you that the Lagrangian formulation is natural. You will be convinced of that as you continue to study more physics, and if you expect to be convinced of it all at once, you are going to be disappointed. It is enough for now that you understand what you've been taught, and it's good that you're thinking about it. But I doubt anyone can quickly change your mind. You'll have to change it for yourself over time.

That being said, I think the most intuitive way to approach action principles is through the principle of least (i.e. stationary) time in optics. Try Feynman's QED, which gives a good reason to believe that the principle of stationary time is quite natural.

You can go further mathematically by learning the path integral formulation of nonrelativistic quantum mechanics and seeing how it leads to high probability for paths of stationary action.

More importantly, just use Lagrangian mechanics as much as possible, and not just finding equations of motion for twenty different systems. Use it to do interesting things. Learn how to see the relationship between symmetries and conservation laws in the Lagrangian approach. Learn about relativity. Learn how to derive electromagnetism from an action principle - first by studying the Lagrangian for a particle in an electromagnetic field, then by studying the electromagnetic field itself as described by a Lagrange density. Try to explain it to someone - their questions will sharpen your understanding. Check out Leonard Susskind's lectures on YouTube (series 1 and 3 especially). They are the most intuitive source I know for this material.

Read some of the many questions here in the Lagrangian or Noether tags. See if you can figure out their answers, then read the answers people have provided to compare.

If you thought that the Lagrangian approach was wrong, then you might want someone to convince you otherwise. But if you just don't feel comfortable with it yet, you'd be robbing yourself of a great pleasure by not taking the time to learn its intricacies.

Finally, your question is very similar to this one, so check out the answers there as well.

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Beautiful response. It reminds me of the Rambam's response to the request to explain the Torah while standing on one foot. "Do not do to others what is hateful to you. The rest is commentary. Now go study." –  AdamRedwine Oct 19 '11 at 13:28
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A philosophical answer but a good one. ;-) –  Luboš Motl Oct 22 '11 at 7:32
    
"In mathematics you don't understand things. You just get used to them." John von Neumann –  Tom Sep 18 '13 at 9:35

What does "principle" mean? It has many definitions but in the context of this discussion it has the power that "axiom" has in mathematics: a very basic assumption, which, if changed, the whole construct theory shifts or is destroyed.

Physics has adopted this from the geometric observation that : the shortest distance between two points is a straight line which logically led to "minimum time taken" and the search for the shortest distance when unknown.

that it is so natural that we would not even dare abandon these ideas?

If somebody brilliant enough can come out with another principle for the system of mathematical formulation of classical mechanics and subsequently quantum field theory that does not follow least action but incorporates perfectly the large data base / existing equations etc there is no problem. It might even be adopted generally if it could predict new spectacular results. Otherwise, from economy of mental effort( another principle :) ) the system developed under the principle of least action will still prevail.

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