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Newton tried to do it with three laws/statements. While the first can be derived from the second, the three form a pretty nice framework.

Later on, I've encountered Lagrangian Mechanics, which involves, from what I gather, one statement:

  • Objects seek a path that minimizes total action.

Which, while sort of simple, doesn't sound too natural or fundamental.

I've heard some of these rather elegant statements:

  • The universe is translation-invariant
  • The universe is invariant under different inertial frames of reference

And from either of these, one may derive Newton's laws and F=ma. However, I'm not completely sure how this can be done. Were these claims true? How is this possible?

Has anyone ever come across a surprisingly/particularly stunningly elegant/fundamental formulation of classical mechanics, and a proof that they are equivalent to Newtonian Mechanics?

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@Justin: could you give a reference to statements you post? I would not know how you could construct some sort of formulation of mechanics from these assumptions on symmetries/conserved quantities. I think that the principle of stationary action might be already quite fundamental as it seems to explain most of the physics we know of today, from quantum field theory to general relativity. –  Robert Filter Feb 23 '11 at 9:01
"Which, while sort of simple, doesn't sound too natural or fundamental." -> it's not fundamental because it's derived from quantum mechanics but on the classical level you can't get anything better than this. It's the single most unifying principle in classical physics and you can use it for everything, not just mechanics. If you can somehow guess the right Lagrangian, the rest of the physics falls out. –  Marek Feb 23 '11 at 9:01
@Robert: quantum field theory? In quantum theory the statement definitely doesn't hold, even non-classical trajectories contribute to the final amplitude. You are perhaps thinking of free fields (where the only contribution to path integral is classical thanks to their being gaussian) and instantons? –  Marek Feb 23 '11 at 9:04
@Robert - I can't, really; I think I picked it up somewhere as an offhand statement. But from the Lagrangian, it can be derived that a translation-invariant system is one that conserves momentum, so I figured they weren't too off. Perhaps they are completely wrong; I just appreciated the elegance of the statements as a formulation of the natural laws. –  Justin L. Feb 23 '11 at 9:05
@Marek - Maybe it's just my own personal bias, but there seems to be just something less-natural about the principle of stationary action, than something like "objects resist changes to their velocity proportional to their mass". –  Justin L. Feb 23 '11 at 9:07
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3 Answers

The principle of least action is much more elegant and fundamental than the explicit formulation of the laws of mechanics using three laws of Newton which disproves your intuition that the principle of least action is not elegant or fundamental.

The two symmetries you mention are symmetries that constrain the evolution of physical systems but they don't totally determine what the evolution looks like. It is not true that the symmetries you mention are enough to derive the forces among a set of objects. $F=ma$, without specifying how $F$ depends on the positions (e.g. gravitational force), is just a definition of $F$ and it is a vacuous statement so you may derive it from anything - even from no assumptions at all.

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I feel that there is something about the principle of least action's elegance that I am not understanding. Can you explain the naturalness in it? I've so far taken no courses with it; I've only worked with the mathematics and found out that it worked. –  Justin L. Feb 23 '11 at 9:15
š Not that I disagree with the elegance of least action, but your opening paragraph is (unusually for your answers) rather empty. You just state that least action is elegant and fundamental, then claim that this statement disproves the idea that it is not. You might want to consider editing out the second half of that sentence, at least. –  Mitchell Feb 23 '11 at 14:35
Dear Mitchell, well, the elegance or non-elegance is obviously an aesthetic issue, and I didn't start with the evaluation of it. I just gave an answer to a question which contained a proposition that it was not elegant. The more properly one understands classical physics, the more he knows that the principle of least action is more elegant than Newton's laws themselves. It generalizes to a universal (Feynman) rule in quantum mechanics, and so on. The point of my paragraph was just that aesthetic prejudices may be showed wrong, and this was the case here, too. –  Luboš Motl Feb 24 '11 at 6:36
Dear Justin, I don't understand what you mean by "naturalness of the principle" and how such a naturalness may be "showed". The principle of least action is natural in the colloquial sense because experiments show that its predictions are obeyed in Nature; it is natural in the physics sense because it doesn't introduce new free parameters with values very different from one; and for a similar reason, it is a robust and universal principle. I am afraid that every other interpretation of "naturalness" you may have adopted is just totally unscientific. Maybe you just meant you're not used to it? –  Luboš Motl Feb 24 '11 at 6:39
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The most geometric formulation of classical mechanics is in terms of symplectic geometry. However in terms of the question asked about the principle of least action, the issue is that this formulation

Objects seek a path that minimizes total action.

is mathematically vague, and that the phrase (and definition of action) can have different variational forms. Some of these are more intuitive and some are more general than others.

The most general formulation is Hamilton's Principle:

The motion of the system from time $t_1$ to time $t_2$ is such that the line integral

$I = \int_{t1}^{t2} L dt$

where $L= T - V$ is a stationary point for the path of motion.

The elegance of this is that it identifies one path from amongst all those between initial position at time $t_1$ and final position at time $t_2$ which extremizes the path. The points and paths here belong to the multidimensional configuration space of the system. Not only that but the remaining equations of motion can now be derived from this for all classical systems, with an extension into quantum mechanics as well.

However there are other (older) formulations of similar variational principles which can also refer to "action" and which have use in classical mechanics too. The classic text Goldstein introduces a "Principle of Least Action" which is defined a little differently. If we introduce the Hamiltonian of a system H and assume that it is conserved:

$\Delta \int_{t1}^{t2} \Sigma p_i q_{i}^{dot} dt = 0$

The term inside the integral is also known as the Action and the variation involved (which is minimized) is denoted by $\Delta$.

To understand this formulation let us consider only the simplest case of it: assume non-relativistic mechanics with no time dependence and no external forces. Then this expression reduces to:

$\Delta(t_1 - t_2) =0$

This says that of all paths possible between two points, consistent with conservation of energy, the system moves along that particular path for which the time of transit is the least.

Other variations of this formulation can have the particle moving along geodesics within configuration space. Some of these formulations are similar to Fermat's principle of Geometric Optics.

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the most elegant form of classical mechanics comes from newton's laws:

  1. f =ma
  2. action = reaction
  3. bodys remain at rest unless acted on by other forces.
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