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Wittgenstein says in the Tractatus:

6.3211 Men had indeed an idea that there must be a 'law of least action', before they knew exactly how it ran. (Here, as always, what is certain a priori proves to be something purely logical).

What evidence does he have for this?

For example, Hero of Alexandria showed that the motion of light followed the path of least distance.

Was there any speculation at the time or post this, or before, that this law could be generalised?

Could one say that Leibniz had a least principle? The 'reflection' of least is best. And he did say that God created the 'best of all possible worlds'.

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One way to interpret the law of least action is that nature tries to convert potential energy into kinetic one as fast as possible. An apple, severed from the tree, doesn't hang in the air for a while, deciding whether to fall or not; it falls down as fast as it possibly could, given the amount of energy it had.

One can put a lot of philosophical or psychological spin on the above, even though it means applying the concept where it doesn't belong. Usually it goes along the lines of a drive to realize one's potential. One politician even claimed that well-trained army cannot stand idle for very long: it's a part of the army's nature to realize its brewing potential energy into a forward moving momentum.

I cannot tell what exactly was in Wittgenstein's mind at the time he wrote it, but, generally speaking, philosophers and psychologists tend to apply the concept of the law of least action in non-physical context along the above lines.

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  • I'm not sure that the phsyical interpretation is quite correct. A falling apple falls fastest if it follows a Brachistone curve; and what constraints is preventing the change of potential energy to kinetic in a split-second? Having said that its an intriguing way to put the least action principle in physics - I haven't come across it before. Can you expand a bit more on it?
    – Mozibur Ullah
    Commented Nov 27, 2013 at 10:34
  • @MoziburUllah: brachistochrone curve is not applicable to a vertical fall. An apple cannot "change" potential energy to kinetic one instantaneously because in order to yield the amount of potential energy that corresponds to reduction of its height by H in needs to physically fall by the height H, and if the corresponding energy can accelerate it to speed V the time required for the conversion cannot be smaller than H/V by the very definition of speed. Actually, upon more careful calculations, the time required would be twice as much.
    – Michael
    Commented Nov 27, 2013 at 17:25
  • @MoziburUllah: according to your bio on this site you have degrees in Math and Physics. Would you recall what "action" and Lagrangian and principle of least action mean in Classical Mechanics? If you do I believe you would confirm that the interpretation of said principle in my answer is basically correct.
    – Michael
    Commented Nov 28, 2013 at 0:18
  • The way I would look at the problem, in Lagrangian Mechanics is to fix two points - the starting & end position. In your example, we choose the end-point vertically below. We then consider all paths connecting these two points and look at the action, which we want to minimise. Because mass is a constant, this is the same as minimising the time taken to traverse the path. Its fairly obvious, without doing any calculations, and going by our experience (after all Physics is an experimental science) that the path with the shortest time taken is the straight line.
    – Mozibur Ullah
    Commented Nov 28, 2013 at 4:42
  • To answer your conjecture in the affirmative, one would need to consider every other path and show that the starting speed is always lower than the one taken falling straight down. Your characterisation may be correct, as I said its an interesting because its not standard. But I'm not proposing to the calculation myself :)
    – Mozibur Ullah
    Commented Nov 28, 2013 at 4:51
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The idea of a formal or theoretical physics presupposes the principle of least action. The principle expresses the possibility of conservation laws. The principle was stated by several 17th Century thinkers including Leibniz but it was Helmholtz who made mathematical expression that linked the principle to the principle of the conservation of energy. See Max Planck´s 1908 paper on "The Unity of physical Wold Pictures" and Ernst Cassirer´s book Determinism and Indeterminism in Modern Physic. Wittgenstein explicitly mentions the principle in the TLP. The principle is a hinge sentence in the sense of Rom Harré. The principle says : Nature not only has an economy, nature operates on itself in an economical manner. Steen Brock

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  • Its only locally true, globally its a principle of stationary action.
    – Mozibur Ullah
    Commented Mar 13, 2019 at 11:16
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Rereading the Tractatus

6.3 The exploration of logic means the exploration of everything that is subject to law. And outside logic everything is accidental.

6.32 The law of causality is not a law but the form of a law.

6.321'Law of causality'—that is a general name. And just as in mechanics, for example, there are 'minimum-principles', such as the law of least action, so too in physics there are causal laws, laws of the causal form.

6.3211 Indeed people even surmised that there must be a 'law of least action' before they knew exactly how it went. (Here, as always, what is certain a priori proves to be something purely logical.)

"Causality" is casually expressed as "there should be a cause" : "X causes Y" is the form, "cold causes shrinking" is a law. Logic is interest in the form, physics is interest in the content given. According to Wttg the "minimum p-ples" are the form, "least action" is a law. What people surmised is a principle and that is indeed the use/meaning of that word.

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