![eq203[1].gif](https://i.imgur.com/IRgy1yv.gif "eq203[1].gif")
How come this is more fundamental than Newton's laws of motion? Some textbooks goes as far as claim that particles "smell" their way beforehand and you need quantum physics to explain it.
But when I solve problems, I usually get (with a constraint [math]g(x,y,z)=0[/math]):
[math]- \nabla U -m \ddot \bar{r}+\lamda (t) \nabla g=\bar{0}[/math]
But [math]\nabla g[/math] is just a normal vector to the path the particle takes, so [math] \lamda (t) \nabla g[/math] is a force that is normal to the path, which makes sense as you need some force to keep the particle on the path. And clearly [math]- \nabla U [/math] is the conservative force. I don't see how this has more than Newton's laws of motion. Yes, it is easy to solve the variation problem with more generalized coordinates but that's just change of coordinates.
[math]- \nabla U -m \frac{d^2}{dt^2} \bar{r}+\lambda (t) \nabla g=\bar{0}[/math]
>>8016099
Because it holds in so much more general situations.
i.e. Works for this [math] S = \int {\operatorname{dt} \left[ {\frac{1}{2}m{{\dot x}^2} - V\left( x \right)} \right]} [/math]
And also works for this, [math]S = - {T_{Dp}}\int {{d^{p + 1}}\sigma } {e^{ - {\Phi _0}}}\sqrt { - \det \left( {{g_{\alpha \beta }} + {B_{\alpha \beta }} + k{F_{\alpha \beta }} + k{b_{\alpha \beta }}} \right)} + {\mu _p}\int {{{\left( {C \wedge {e^{B + kF}}} \right)}_{p + 1}}} [/math]
>>8016491
Even if it works, can it still be fundamental?
It's dubious what you're asking here.
The dynamics for a given classical system are the same for both. Fixing a Lagrangian has more information in that you can evaluate paths.
Everything with "smell" is silly, imho. There will always be the option to ask a "why" that has no answer.
![Snells_law_simple_schematic[1].png](https://i.imgur.com/gu8tZT6.png "Snells_law_simple_schematic[1].png")
>>8016513
http://www.feynmanlectures.caltech.edu/II_19.html
Mr. Feynman starts to talk on this subject about "smelling the path" and about quantum mechanics. My professor told that this integral is an axiom that is proven by physical experiments. But I think this can be derived directly from Newtons laws by using calculus.
Take, for example, Snell's law. Surely light doesn't know how to route itself, but the solution is same as variational problem (Fermat's least time). Modern knowledge uses either Maxwell's lequations or Huygen's principle to derive this result.
>>8016506
>>Even if it works, can it still be fundamental?
yes, physicists claim that this is a physical equation, therefore that the action is a real entity in the universe.
Nobody has ever observed an action though....
>>8017203
Didn't they see it at the LHC? I think they did detect some of the terms, but are still looking for some of them.
IIRC the also saw some higher order terms in it and are trying to confirm.
>How come this is more fundamental than Newton's laws of motion?
Because quantum fields don't follow Newton's laws of motion, but there is a Lagrangian formalism that explains them very well.
Also Noether's theorem.
>>8017598
they do not detect the action nor the lagrangians. they gather numerical values then they claim that, because those numerical values stem, in their model, from some observables (pertaining to the model) which are deduced from the action (and the principle of least action) and other assumptions, then the *reality* is indeed described by the model and whatever is in it: the action, the ''laws'' of motion that you get form the principle of least action and so on, like other concepts such as spin, isopsin, quarks, SU[2] , symmetry breaking and anything related to the standard model.
All this is a story where you apply rules of inferences, that you call rules of deductions, from your assumptions.
This is called the faith in the scientific realism. If you want to know more, then take a course in philosophy of science and history of science.
Also, the principle of least action is a deduction form some other assumptions (in classical logic) in classical mechanics
https://en.wikipedia.org/wiki/D%27Alembert%27s_principle
>>8017617
>>Also, the principle of least action is a deduction form some other assumptions (in classical logic) in classical mechanics
>https://en.wikipedia.org/wiki/D%27Alembert%27s_principle
or rather that euler-lagrange equations stem deductively form some other principles.
>>8017661
http://physics.stackexchange.com/questions/78138/is-there-a-proof-from-the-first-principle-that-the-lagrangian-l-t-v
http://physics.stackexchange.com/questions/131363/why-are-dalemberts-principle-and-the-principle-of-least-action-related/131392#131392
so take your pick: in any case, it does not matter, because physicists always claim that what is real is what eases their life.
Scientists are not famous to think about what they say, so it leads them to claim that some mathematical concepts are more real than others; if it suits them to think that the ''universe'' manages to apply deductive rules to the some principle as soon as they make an experiment, then they will claim thus. The next generation of scientists will have faith in other formalism and other principles...
>>8017617
Are you literally autistic? and I dont mean this in the internet-insult way, I mean it in the medical sense.
https://www.youtube.com/watch?v=3znjCuLlf8E
First, what’s supposed to be so scandalous about Nietzsche. Nietzsche is supposed to hold the scandalous view that knowledge is a form of power. Now that is scandalous because knowledge is knowledge. It’s objective. You know, like journalism. And it would be scandalous to show that wherever we find knowledge, we will find it structured and constructed around a system (or systems) of power. Won’t find one without the other. Now, one can think of this along the simplest pedagogical models. By that I mean the classroom models. I mean, I ought to know this from teaching the university. I know how to pass along knowledge.
To get someone to believe me in the last analysis, I give them an “A”, which I could replace with a “happy face”. They are used to that, it’s from kindergarten. They are both just symbols, right, of achievement. They’re not getting paid for this stuff. Just give them a little “A”, they smile. That same system starts in kindergarten: “happy face”… “A”… runs through to “F”. “F”, no face… blank. The same thing would work in kindergarten. That form I used looks fair. I mean, I am grading objectively. But the point is deeper. That what the knowledge is based on is my spot of power as the teacher. That’s what it’s based on. Now, you would go: “oh no – it’s based on what’s really true!” Yeah, but… but… how does that get meted out and parsed out? Who decides that? Well the blunt and ugly answer is: we do. The teachers do. We decide.
Now there are clear counter examples to Nietzsche’s argument. In mathematics at its simplest levels, I will grant you, that if we are doing a mathematics course, I could grade objectively. But I will also grant you that nothing of great importance to human values hangs on truths that everyone can accept. That two plus two equals four, that A is A, are all acceptable, and they are acceptable precisely because nothing of very great human importance hangs on them. The moment you go a little beyond that in any direction, even in math class, when you discuss for example the philosophy of mathematics, then the disputes start, and then power at some point has to insert itself and decide.
So, an important part of Nietzsche’s investigation is in the interconnection between the forms of knowledge and power. Forms of… and for the purposes of our course… forms of ethical behaviour and power, ah, are the subject of his most important book. Well, maybe not his most important, but certainly the one that is the most coherent: “On the Genealogy of Morals”, by Nietzsche.
