on the principle of least action

[Anonymous No.9787755]

pic completely related,

right now im studying maths, but i got interested on something a friend of mine told me; that the behaviour of kinetic and potential energy in a falling object is such that if the subtraction of them through time creates a curve which's area under it is minimal.

i came out with a caveman wojak-worthy counterexample (if kinetic and potential energy curves behaved logarithmically, or like a +2th order ecuation, action will be minimized even further than what we see in experiments)

so. physicist friends, ¿why aren't projectile's trajectories drawing logarithmic curves?

[Anonymous No.9787772]

why should they?

[Anonymous No.9787776]

>>9787772
because action should be minimal

also i put that L=U-K in my pic, it's l=K-U

[Anonymous No.9787803]

>>9787755
Physical laws.

[Anonymous No.9787936]

>>9787755
Because they both go with the square of velocity, non with the log of velocity

[Anonymous No.9787948]

Because kinetic and potential energy don't don't have that form. The kinetic energy is a quadratic form, meaning it is proportional to the square of the velocity, and potential energy usually only depends on distances between objects. These quantities are defined independently and the classical formulation of the PLA says that given those quantities, the trajectory of a system is such that the action is extremized.

[Anonymous No.9788242]

>>9787948
then is the principle of least action supposed to be as follows?:

from t(0) to t(f)

given U,K quadratic curves with their vertices in t(0).

and we know the value of U(f) XOR K(f)

then the curve L=K-U determines the value U(f)/K(f) that we don't know as it minimizes the area under itself.

being the conservation of energy a conclusion of this, and not an axiom which the principle of least effort lies on.


i have to say that this formulation is quite a lot less impressive than what i thought physicists referred when saying that the principle of least action predicted the behaviour of trajectories, as it's limited to U,K being quadratic (and with vertices in the same value of t), predicting only the ratio of both function's a value (remember a quadratic formula is of the form ax^2+bx+c

[Anonymous No.9788337]

The thing is, the principle of least action is about all the possible trajectories, not all the possible forms for the kinetic and potential energy.

The falling ball could move along any trajectory, but in reality the system evolves to minimise the action, we then find that if you drop a ball from rest, the path of least action is a straight down fall. If the ball has some initial sideways speed, then the path of least action is an upside down parabola just as in reality. We find that the path of least action only corresponds to reality if we make the kinetic energy be 1/2mv^2, and the potential energy to be whatever works, in the case of gravity, mgh

[Anonymous No.9788348]

>literally no one knows the correct answer
Is this a joke? How fucking stupid are you idiots?

[Anonymous No.9788353]

>>9788242
You're grossly overthinking this concept, the Principle of Least Action is most completely answered purely conceptually, "everything should be made as simple as possible but not simpler" etc. etc.

The Principle of Least action states that given any path or choice or model that could possibly exist, the universe chooses the "easiest" or least entropy-intensive option.
Because potential and kinetic energy vary quadratically and not logarithmically the logarithmic energy graph isn't a possible state to choose from.

It is not limited to U and K being quadratic - U and K ARE quadratic. If they weren't it would still apply, but they aren't so it doesn't.

[Anonymous No.9788356]

>>9787755
Right side isn't a local extremum, so it doesn't happen

[Anonymous No.9788393]

>>9788356
this path of reasoning sounds promising,

[Anonymous No.9788520]

You're all fucking idiots. The reason is that the action must obey the symmetries of the system, and only a quadratic form in the velocity (up to constant factor) is invariant under the symmetries of the free particle, i.e. the group of Euclidean rigid motions.
1. Translation invariance implies only velocities can appear
2. rotational invariance implies only inner products can appear
Put 2 and 2 together and you have [math]v^2[/math] (and only that) in the kinetic energy. For exactly the same reason the potential must depend only on the magnitude of the coordinates or their derivatives.
How fucking stupid is this board? Like honestly?

[Anonymous No.9788549]

>>9788520
Why can't kinetic energy be log(v^2) or exp(v^2) then? You also completely misunderstood the question. He's not asking if kinetic and potential energy can be logarithmic functions of position, velocity. Read the OP again, brainlet.

[Anonymous No.9788564]

>>9788549
>Why can't kinetic energy be log(v^2) or exp(v^2) then?
Do you not understand what an inner product is, retard?

[Anonymous No.9788569]

>>9788564
Any function of the inner product will also be translationally and rotationally invariant. How can you be this dumb lol. Also, you ignored the rest of my post. Good job.

[Anonymous No.9788574]

>>9788569
>any function
Which needs to ALSO transform covariantly under the symmetries. The Euclidean motions are INTERNAL symmetries that leaves the Lagrangian invariant. If the kinetic energy contains a non-trivial function of the inner product then any group action, as a diffeomorphism on coordinate space, pulls back to 0-forms (i.e. smooth functions to a complete idiot like you) according to some non-trivial representation of the group, and this non-trivial representation fucks up the fact that Euclidean motions is supposed to be an internal symmetry. The only function that admits a trivial rep for ALL action in the Euclidean group is the identity.
Also if you pick log then the pull back can actually pick up holonomies which fucks up the invariance of the ACTION itself, so it's not even an external symmetry anymore, you fucking retard.
Jesus fucking Christ this board is basically Dunning-Kruger central.

[Anonymous No.9788586]

>>9788574
>Which needs to ALSO transform covariantly under the symmetries. The Euclidean motions are INTERNAL symmetries that leaves the Lagrangian invariant. If the kinetic energy contains a non-trivial function of the inner product then any group action, as a diffeomorphism on coordinate space, pulls back to 0-forms (i.e. smooth functions to a complete idiot like you) according to some non-trivial representation of the group, and this non-trivial representation fucks up the fact that Euclidean motions is supposed to be an internal symmetry. The only function that admits a trivial rep for ALL action in the Euclidean group is the identity.
I don't really understand this but whatever. Your original post was off topic anyway as you obviously can't read.

[Anonymous No.9788598]

>>9788586
>literally answers the OP
>off-topic
Fucking retard.
>don't understand this
No shit. For any function f that maps your coordinate space into some other space (possibly itself) f pulls back onto an endomirphism f* on the cotangent bundle, and hence an endomorphism on smooth functions. This f^*(exp), for instance, is how a kinetic term like exp(v^2) will transform under ANY map f. Considering the group action g(x) as a diffeomorphism of the coordinate space into itself for a group element g in the symmetry group G, it also pulls back g^* to smooth functions given an appropriate representation of g. g^*(exp) is in general not trivial and for torsionless groups G like the Euclidean group this fucks up the invariance of the Lagrangian.
Also if you pick log then g^*(log) can pick up the singularity at the origin 0 of the coordinate space, leading to extra terms in the ACTION that fucks up even the invariance of the action.
Just stop fucking talking, undergrad.

[Anonymous No.9788600]

>>9788353
>the Principle of Least Action is most completely answered purely conceptually, "everything should be made as simple as possible but not simpler" etc. etc.
>The Principle of Least action states that given any path or choice or model that could possibly exist, the universe chooses the "easiest" or least entropy-intensive option.
No it doesn't, it says that the quantity called the action, which defines the theory, is minimised by the actual evolution of the system, from a fixed initial state to a fixed end state.

[Anonymous No.9788605]

>>9788598
>>literally answers the OP
The question in the OP was this: Why can't the trajectory be like the RHS as it has a lesser value of the action than on the LHS?

[Anonymous No.9788616]

>>9788605
And the answer is because the Lagrangian and action need to be invariant under symmetries, which is exactly the point of the original post: symmetries.
Didn't I tell you to stop talking, stupid freshman? Or are you too stupid to take instructions?

[Anonymous No.9788626]

>>9788616
You're explaining why the Lagrangian is what it is. Literally no one cares about that. Dumb anime poster. I'm not explaining this shit to you again.

[Anonymous No.9788629]

>>9788626
>You're explaining why the Lagrangian is what it is.
Which is literally why whatever OP says isn't allowed, you stupid fucking retard.
>explain in
Yeah walk away. Don't ever try to explain shit you don't understand again. I hope you've learned your lesson, dumbfuck.

[Anonymous No.9788672]

>>9788616
OP was obviously having trouble with the formulation. He thought the functional itself could change if that minimized the "area" i.e. the action. The poibt here is that what you minimize is the trayectory the system follows because, even if the kinetic energy must be quadratic with the velocity, the potentiak energy can be pretty nasty, velocity dependent or shit that doesn't make the Lagrangian convex, but the poin is that for a given system you have one lagrangian that depends on the path (the literal path parametrizied by time) and you do variations of these path to get the result. But yea, hide behind your basic ideas from lie groups to answer something that was completely irrelevant or doesn't need all that machinery as there are many ways one can explain the reason for proportionality to the square of the velocity which still, doesn't answer anything as to how the potential energy needs to be.

[Anonymous No.9788675]

>>9788672
Wgat you use for minimizing is the trayectory, but the functional form of the lagrangian is mantained*

[Anonymous No.9788709]

>>9788672
>as there are many ways
All of which are wrong. Unless you understand the symplectic geometry behind analytical mechanics you do not understand analytical mechanics at all, my sweet summer undergrad.
>doesn't answer anything as to how the potential energy needs to be.
I never said anything regarding the potential energy, which is completely determined by physical model. In fact log(|x-y|) shows up in 2D Coulomb on the configuration space of 2 fermions.
Why do you think putting words in my mouth is gonna help your case, dumbass?

[Anonymous No.9788784]

>>9788709
Lol, you can't understand that the problem in OPs comprehension was with what it means to minimize a functional and that he thought the functional form of the Lagrangian could change because the principle of least action only demands minimizing the "area".

>All which are wrong
Oh so a classical newtonian treatment cannot explain why the kinetic energy must be of that form? Hiding under "complexetiy" as a way to show comprehension and knowledge is the best way to spot an insecure retard, worse that it happens in a fucking anonymous forum.

>Why do you think putting words in my mouth
Lol classical response you insecure retard. The point was to show that the general form of the Lagrangian has nothing to do with what OP asked, as the fucking point is that the functional form of the Lagrangian is set for a given configuration. More mathematically for your autism, your first start defining a configuration manifold that gives you the complete information of the position of all the components of your system and then define a map from the tangent budle to the real numbers, and then using the natural pairing of a curve in the manifold and it's tangent vector you define the Lagrangian functional as the action. The form of the Lagrangian isn't even assumed in many cases considering the different amount of cases you could have, but in general the kinetic part IS a quadratic form. But with velocity dependent potentials and other fuckerys you could have a Lagrangian that is more similar to what OP draw, but that doesn't matter because the point is that you are not going to change the Lagrangian to solve the problem but the paths that move through the tangent bundle.

[Anonymous No.9789051]

>>9788574
so, the conservation of energy, which is nothing but the consequence (or cause, idk) of symetry throughout time, implies that the energy curves in my shitty graph can only be polinomial ones.

still im curious about why is it a quadratic behaviour and not a (for example) poinomial equation of 10000000th order

[Anonymous No.9789077]

>>9787755
>U and K are paraboloid for free-falling objects.
>citation needed

[Anonymous No.9789091]

>>9789077
E=U+K is constant
U depends directly from the heigh of the falling object, and since in nature projectiles draw an upside down parabola in their trajectories, U behaves the same way.

since E is constant K must behave quadratically as well

[Anonymous No.9789107]

>>9789091
What's the x axis in your graph?

[Anonymous No.9789110]

>>9789051
No, what that autists is yelling is that from the first law of mechanics, i.e., that the physics is invariant under galilean transformations i.e. it is galilean covariant, then the Lagrangian must be quadratic in the velocity component.

[Anonymous No.9789122]

>>9789110
yes, what's wrong with that? sounds legit

>>9789107
the horizontal one ofc, duh

[Anonymous No.9789135]

>>9789122
>the horizontal one ofc, duh
Do you know that the action is an integral over time?

[Anonymous No.9789150]

>>9789135
stop wasting my time, if you are not going to answer the question then you should leave, as i can tell for your only 3 comments that you are not very bright

[Anonymous No.9789168]

>>9789150
>3 comments
Wrong.
Can you even write down the expressions of the potential and kinetic energy at different instants of time and prove that you'll get parabolas? Your reasoning is all over the place.

[Anonymous No.9789186]

>>9789168
take the formula for the height of an free falling object and multiply it by mass and gravitational acceleration.

K=E-U with E constant

[Anonymous No.9789364]

>>9788784
>functional form of the Lagrangian could change
Except this is exactly what my comments addressed, dumbfuck.
>Oh so a classical newtonian treatment cannot explain why the kinetic energy must be of that form?
No, because Newtonian treatment (which isn't even analytical mech btw, that's Hamilton-Jacobi) doesn't explain shit. It just says "here's how things work". The explanations you've given so far are all handwaving bullshit fed to you by high school physics teachers who had at most a masters in engineering. Perfect for your IQ range though, I'd say.
>system and then define a map from the tangent budle to the real numbers, and then using the natural pairing of a curve in the manifold and it's tangent vector you define the Lagrangian functional as the action.
Are you retarded? The "natural pairing" (which isn't even natural, it's canonical in the sense of de Rham) IS the map to the reals. That's the functional defined via an integration. You literally have no idea what you're trying to talk about.
>paths that move through the tangent bundle
The path parameterized by the time interval lies in the symplectic manifold, not its tangent spacebar its bundle you fucking retard. Again, you have no idea what you're talking about.
>but that doesn't matter because the point is that you are not going to change the Lagrangian to solve the problem
Except minimizers are unique for a given functional, so any physical configuration that arises out of minimizing the action can only change if you change the Lagrangian itself. Again again, you have no fucking idea what the fuck you're talking about.
You're an actual brain dead retard, not just for misunderstanding the basic concepts of symplectic geometry and skimming Wikipedia for the jargon I've put out, but also for digging yourself this hole deeper and deeper and not knowing when to stop. It's fucking pathetic and embarrassing to watch.

[Anonymous No.9789367]

>>9789364
>spacebar
space nor*

[Anonymous No.9789387]

>Guy comes in and provides the exact reason why the functional in a principle of least action argument has the form it has
>"Stfu retard"

Never change, /sci/

[Anonymous No.9789419]

>>9789387
Yeah , it's a fucking panto how stupid /sci/ has become at this point.

[Anonymous No.9789455]

>>9789364
>Except this is exactly what my comments addressed
Lol no, the functional form a lagrangian is not changed when applying E-L equations you retard. The principle of least action talks about variations in the paths. That there are transformations that leave the equations unchaged is another, given, important topic.
>No because Newtoniana treatement doesn't explain shit
Really? https://en.wikipedia.org/wiki/Work_(physics)#Work%E2%80%93energy_principle Do you really think knowing symplectic geometry is the only way to view mechanics? How fucking ignorant and retarded.
>The natural pairing is the map to the reals
Holy fucking shit, you understand the difference between hamiltonian and lagrangian mechanics? The natural pairing is to take the point the pairs of the curve that lies in the configuration manifold and it's tangent vector, so that pair is in the tangent bundle you symplectic mong. The hamiltonian view is equivalent, but mathematically different you retard.
>Except minimizers are uniequ for a given functinal
Not exactly true as existence and uniqueness has subtle but important problems that are physically important, in particular when the lagrangin is not convex, the thing however, if you could actually read what OP tried to say is that he gave a "counter example" of the principle of least action which involved changing the fucking functional itself, but what you are actually "varying" is the possible trajectories in your configuration manifold and you are leaving the lagrangian the fuck alone. I.e. you proceed as knowing what the Lagrangian is to give an action functional which you then take the functional derivative of it to get the E-L equations that are the equation of motions. The symplectic structre is a whole different perspective and it has nothing to do with what you say which was more about equivalent functionals for a given set of transofrmations. Maybe canonical or something but I don't know since you fail to understand the issue here.

[Anonymous No.9789509]

>>9789455
Just let me be clearer. OP basically talked about the "type" of curves that could potentially be minimal compared to the example he was give. In this case the problem is that he is drawing curves with axes that show the lagrangian, and how it changes depending of the change of T and K understood as the kinetic and potential energy. However this curves are not the curves you are varying when considering the principle of least action, because you are no longer minimizing the variables as in calc, you are talking about minimizing a function that comes from an infinite dimensional space, so how the energy is varying cannot even be visualized as a fucking graph and it doesn't matter for the purposes of minimizing that the form of the kinectic energy is quadratic on the velocity. That has to do with other shit, but the idea of analyzing the PRINCIPLE OF LEAST ACTION cannot be done by the picture OP draw as your particular trajectory can then give a different graph if you now graph the energy of the particular solution. For example, the fucking harmonic oscillator. Looking at the graph of potential energy, the hamiltonian or lagranigan gives you a pure QUALITATIVE way of understanding how solutions may evolve, but it's completely different than trying to visualize minimizing the action which absolutely most deal with trajectories that are parametrizised by time so you have a zoo of possibilities.

As an exercise to anyone, try to "visualize" the arc length functional. The functional gives you the arc length of a given piece wise curve. You basically have no restriction to what curves are possible and so you really have no graph to talk about this shit, because the functional (in it's not so general form) works for all C^1 curves. In the OP, the picture basically has the same problem, that the functional itself cannot represent the process of doing a derivative in infinite dimensions, but it is not solved by saying anything about the in variance of the functiona

[Anonymous No.9789559]

>>9789455
>Do you really think knowing symplectic geometry is the only way to view mechanics?
I said it's the only CORRECT way.
>The natural pairing is to take the point the pairs of the curve that lies in the configuration manifold
Are you even listening to yourself? This is literally word salad. Write it down with TeX or fuck off, dumbshit.
>pairing M and TM gets you back TM
Fucking retard.
>implying this has anything to do with Hamiltonian vs Lagrangian
Dumbass, a Hamiltonian vector field is defined by a formula in terms of the symplectic form, not with respect to some curve you stupid dumbfuck. The Lagrangian is then defined through this. They're literally mathematically equivalent, stupid cunt.
>Not exactly true
I agree. However once the kinetic energy term has been fixed by the symmetries then all it requires is for the potential to be Caratheodory for the direct method to get you unique extremizers. I dare you to give me a physical example in which the potential is not Caratheodory.
>but what you are actually "varying" is the possible trajectories
Which means you evaluate the action functional at different points in the function space, so those are no longer minimizers. What's your fucking point, tardgargler?
> take the functional derivative of it to get the E-L equations
You do know that Frechet differentiable functionals are a subclass of Caratheodory ones right? Oh wait, of course you don't. If you're gonna give me shit for assuming the existence and uniqueness of extremizers at least be consistent and not talk about (weak) EL. Fuckin dumbshit.
>The symplectic structre is a whole different perspective
Completely wrong. The Jacobi action functional is defined through the Hamiltonian vector field, which in turn is defined through the symplectic form. And its existence depends on caustics along the Hamiltonian vector field. Fucking idiot.

[Anonymous No.9789606]

>>9789559
>I said it's the only CORRECT way
Well then you are clearly retarded.
>This is literally word salad
Well, there is no absolute definition of a configuration manifold as it depends on the physical system you are analyzing and it would be pretty dry to just say "any general submanifold in 3N dimensional coordinate space". I don't know how someones who claims comprehension cannot even see a simple concept. For a retard like you let me tell you about a basic example. The double pendulum is completely determined by the angles each pendulum does with a given plane in each joint, which in principle can take all values so (considering the lenghts of the pendulums are the smae) the configuratioj manifold is the torus S^1XS^1 (latex isn't working for me, but I shouldn't have to explain this if you are supposed to know what the fuck are you talking about).
>>Pairng M and TM gets you back TM
Are you this fucking stupid? you have a fuction from some interval in the real R-->M which you call a curve in the manifold, the derivative of this map defines a function to the fucking tangent bundle that is just the (point,tangent vector at the point), which let's you talk about a space of paths in the manifold and formally talk about the "velocity" variable.
>Dumbass , a Hamiltonian vector field is defined
Who the falk talked about the hamiltonian? Are you actually this blind and dogmatic about a particular field of diff geometry? The lagrangian is defined however the fuck you want, and it certainly can be defined as I told you.
>Which means you evauate the action functional at different points in the functions space so those are no longer minimizers
So you see exactly that the variation of the action comes not from what OP draw then you retard.
>non caratheodory potential
>Frechet differentiable are a subclass of Caratheodry
Retard, I'm was talking about situations in which you cannot even assume the lagrangian is convex.
Cont

[Anonymous No.9789609]

>>9789559
>>9789606
Non convex lagrangians, make the correspondace with the hamiltonian, maybe not even well defined. This comes in gravitational optics, so it's not classical, but the point is that in general there is not really that much assumption on the general form of the lagrangian.
>Completley wrong. The jacobi action functional is defined through the Hamiltonian vector field.
Again, no matter how much you shout, there is no universally agreed "axiomatization" of mechanics, and nothing is readily defined through a single object. I can't understand the autism you must have too delude yourself into thinking a pure Lagrangian approach is not CORRECT, which in this case shows why it is better to solve what OP was having trouble as you just spouted some irrelevant point. You have a general submanifold or manifold by itself with a quadratic form on the tangent space defined through a riemannian metric. You define a functional and a space of trajectories and apply basic results from the calculus of variations. The minimizer property comes exclusively from this point in the most standard way of formulating it and the OP was talking about that, but for some reason drawing L vs T, K which is not a correct way of describing variations in an infinite dimensional space.

[Anonymous No.9789617]

>>9788520
Turbo undergrad dunning-kruger right here. Keep reading to get smarter, since thinking is not your strong suit.

[Anonymous No.9789618]

>>9789606
>Well then you are clearly retarded.
Coming from a complete idiot like yourself? That's a compliment.
>Well, there is no absolute definition of a configuration manifold
Yes there is. A configuration space is the space on which particles move without the diagonals (i.e. coinciding positions), if the particles are indistinguishable. This is NOT the symplectic manifold by the way, so you using retarded terminology that is literally irrelevant to the convo just makes you look worse, if such a thing is even possible.
>"any general submanifold in 3N dimensional coordinate space".
THIS is what you think a configuration space is? LMFAO holy shit.
>I don't know how someones who claims comprehension cannot even see a simple concept.
I cannot see the retarded terminology you're using, not the concept. If you actually learn the material instead of skimming shit you'd know the right words to use. Until then you can sit in your tard corner.
>the derivative of this map defines a function to the fucking tangent bundle
And you call this a pairing? LMFAO you're literally just evaluating the tangent vector at some point, this is by no means a pairing. Again, learn the right words, dilettante.
>Who the falk talked about the hamiltonian?
You did. The Hamiltonian formalism is built FROM a Hamiltonian [math]H\in C^\infty(M)[/math] as an a priori given. Fuck's sake it's like talking with a neckbeard undergrad who wears a Schrodinger's cat shirt.
>The lagrangian is defined however the fuck you want
LMFAO. The entire point of symplectic geometry is to put analytical mech on rigorous grounds. If the Lagrangian isn't defined in those terms then it is NOT analytical mechanics. Are you actually this fucking dense?
>So you see exactly
I see exactly that the reason I gave completely breaks what OP was trying to do, since he HAS to change the Lagrangian in order for it to admit what he drew as a extremizer. Are you actually this stupid?

[Anonymous No.9789628]

>>9789609
>I'm was talking about situations in which you cannot even assume the lagrangian is convex.
Fucking retard, the convexity matters to existence-uniqueness in a very small class of potentials, which was the entire point of your paragraph.
>I'm was
Try speaking English for once, fucktard.
>Non convex lagrangians, make the correspondace with the hamiltonian, maybe not even well defined.
You mean the Legendre transform, which has absolutely nothing to do with existence-uniqueness? Did you get confused about the tabs of Wiki articles you're skimming? LMAO fucking pathetic idiot.
>Again, no matter how much you shout, there is no universally agreed "axiomatization" of mechanics,
Yes there is, it's called symplectic geometry. If you actually LEARN what it is you'd understand how powerful it is. To give a quick rundown:
>holonomic constraints = symplectic reduction
>constant energy surfaces = leaves of foliations
>phase space volumes = Liouville measure
>gauge groups = coadjoint orbit
>localization = moment map
and many more. Again, the entire point of symplectic geometry is to generalize and mathematically formalize analytical mechanics; just because you know jack shit about it doesn't mean it doesn't exist, fucking retard.
Calling your tardwrangler now, gimpboy.

[Anonymous No.9789678]

>>9789618
>>9789628
I talked about the hamiltonian? Again, the problem is not my, poor use of words and my poor english, but you are the dense retard who thinks that sympelctic geometry and the hamiltonian formalism are literally the only correct form of understanding mechanics.

Exactly, OP was changing the Lagrangian, but your response is obviously misguided and retarded, because it's not about the fucking form of the Lagrangian, because the principle of least action assumes that the Lagrangian IS FIXED FROM THE START so it has absolutely nothing to do with the functional equation, but with the fucking procedure using variational calculus ffs.

Sorry if you understood natural pairing with canonical, but I really have nothing to say to someone who can't understand that fucking symplectic geometry is not used in Lagrangian formalism.
>Holonomic constraints = submanifold by regular value theorem
>lagrangian, function from the tangent bundle to the reals identified as kinetic energy - potential energy, the kinetic energy for general holonomic constraints is taken from the cartesian counterpart
>action functional from space of curves in the tangent bundle defined through the lagrangian
>E-L equations, equations of motion using the principle of least action

To summarize the problem here are two. You don't understand how the form of the lagrangina is irrelevant to the princle of least action, as the action is defined with a FIXED functional, so the issue has to do with interpreting calculus of variations. And for some reason you don't understand that Lagrangian mechanics and Hamiltonian are fucking DUAL and you can begin with one and derive the other with a natural identification of the tangent and cotangent bundle and they are both formal and equivalent. I didn't want to use holonomic constraints as it's too strong so I thought your fucking autism would be triggered, so I just assumed a fucking smooth manifold.

[Anonymous No.9789719]

>>9789678
And ok, to be more general, when you are talking about holonomic constraints, not necesarily you fix a value, so it could be also just a change of charts of usual coordinate space. But in most cases you usually use them when there are workless restrictions which is then just saying, the possible values for positions of the particles is restricted and it's usually a submanifold. I understand how this can be restrictive as you could have more general constraints, such as the particles only moving inside a cube, which has some issues with collisions with the boundry. and time dependent constraints make it even more difficult as a proper manifold has to consider this so it's not identified soley with position coordinates. This is just an attempt at formalization of the shakey notion of "generalized coordinates".

[Anonymous No.9790781]

>>9789051
Why are you posting that fucking right-wing idiot?

[Anonymous No.9790813]

>>9787755
When we stop talking and make antialiased graphs publicly aviable, we will understand.

[Anonymous No.9790885]

Brainlet here.
Why should the area under the curve be minimal? I understand that nature tends to do things as simply as possible, but what exactly does the area under the curve represent? Why should it be minimal?
Also the orange part is considered negative area, yes? What does a negative area mean in this context?

[Anonymous No.9790942]

Engineering major.

We derive the laws of motion from experiment, then construct a theory to fit the experiment along with human concepts that help us make future predictions.

In this line of thinking, energy is not "real", but a human budgeting concept. In this case, from the motion curve s(t) = 1/2*g * t2 + v(0) * t + s(0) (s(0) and v(0) being the initial position and velocity) along with newton's laws that link acceleration to force, and the energy as the path integral of force along the direction of motion, we arrive at the curve on Op's left picture.

The fall of a ball could also be instantaneous (arguably the extreme case of principle of least action), then our formula above would not apply. But in the macroscopic world, it is not.

[Anonymous No.9791146]

>>9790885
The area used in the OP is a way of saying integration. But as it's been discussed here it's not a simple. Basically if you have some sort of weird fubction that gives you possible trajectories and gives you back the integral of it's kinetic energy - potential energy. And the trajectory of the physical system must minimize this. The exact reason to this is somewhat mysterious, because the thing is that it works for all fucking systems we know, but a pure classical perapective, you can sort of derive the result from a more tangible princple https://en.m.wikipedia.org/wiki/D%27Alembert%27s_principle

[Anonymous No.9792947]

jesus this thread. I particularly like the posters that obviously don't even know what the principle of least action is, yet still feel the need to post an 'answer'

[Anonymous No.9792950]

>>9792947
So, care to post your views/interpretation in laymans or correct any of the posters?

[Anonymous No.9793919]

>>9792950
no, I was too late, there are already people who did it itt.

[Anonymous No.9794013]

>>9788348
Enlighten us faggot