Why does the principle of least action.. work

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Other than its maths why does it intuitively work.It seems such a weird way of deriving F=ma (from the langrange).Furthermore it seems this was just someone throwing formulas and seeing what came back from it.Damnt Euler was a genius.
 

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Other than its maths why does it intuitively work.It seems such a weird way of deriving F=ma (from the langrange).Furthermore it seems this was just someone throwing formulas and seeing what came back from it.Damnt Euler was a genius.

Can u explain in laymen terms? how is it applied
 

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Much of the notation used by mathematicians today - including e, i, f(x), ∑, and the use of a, b and c as constants and x, y and z as unknowns - was either created, popularized or standardized by Euler. His efforts to standardize these and other symbols (including π and the trigonometric functions) helped to internationalize mathematics and to encourage collaboration on problems.


This makes sense. He set up a syntax rule that anyone can understand. Constant numbers like the measurement of a certain distance, time, shape, angle, circumference, things u have a definite number for after measuring it. Then the unknowns with a certain syntax the things u wanna find out based on your measurements or what your applying the maths too like taking a leaf shape and applying it to a garden. The unknown is the garden, the leaf can be measured unless there is some unknown in there somewhere. But it breaks ur problem down into unknowns and knowns really.

I simply skip this part when really doing the maths but if I need help on it I will definitely use it so it brings colloboration on the problem from others.

Mind you that is not solving anything the syntax. It's just presenting the problem into a language people can understand, it isn't going to magically give u an answer to your problem but sets your problem into an algorithm that people can understand, this is where I respectfully disagree with @Naissur

Syntax I think he thinks will give him an answer, trust me real maths is 'paper loads' of figures and confusion untill it's simplified into a nice equation or answer. Real maths isn't sitting there and looking at the problem and deciding what is known figures and unknown figures and assigning it the right letter. Come on god dammit, @Naissur I really thought u were better then this. Me and u can figure out what is known and unknown in a leaf and garden pretty quickly and assign letters cause we know what we wanna 'do' which is translate the leaf design into the garden size. We know the garden size and leaf size are different. So an unknown clearly is adjusting leaf measurements to the garden itself. Other unknowns can be accurately measuring the curve on the leaf, and measuring the different points of the leaf cuz it gets smaller and bigger in different areas, the over-all space within the leaf. U can look at what's stopping u from achieving ur goal of making the garden into a leaf in 5 minutes and probably another 5 minutes breaking into the unknowns and known language of algebra.

I am disgusted with my brother @Naissur and refused to respond to him in the science thread.
 
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Can u explain in laymen terms? how is it applied
That's what I'm asking people to explain to me.The best way I can explain it without intense maths is that say you have a quantity S which is action.Now assume you have a "path" throughout that path you have certain relations.Or basically at a point you have a certain amount of energy, certain amount of difference in energy, certain amount of some property.Now the if you add up all the relations in the path , so at 1 you have 5, at 2 you have 6.Now sum them all up with respect to some axis(time, etc doesn't matter).Finally take a path which is the absolute minimum of that action.That path is always preferred.

For example light always gets a to b in extremely short amount of times when not affected by gravity , why?

why is dT/dx = 0?

It is the basis of all forms of classical physics that is the basis of Hamiltonian,langragian and Newtonian physics.It was also used in general relativity to formulate the theory effectively.It seems to be part of the universe.
 
Much of the notation used by mathematicians today - including e, i, f(x), ∑, and the use of a, b and c as constants and x, y and z as unknowns - was either created, popularized or standardized by Euler. His efforts to standardize these and other symbols (including π and the trigonometric functions) helped to internationalize mathematics and to encourage collaboration on problems.


This makes sense. He set up a syntax rule that anyone can understand. Constant numbers like the measurement of a certain distance, time, shape, angle, circumference, things u have a definite number for after measuring it. Then the unknowns with a certain syntax the things u wanna find out based on your measurements or what your applying the maths too like taking a leaf shape and applying it to a garden. The unknown is the garden, the leaf can be measured unless there is some unknown in there somewhere. But it breaks ur problem down into unknowns and knowns really.

I simply skip this part when really doing the maths but if I need help on it I will definitely use it so it brings colloboration on the problem from others.

Mind you that is not solving anything the syntax. It's just presenting the problem into a language people can understand, it isn't going to magically give u an answer to your problem but sets your problem into an algorithm that people can understand, this is where I respectfully disagree with @Naissur

Syntax I think he thinks will give him an answer, trust me real maths is 'paper loads' of figures and confusion untill it's simplified into a nice equation or answer. Real maths isn't sitting there and looking at the problem and deciding what is known figures and unknown figures and assigning it the right letter. Come on god dammit, @Naissur I really thought u were better then this. Me and u can figure out what is known and unknown in a leaf and garden pretty quickly and assign letters cause we know what we wanna 'do' which is translate the leaf design into the garden size. We know the garden size and leaf size are different. So an unknown clearly is adjusting leaf measurements to the garden itself. Other unknowns can be accurately measuring the curve on the leaf, and measuring the different points of the leaf cuz it gets smaller and bigger in different areas, the over-all space within the leaf. U can look at what's stopping u from achieving ur goal of making the garden into a leaf in 5 minutes and probably another 5 minutes breaking into the unknowns and known language of algebra.

I am disgusted with my brother @Naissur and refused to respond to him in the science thread.
I don't see maths as a language as that implies its singular, rather think of our maths as being a language.Maths itself could have used completely different symbols but essentially if you can translate one maths to another maths it is consistent. Mathemetics has certain axioms which for some reason describe the universe beautifully.Albeit someone without a maths language as no hope of understanding the universe.maths is the most efficient way into understanding the universe , however it seems the universe is limited.Maths goes on to infinity , pure maths comes into play here.
 

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I don't see maths as a language as that implies its singular, rather think of our maths as being a language.Maths itself could have used completely different symbols but essentially if you can translate one maths to another maths it is consistent. Mathemetics has certain axioms which for some reason describe the universe beautifully.Albeit someone without a maths language as no hope of understanding the universe.maths is the most efficient way into understanding the universe , however it seems the universe is limited.Maths goes on to infinity , pure maths comes into play here.

Well we agree at least numbers go into infinity either direction up and big like the universe and into small grains of a beach sand. U agree the numbers are going into infinity either direction, I hope? plus do we agree any maths or problem will use arimethic functions addition/multiplication/division/substraction to find the answers? Do we accept those arithmetic functions is what we are going to use for any problem? The problem might be different chemicals, heat, energy, cold, wind, air, time, shapes, distances, etc all these and many more other forms of problem are the problem only and not the answer?
 
Well we agree at least numbers go into infinity either direction up and big like the universe and into small grains of a beach sand. U agree the numbers are going into infinity either direction, I hope? plus do we agree any maths or problem will use arimethic functions addition/multiplication/division/substraction to find the answers? Do we accept those arithmetic functions are we are going to use for any problem?
Numbers going to infinity mean nothing.Its just our curiousity manifesting into an extremity.

Nothing special about arithmetic properties they are like commas, fullstops etc.The more advanced our mathematical skils the more operators we have.Calculus was the invention of more operators.This is why it propelled society into a new golden age of physics/mathematics.
 

DR OSMAN

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Numbers going to infinity mean nothing.Its just our curiousity manifesting into an extremity.

Nothing special about arithmetic properties they are like commas, fullstops etc.The more advanced our mathematical skils the more operators we have.Calculus was the invention of more operators.This is why it propelled society into a new golden age of physics/mathematics.

I am not sure what influenced Calculus, but I suspect a flat ruler for example can't measure a curve cause the curve or circles or any shape that doesn't have some sort of flat surface, you needed some sort of other mechanism to measure it accurately and I guess they came up with points like little dots or | | or markers cause u can measure that piece by piece in the curvature and come with an over-all sum at the end but also can see the difference in numbers at each point cause the curve goes up and down not at the same dip. {{{{{{{ i mean look at that shape, how is a straight ruler going to answer it cause the figure clearly shows different points.

I guess this also applies to anything, the ingredients in a beach sand inside of it could be highly complex and different levels of different matter. I am not sure u understand the importance of infinity if u know if it can go to a grain in the sand and all the way up and out into the universe. U know the benchmark your playing in and everything else falls in somewhere in between. U realize how u measure things will be tiny or damn large and need to come up with ways that are innovative and accurate.
 
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Other than its maths why does it intuitively work.It seems such a weird way of deriving F=ma (from the langrange).Furthermore it seems this was just someone throwing formulas and seeing what came back from it.Damnt Euler was a genius.
Find Feynman's lectures on physics. He gives a very nice explanation. But in general, it's not very intuitive result (though of course beautiful)! Also, always keep in mind John Vaughn Newman's remark: "In mathematics you don't understand things. You just get used to them."
why is dT/dx = 0?
Obviously I don't have the context, but isn't this from when they are finding the minimum?

If so the derivative isn't equal to zero; rather, they set it equal to zero to find the minimum.
 
I am disgusted with my brother @Naissur and refused to respond to him in the science thread.
Isn't this bit of an overreaction, brother? :icon mrgreen: I enjoyed reading that thread, and didn't mean to cause any 'disgust'.

My point was that this stuff is not trivial. It's very hard to solidify. For example, how does one arrive at a coherent concept of real numbers? Forget about curves first, because before you even get to that, your tools have to be solid. What are numbers? All the constructions of the real numbers I know (Dedekind cuts, Cauchy sequences etc) require seemingly prophetic intuition and took some of the most brilliant minds ever to come up with them.

So surely it makes sense when I suggest you learn them instead of trying to rediscover them yourself? Take infinity for example. You absolutely need to define what this is meant by to resolve seemingly paradoxical scenarios (like Zeno's paradox), and for calculus to make complete sense. Take for example when you were slicing up curves then adding up the slices to find the area. You don't get the area unless you take limit that goes to infinity (to make sure your slices "fill" the entire curve). So again, you need to define limits.

None of this is easy to come up with. And it's not like mathematics is short of interesting results to rediscover. You can try to rediscover some amazing results after you have covered the foundations. There are actually schools of thought dedicated to this type of learning, but none that I'm aware at the foundation level. There is something called The Moore Method, whereby students are instructed to come up with hard theorems by themselves, and books written for that method. So you can try things like that. I hope this makes sense.
 
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Find Feynman's lectures on physics. He gives a very nice explanation. But in general, it's not very intuitive result (though of course beautiful)! Also, always keep in mind John Vaughn Newman's remark: "In mathematics you don't understand things. You just get used to them."
Obviously I don't have the context, but isn't this from when they are finding the minimum?

If so the derivative isn't equal to zero; rather, they set it equal to zero to find the minimum.
Are they on youtube?

A quick flick found that this was due to the quantum world , but still hardly intuitive.
 
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