Calculus of Variation - Classical Mechanics

In summary: Well, thank you very much! In summary, this conversation is discussing the 6th chapter of Classical Mechanics (Taylor), which covers the basics of calculus of variations. The discussion focuses on the derivation of the Euler-Lagrange equations and various confusions and questions about the process. The conversation also includes links to helpful resources and the offer of a virtual cookie as a reward for helpful explanations.
  • #1
Astrum
269
5
I'm reading Classical Mechanics (Taylor), and the 6th chapter is a basic introduction to calculus of variations. I'm super confused :confused:

I've tried to go to other sources for an explanation, but they just make it even worse!

So, let me see if I can get some help here.

[tex]\int^{x_{2}}_{x_{1}} f(y(x), y'(x), x)dx[/tex] - the integration of a function of three variables. y(x) is an as yet unknown curve. I understand that although f(y, y', x) is a function of three variables, it is only dependent on one variable, x. (where do these come from, exactly?)

Taylor then defines Y(x) = y(x) + η(x) is the WRONG path, where y(x) is the correct one. η is the variation of Y(x) from y(x). - why do we need to introduce the INCORRECT path?

Next, he introduces α into Y(x) = y(x) + αη(x). If we set α = 0, we will have Y(x) = y(x) - why do we need α?

Our integral now becomes: [tex]\int^{x_{2}}_{x_{1}} f(y(x) + αη(x), y'(x) + αη'(x), x)dx[/tex] - we're assuming that α is equal to 0? I'm not sure I 100% understand this step.

We need to check that [itex]\frac{dS}{d\alpha} = 0[/itex]- is this to check that α is a constant? or used as a way of making sure α = 0?

Take partial derivative: [tex]\frac{\partial f ((y(x) + αη(x), y'(x) + αη '(x), x)}{\partial \alpha}= \eta \frac{\partial f}{\partial \alpha}+ \eta ' \frac{\partial f}{\partial y'} [/tex] - because of the chain rule [tex]\frac{dS}{dα}=\int^{x_{2}}_{x_{1}}\frac{\partial f}{\partial α}dx = 0[/tex] -

Next he works some voodoo magic by using integration by parts on the integral. I haven't worked this step out myself, but I assume it's straight forward.

So, in the end, we get: [tex]\frac{\partial f}{\partial y}-\frac{d}{dx}\frac{\partial f}{\partial y'}= 0[/tex]

So, I'm pretty lost. I think it would help if I understood the idea of what we're really doing here. This is essential arc length along the shortest curve, but all the additional variables and what not are confusing the hell out of me.

Sorry for the long post, but any help is much appreciated. I'm so desperate, I'm offering a reward of one (1) virtual cookie to the first helpful post.

NB - I put this in the Classical Physics section, because I'm more concerned with how this is used in mechanics right now. Although I'm interested, in what course is Calculus of Variation taught in at a rigorous level?
 
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  • #3
What part of Taylor is this exactly? Is it related to a problem? Or is it the derivation of the Euler-Lagrange equations? If it is the latter then I must say it is one of the most convoluted ways of deriving the equations I've ever seen; the standard method of taking a first variation is a lot more intuitive and straightforward.
 
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  • #4
jedishrfu said:
This writeup on the Brachistochrone problem may help:

http://www.hep.caltech.edu/~fcp/math/variationalCalculus/variationalCalculus.pdf

Thanks, checking it out now.


WannabeNewton said:
What part of Taylor is this exactly? Is it related to a problem? Or is it the derivation of the Euler-Lagrange equations? If it is the latter then I must say it is one of the most convoluted ways of deriving the equations I've ever seen; the standard method of taking a first variation is a lot more intuitive and straightforward.

This is the derivation of the Euler-Lagrange equations. I'm über confused about it.

Starts on page 218

Here's your virtual cookie, by the way [Broken]
 
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  • #5
See here instead: http://www.colorado.edu/engineering/CAS/courses.d/ASEN5022.d/Derivation%20of%20%20E-L%20Equation.pdf
 
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  • #6
I'm looking into the link first posted. I'm befuddled by line 3 on page 4. But other than that, It it explained it much better than in Taylor's book.

Although I'm still confused by why the function is f(y,y', x). y = this is the curve, of which the constraints are on. y' is the rate of change of this curve, and x is what all of this is dependent on.

[tex]\int^{x_{2}}_{x_{1}} f(y(x), y'(x), x)dx[/tex] this gives us the length of the shortest path? What exactly is f(y,y', x)?
 
  • #7
WannabeNewton said:
See here instead: http://www.colorado.edu/engineering/CAS/courses.d/ASEN5022.d/Derivation%20of%20%20E-L%20Equation.pdf

That looks like a really good link too.
 
  • #8
Astrum said:
Thanks, checking it out now.




This is the derivation of the Euler-Lagrange equations. I'm über confused about it.

Starts on page 218

Here's your virtual cookie, by the way [Broken]

Oh nooo! WannaBeNewton's going to get to the cookie first. Are there seconds?
 
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  • #9
jedishrfu said:
That looks like a really good link too.
Apologies if you thought I meant to take a look at my link as opposed to yours. I meant to take a look at my link as opposed to the explanation in Taylor's text, which I found to be atrocious.
 
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  • #10
WannabeNewton said:
Apologies if you thought I meant to take a look at my link as opposed to yours. I meant to take a look at my link as opposed to the explanation in Taylor's text, which I found to be atrocious.

No apology needed. Can we split the cookie? I liked your link too. It was more to the point.
 
  • #11
Astrum said:
What exactly is f(y,y', x)?
It is a ##C^{1}## real valued function of ##y(x)##, ##y'(x)## and ##x##. It can be various things depending on what problem you are working on. For example, it could be the arc-length ##f(y,y',x) = \sqrt{1 + y'^{2}}## of a curve ##y(x)## or, more pertinent to mechanics, the Lagrangian ##L(q(t),\dot{q}(t),t) = \frac{1}{2}m\dot{q}^{2} - U## of a particle of mass ##m## and trajectory ##q(t)## interacting with a potential ##U##.
 
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  • #12
jedishrfu said:
No apology needed. Can we split the cookie? I liked your link too. It was more to the point.
I'd rather you take the whole thing because I'm eating chips ahoy as we speak xDD :tongue:
 
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  • #13
WannabeNewton said:
I'd rather you take the whole thing because I'm eating chips ahoy as we speak xDD :tongue:

Okay, thanks. I just finished my peanut butter sandwich (midnight snack).
 
  • #14
Many thanks for those links, they really cleared things up for me!

As for cookies, have some chemistry cookies:
partyperfect_science%2Bparty.jpg


I need to go and work some problems out, if I have any other questions, I'll be sure to post.
 
  • #15
Chemistry cookies? I feel cheated :[
 
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  • #16
WannabeNewton said:
Chemistry cookies? I feel cheated :[

http://www.math.umn.edu/~olver/am_/cvz.pdf this link is what I tried to use originally, was kinda confusing.

Alright alright. Have some Einstein chocolate
5754428326_0631d7f69d_z.jpg


Never happy!

So, what kinda of class is this taught in, anyway? In a rigorous mathy style
 
  • #17
In a class called calculus of variations, believe it or not lol.
 
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1. What is the purpose of Calculus of Variation in Classical Mechanics?

Calculus of Variation is a mathematical tool used to find the optimal solution to a functional, which represents a physical quantity in classical mechanics. It helps us to determine the path or trajectory that a system will take to minimize or maximize a specific physical quantity, such as time or energy.

2. How does Calculus of Variation relate to the principles of classical mechanics?

Calculus of Variation is based on the principle of least action, which states that a physical system will follow the path that minimizes the action integral. This principle is closely related to the fundamental principles of classical mechanics, such as the principle of least potential energy and the principle of least kinetic energy.

3. What are the key concepts in Calculus of Variation in Classical Mechanics?

The key concepts in Calculus of Variation include the functional, the action integral, the Euler-Lagrange equation, and the boundary conditions. The functional represents a physical quantity, the action integral is the integral of the functional over a certain time interval, the Euler-Lagrange equation is used to find the optimal solution, and the boundary conditions define the starting and ending points of the system.

4. What are some real-life applications of Calculus of Variation in Classical Mechanics?

Calculus of Variation has numerous applications in classical mechanics, including the study of motion of particles, oscillatory systems, and rigid bodies. It is also used in the development of mathematical models for various physical systems, such as pendulums, springs, and celestial bodies.

5. What are the limitations of Calculus of Variation in Classical Mechanics?

Although Calculus of Variation is a powerful tool in classical mechanics, it has some limitations. It can only be applied to systems that can be described by a single parameter, and it assumes that the system follows a continuous path. Additionally, it does not take into account external forces or constraints, which may affect the motion of a system in real life.

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