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?
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?