Calculus of variations Definition and 154 Threads

Homework Statement A rectangular solid of height h increases in density as its height increases, so the index of refraction of the solid increases with height according to: n(y) = 1.20(2y + 1) where y is the distance, in meters, from the origin (see diagram). A beam of light...

Hi, I would like to know if anyone has good ideas for problems involving calculus of variations, other than the classic textbook questions (brachistochrone, Fermat, catenary, etc..) that I could create as a classical mechanics class project? Thank you

I need to find the minimum of this integral F=∫ (αy^-1+βy^3+δxy)dx where α, β and δ are constant; y is a function of x the integral is calculated over the interval [0,L], where L is constant I need to find the function y that minimizes the above-mentioned integral The integral is subject to...

In Lagrangian mechanics, can anyone show how to find the extrema of he action functional if you have more constraints than degrees of freedom (for example if the constraints are nonholonomic) using Lagrange Multipliers? I've looked everywhere for this (books, papers, websites etc.) but none...

The following theorem is called "the 'basic lemma' of the calculus of variations" on page 1 of this book: "If f is a continuous function in [a,b] s.t. ∫abη(x)f(x)dx = 0 for an arbitrary function η continuous in [a,b] subject to the condition that η(a) = η(b) = 0 then f(x) = 0 in [a,b]" If...

For the integral J =\int f(y,y_x,x) dx if f(y,y_x,x) = y^2(x) find a discontinuous solution similar to the Goldschmidt solution. This is the first time I approach the calculus of variations, so I thought of using the Eulere equation f - y_x\frac{\partial f}{\partial y_x} But I...

Hi, I am working on a calculus of variations problem and have a general question. Specifically, I was wondering about what kind of constraint functions are possible. I have a constraint of the form: f(x)x - \int_{x_0}^x f(z) dz \leq K If I had a constraint that just depends on x or...

Hello, this should be an easy one to answer, hope it's in the right place. I'm going through Sean M. Carroll's text on General Relativity, "Spacetime and Geometry." I'm working through calculating Christoffel connections (section 3.3, if you happen to have the book), which Carroll...

I'm trying to find the function f which maximizes this integral: I'm not quite sure how to handle a problem like this with a second variable (r) which is implicitly determined by a constraint. Can anyone help? Thanks.

Calculus of variations problem. I want to make stationary the integral of (1+yy')^2 dx from 0 to 1. I know what the Euler-Lagrange differential equation turns out to be, but how do I interpret the limits of integration as initial conditions for the diff eq? also, can i use laplace transforms to...

It seems like a problem that a physicist would need to solve, but I can't wrap my head around the physical interpretation of it. http://exampleproblems.com/wiki/index.php/CoV7 Also, why do they use u=c*x2? What is c in this case? It says "classical" so it can't be the speed of light, right?

Homework Statement The problem requests to make stationary the integral: \int_{\phi_1}^{\phi_2} \sqrt{\theta'^2 + sin^2\theta}d\phi where \theta'=\frac{d\theta}{d\phi} Homework Equations The Attempt at a Solution I know how to start with the problem, and with two different methods I get the...

Homework Statement Suppose a ray of light travels from (x,y) = (-1,1) to (x,y) = (1,1) in a region where the index of refraction is n(y) = e^y. (a) Find the path. Homework Equations The Attempt at a Solution Is this okay? The positions of the light ray are given by initial...

I saw a nice formulation of the variation on odd dimensional manifolds in the paper of http://arxiv.org/abs/math-ph/0401046" : The referenced book of Arnold uses completely different formalism than this. I don't see clearly the connection between the traditional calculus of variations...

Calculus of variations (HELP!) Hi all! Just a question... How should I visualise geometrically the minimising of definite integrals, and what is the significance of finding stationary points of definite integrals? (Can someone provide me with an intuitive explanation?) Thanks so much!

Homework Statement Show that the shortest distance between two points in three dimensional space is a straight line. Homework Equations Principally, the Euler Lagrange equation. The Attempt at a Solution I understand how to do this for a plane, but when we move into three...

I can't get rearrange the last equation into a nice form to integrate with respect to x to minimize the surface area. http://i111.photobucket.com/albums/n149/camarolt4z28/2010-10-17122502.jpg?t=1287336976...

Homework Statement Fermat's principle establishes that the path taken by a light ray between 2 given points is such that the time that the light takes is the smallest possible. 1)Demonstrate that a light ray which propagates through a medium with a constant refractive index follows a straight...

Hi, I've seen the words "Calculus of Variations" mentioned quite a bit but never thought too much about them since it seemed too advanced. Well, I am nearly finished the computational style calculus and am awaiting my Apostol text to get more into the theory but I also picked up a text called...

Can someone please tell me what the best book for learning calculus of variations is?

Hi all. In my notes I wrote down from the blackboard, I wrote [Fundemental Lemma of the Calculus of Variations] Let f : [a,b] -> R be continuous and suppose that \int_a^b f(t)h(t)dt = 0 for all h\in C_{0,0}^1([a,b], R), where C_{0,0}^1([a,b], R) is the space of C1 parametrized...

Homework Statement Consider the functional defined by J(y)=\int_{-1}^1 x^4(y'(x))^2 dx Without resorting to the Euler-Lagrange equation, prove that J cannot have a local minimum in the set S=\{y\in C^2[-1,1]:\ y(-1)=-1,\ y(1)=1\}. The Attempt at a Solution I have thought...

AAAAHHH! Calculus of Variations Homework Statement See attached This is a project for an upper level math methods of physics course. My background is insufficient and ultimately, I don't know what is going on, AT ALL. The work I've provided is the product of the collective efforts of my...

I'm in dire need of help in understanding calculus of variations. My professor uses the Mathews and Walker text, second edition, entitled Mathematical Methods of Physics and, he has a tendency to skip around from chapters found towards the beginning of the text to those nearer the end. I...

for Newton's equation, we have Lagrange function which can give the solution while its variation equates 0. however, what about the situation for a general differential equation. is this method can deal with such general situation? and how can we find out the corresponding "Lagrange function"...

Please allow me to preface: I'm an undergraduate physics student at a small school where upper-level courses are on a two year rotation. So, I'm currently in an advanced Mathematical Methods course for which I lack prerequisites. I'm only concurrently enrolled in differential equations...

1. A uniform string of length 2 meters hangs from two supports at the same height, 1 meter apart. by minimizing the potential energy of the string, find the equation describing the curve it forms and, in particular, find the vertical distance between the supports and the lowest point on the...

Homework Statement Find the curve y(x) that extremizes the functional J[y]= int({1-y'^2}/y,x=a..b) if the end points lie on two non-intersecting circles in the upper half-plane. Homework Equations Euler's equation: if F=F(x,y,y') then Euler's equation extremization is found from...

Hi, Does anyone have any recommendations on an exceptional Calc of Variations text or other resource? The few I've been able to preview at amazon.com didn't impress me. Any recommendations would be appreciated. jf

I asked a question earlier about Calculus of Variation, but the question I gave didn't really highlight my confusion well. I've come across some other questions that I think reveal my misunderstanding. Homework Statement Solve the Euler equation for the following integral: (integral...

Just did this in class today and was doing a problem to see if I understood it and I'm not sure I did. Thanks for any help Homework Statement Solve the Euler equation to make the following integral: (integral from x1->x2) ∫ [(y')² + y²] dx Homework Equations Euler-Lagrange...

Homework Statement "An aircraft whose airspeed is vo has to tfly from town O (at the origin) to town P, which is a distance D due east. There is a steady gentle wind shear, such that v-wind = Vy(x-hat) [the wind shear is in the x direction...]. x and y are measured east and north respectively...

I'm looking for a derivation of the method of Lagrange multipliers as used in the calculus of variations for extremizing a functional subject to constraints. More specifically, I'm trying to understand the relationship between the "method of Lagrange multipliers" from standard calculus and the...

Homework Statement Optimize the following cost integral x(1)^2 + \displaystyle \int_0^1 (x^2 + \dot{x}^2) dx subject to x(0) =1, x(1) is free Homework Equations Now our prof showed us a method of doing this. In general, if we want to minimize f(b,x(b)) + \displaystyle...

Hello! I have a couple of questions concerning Lagrangian and Hamiltonian mechanics. First of all, are generalized velocities dq/dt (t) functionally depenent or independent of generalized coordinates q(t)? We vary them independently while deriving Euler - Lagrange equations, so it would...

In my Classical Dynamics text (Thornton & Marion), there's one step in the derivation of Euler's equation that I don't understand. I think if I understood it, I'd be able to derive the equation on my own. I wrote out the entire derivation up to the point I don't get, just so you guys would...

Hello everybody. Sorry, I don't know how to use TeX yet, I couldn't find a testing zone. Problem: Let I = \int_0^\infty [(dy/dx)^2 - y^2 + (1/2)y^4]dx, and y(0) = 0, y(\infty) = 1. For I to be extremal, which differential equation does y satisfy? Solution: The condition is that \delta I...

This is nearly vacuous thing to say, but there was just a post about the rigorous definition of area under a curve, and so I decided to go ahead and mention this.. Given a path (just say a continuous function) p(t):[a,b] \rightarrow \mathbb{R}^n , the "length" of the path is defined as...

Older textbooks on the Calculus of Variations seem to define the first variation of a functional \Pi as: \delta \Pi = \Pi(f + \delta f) - \Pi (f) which looks analogous to: \delta f = \frac {df} {dx} \delta x = lim_{\delta x \rightarrow 0} (f(x+ \delta x) -f(x)) from...

Hi everyone, I'm already familiar with, and have used Lagrangians and Euler-Lagrange equations. I'm interested in calculus of variations, but if it all boils down to solving euler-lagrange equations (and this is probably the part where I'm mistaken), then what's the point? Please tell me if...

Are there any advanced resources on the topic that go beyond the basic concepts. I'm interested in learning the more advanced applied and theoretical concepts(beyond euler and lagrange).

Light travels in a medium in which the speed of light c(x,y) is a function of position. Fermat's principle states that the time required for light to travel between two points is an extremum relative to all possible paths connecting the two points. 1) Show that the time for the light to travel...

I've been looking at this example for a while now. Could someone help? "Take the functional to be J(Y) = \int_{a}^{b} \( \alpha Y'^2 + \beta Y^2) dx For this F(x,y,y') = \alpha y'^2 + \beta y^2 and p = \frac{ \partial F}{\partial y'} = 2 \alpha y' \Rightarrow y' =...

I need a good calculus of variations book. I would like something that is clear but not devoid of mathematical rigour.

I am trying to learn the calculus of variations, and I understand the mathematical derivation of the Euler-Lagrange equation. As I understand it, the calculus of variations seeks to find extrema for functions of the form: S[q,\dot{q}, x] = \int_{a}^{b} L(q(x),\dot{q}(x), x) \,dx. Here is my...

I need to find the maximum value of A[y(x)]= \int_{0}^{1}y^2 dx with boundary conditions y(0)=y(1)=0 and \int_{0}^{1}(\frac{dy}{dx})^2=1 Do I have to use the Euler lagrange equations? I thought that found the minimum value?? Any hints on the steps to take would be appreciated.

I'm doing my final year in maths and am just away to start my 4th year project. It involves learning a subject on my own then submitting a report and doing a presentation. The topic I have to do is "Calculus of variations". I've been reading about the topic briefly on a few webpages and it...

Could anybody recommend any texts on Calculus of Variations? Unlike most areas of mathematics I'm finding it difficult to obtain standard texts.

this is one of those things that looks like it should be really simple but for some reason i just don't get it :confused: I've looked at a few books and they all start explaining calculus of variations in the same way. i'll quote a paragraph from feynmann lectures II (concerning finding the...

Presume the Earth is spherical, homogeneous and of radius R. What should be the shape of a tunnel connecting two points on the surface in order to minimize the time it takes for a particle to travel between the two points. I have had a go at doing it in both polar and cartesian co-ordinates...