Questions on the calculus of variations, a subfield of calculus that deals with the optimization of functionals.
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Existence of variation
Let $I[w] =\int_U L(Dw,w,x) dx$. Let $1<q<\infty$, and there exist constants $\alpha>0$,$\beta\ge0$ such that $$L(p,z,x)\ge \alpha |p|^q - \beta$$
This implies that if $I[w]$ exists,
$$I[w] ...
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vote
1answer
27 views
Finding Euler-Lagrange equations
Maybe you can help here. There is kind of a lengthy setup to understand what the question is asking. There is a paper I'm reading, and in one section of it I can't make heads or tails of the result. ...
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0answers
35 views
Bolza example like Question
I have to find $u$ minimizing $\int_0^1 F$ with $F(x,u(x),u'(x)) = (1-(u'(x))^2)^2+(u(x))^2$ with $u(0) = 0$ and $u(1) = 1$.
I'm relatively new to CoV and got told i should try ...
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0answers
59 views
minimization of function $F(a) = \int_0^1 (G(x) - P_a(x))^2\,dx$?
I have the following questions referring to this link to a previous question on this site : Approximate a function over the interval $[0, 1]$ by a polynomial of degree $n$ (or less).
a) Explain why ...
1
vote
2answers
37 views
Vector Field Generating Variation Along Curve
I'm learning a proof of the fact that length extremising curves are geodesics of the Levi-Civita connection, and have found something I don't understand. The argument states the following.
Suppose ...
1
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1answer
51 views
Curvature and Torsion problem
Calculate the curvature and torsion of
$$x= e^t\sin(t),\quad y= e^t\cos(t),\quad z= e^t$$
I'm not sure if I am doing this correctly since I am getting quite complicated results.
But I understand ...
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0answers
17 views
Question on a third-order boundary value problems
This is the corollary $2.1$, from the article "Positive solutions of third order semipositone boundary value problems"
if $$u'''=\lambda \left(\sum_{i=1}^m c_i(t)u^{\mu_i}-d(t)\right)+e(t), t\in ...
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2answers
58 views
Euler lagrange equation is a constant
I'm working through exercises which require me to find the Euler-Lagrange equation for different functionals.
I've just come across a case where the Euler Lagrange equation simplifies to
$$1=0.$$
...
1
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1answer
57 views
+200
Who was responsible for finding sufficient conditions for functional extrema?
In the calculus of variations, there is a well-known sufficient condition for weak functional extrema, involving conjugate points and the strengthened Legendre condition ($f_{y'y'} > 0$). Who was ...
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0answers
16 views
Calculus of Variations statement of a Singular Value Decomposition?
My previous question on SVD gained very little traction, so I thought I'd try a different version that hopefully has an explicit solution. As noted in the linked question, I am taking a function of ...
1
vote
1answer
35 views
Finding the Euler Lagrange equation - differentiation
I'm teachin myself the basics of Calculus of variations. So far I know how to calculate the Euler Lagrange equation for simple functionals.
I'm now stuck on how to compute the total differentiation ...
3
votes
1answer
53 views
Local and global extremes
I Wrote problems and solutions, I need just few explanations.
1.Let
$$J(x)=\int_{0}^{1}x'^{2}dt,\quad x(0)=0, x(1)=1. $$
Find the extrema value for $J$.
I'm doing this using Euler equation ...
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0answers
35 views
Green's function
Please can someone told me how to find the Green's function $G(t,x)$ of BVP : $$u'''(t)=0 , \quad t\in (0,1)$$ and BC : $$u(0)=u'(p)=\int_q^1 w(s)u''(s) ds =0 $$ where $\frac12 < p<q<1$ are ...
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23 views
Need an application of Morse theory for second-order differentialle systems
I'm looking for some applications of Morse theory for the second order differentialle systems,
Someone can help me with a pdf or a book or an article which has these applications ?
Please
Thank ...
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0answers
42 views
Levi-Civita Connection for 2-dimensional Riemannian manifold
I'm trying to show the following. Suppose $(M, g)$ is a $2$-dimensional Riemannian manifold with connection $\nabla$. Suppose also that $\nabla$ is metric compatible, and that length extremizing ...
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0answers
19 views
Second variation positive definite but not weak local minimum?
Consider a functional $J: S \to \mathbb{R}$ where $S \subseteq C^2[a,b]$. Let $J(y) = \int_a^b f(x,y,y') \, dx$, let $y$ be an extremal (solution to the Euler--Lagrange equation) for $J$, and suppose ...
1
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1answer
27 views
Analysing functionals having no local extrema
In the calculus of variations, how do we analyse functionals for which there are no local extrema? In basic calculus, functions not having local extrema can often be seen to diverge to an infinity ...
2
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1answer
34 views
Multiple Integral Equation
$$f(x) = 2a \int_{0}^{x}{f(t)\;dt} - \left(\frac{b^2}{2}\right)\int_{0}^{1}{|x-t|f(t)\;dt}$$
where $0<a<b$
My task is to solve for $f(x)$. I'm having difficulty solving this integral equation. ...
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0answers
23 views
Prove the property that integral of variation of function is equal to variation of integral of function.
Prove this property that variation of integral of a function is equal to integral of variation of that function.Note Here function is actually a functional i.e. function of functions.
2
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1answer
79 views
Sufficient conditions for functional extrema
In the calculus of variations, we can develop a sufficient condition for a functional $J: S \to \mathbb{R}$,
$$J(y) = \int_a^b f(x,y,y') \, dx$$
to have a local maximum or minimum, where $S \subseteq ...
3
votes
2answers
83 views
Derive the solution to the Lagrangian $ \mathcal L= y(x)\sqrt{1+y'(x)^2}$
I am supposed to derive the solution to the Lagrangian $$ \mathcal L= y(x)\sqrt{1+y'(x)^2}$$
Unfortunately I am unable to solve both, the Euler Lagrange equation or the Beltrami equation. It may be ...
2
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0answers
71 views
Derivation of Euler-Lagrange equation
Here is a simple (probably trivial) step in the derivation of the Euler-Lagrange equation.
If we denote $Y(x) = y(x) + \epsilon \eta(x) $, I want to know why is
$\dfrac{\partial ...
3
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0answers
72 views
Green's function for third order boundary value problems
How to find the Green's function $G(t,x)$ for the BVP consisting of the equation :
$$u'''(t)=0 , \quad t\in (0,1)$$
and BC :
$$u(0)=u'(p)=\int_q^1 w(s)u''(s) ds =0 $$
where $\frac12 < ...
0
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0answers
38 views
Minimize a three variable function using Euler-Lagrange theorem
I have to minimize the function $g(x,y,z)=x^2+y^2+2z^2-x-yz$ in two cases. First, with the restriction $x+y+z=35$, and after with $x+y+z\geq35$
I know how to do this using Lagrange multipliers ...
0
votes
1answer
57 views
Gateaux derivative
I have the following definition of Gateaux differentiability
$f$ is Gateaux differentiable at $x_0$ if there is a continuous and linear operator $T$ so that
$$ \lim_{t \rightarrow ...
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0answers
34 views
Calculus of variations-fields and weierstraß excess function.
if i have a lagrangian $$L (t,x(t),y(t),\dot{x}(t),\dot{y}(t))$$ that depends on two functions and one parameter.
Then I will get two Euler-Lagrange equations as a test for extrema. Let us assume ...
0
votes
1answer
23 views
Identity between functions
Let $f$ and $g$ be continuous functions of one real variable. We want to show that $\frac{\mathrm{d}}{\mathrm{d}t}f = g$ on the interval $[a,b]$. I have shown that for any subinterval $[t_a,t_b] ...
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0answers
33 views
Lagrange Multipliers and max area
You have a straight line of length b. You want to connect the ends of this fence so as to enclose a maximum area. You have a cost constraint. In the area between x=0 and x=b/2 costs 1 dollar per ft ...
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1answer
32 views
Why is the weak limit of the derivatives the derivative of the weak limit here?
In [1, chapter 8.2.1.b, p.466] the author uses the following argument:
Let $U \subset \mathbb{R}^N$ be an open, bounded domain with smooth boundary. Given a bounded sequence $(u_k)_{k \in ...
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1answer
25 views
Constrained variational calculus: Are we allowed to make use of the constraint before taking variations?
Suppose that we have a variational problem,
$\int_{t_1}^{t_2}f(\vec{x}(t),\vec{x}'(t),|\vec{x}(t)|)dt$
subject to the constraint:
$|\vec{x}|=1$
where $\vec{x}(t)=\left\{x_1(t),x_2(t),x_3(t) ...
4
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0answers
63 views
Optimizing a functional with a differential equation as a constraint
I am working on solving the following optimization problem. I think it is well-poised but, if not, please give me some pointers that could make the question make more sense.
We have a parametric ...
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Composition of bounded variation functions
Assume $f\in BV[a,b]$ and $g : [c,d]\rightarrow[a,b]$ is increasing, continuous, and onto. Prove that $F:=f\circ g\in BV[c,d]$ and $V^b_a f=V^d_c F$
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1answer
31 views
Euler-Lagrange Query
Given F:
$$ F(x,y,y\prime) = 2\cdot \pi \cdot y \cdot \sqrt{1+(y\prime)^2} $$
We can derive the following Euler-Lagrange equation (I know how to do this part):
$$ \frac{d}{dx}\left(\frac{y\cdot ...
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1answer
35 views
Simple question on calculus of variations: critical point of functional subject to constraint
Let $V$ be the set of smooth functions $f:[0,1]\to \mathbb{R}$ such that $\int_0^1 f(t) dt =k$. If $F:V\to\mathbb{R}$ is given by $F(f) = \int_0^1 f(t)^2 dt$, show that the only critical point of $F$ ...
2
votes
1answer
47 views
Finding the critical point of $\int_0^1(f(t))^2dt$ subject to $\int_0^1f(t)dt=k$
I wish to find the critical point of the functional $F[X]=\int_0^1(f(t))^2dt$ subject to $\int_0^1f(t)dt=k$ for a constant $k$. I read something about using a Lagrange multiplier to convert it to a ...
3
votes
1answer
85 views
Taylor series with functions as parameters (as opposed to variables)
I'm doing my own research on the Euler-Lagrange equation and came across a proof in van Brunt's textbook "The Calculus of Variations". However, there is something I don't quite understand.
Here is an ...
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0answers
23 views
Prove a transformation is a variational symmetry for J
The following problem is from The Calculus of Variations by B.von Brunt (page 215, Exercise 9.2.1)
Let
$$
J(y)=\int_a^b xy'^2\mathrm{d}x.
$$
Show that the transformation
$$
X=x+\epsilon2x\ ...
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0answers
63 views
Extremal condition calculus of variations
if I have a functional with a Lagrangian $L(t,x(t),y(t),x'(t),y'(t))$, meaning two functions x and y of one parameter t. And want to solve the minimization problem $$ \int_0^t L \, dt. $$ Then I get ...
2
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1answer
37 views
Formal Variational Calculus Reference Request
I want to ask for a reference to study Variational Calculus from a formal point of view. What I mean is that many of the references that I've found are inside Physics books, and the authors do not ...
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55 views
Minimum calculus of variation
Hi I am looking for a criterion that is sufficient to prove that a solution to a functional depending on two functions y(t) and x(t) is an extremum.
it is about the following functional$$ \int_0^b ...
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1answer
48 views
Calculus of Variations-question on rotating curve of max volume
My calc of variations is still rusty. I'm assuming implementation of arclength revolution formula is necessary, but how to find y(1/2a)?
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0answers
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Calculus of Variations - Circles with soapy membrane problem [duplicate]
Calculus of variations is coming to me at a crawl pace. Here is a problem on my agenda that I wanted to get solved, but am not quite sure how to approach. I've been thinking about it for a while ...
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0answers
55 views
Calculus of variations: find $y(a/2)$ if $y(x)$ maximizes the volume of rotation
A curve $y(x)$ of length $2a$ is drawn between the points (0,0) and (a,0) in such a way that the solid obtained by rotating the curve about the $x$-axis has the largest possible volume. Find ...
2
votes
2answers
136 views
satisfy the Euler-Lagrange equation
Two circles of unit radius, each normal to the line through their centers are a distance d apart. A soap film is formed between themas shown below; energetic considerations require the filem to ...
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0answers
32 views
DP formulation for the shape of Water-drop?
There are different suggestions for the shape of water-drop such as Joukowskis and Piriform parametrization. I am trying to understand how the water-drop-shape is deduced.
Suppose a trivial case ...
2
votes
1answer
51 views
Show that the path of shortest hyperbolic length satisfies $(x-c)^2+y^2=r^2$
The hyperbolic length of a curve $y:[a,b]\rightarrow\mathbb{RxR}_+$ is given by the functional $$\lambda(y)=\int_a^b\frac{\sqrt{1+y'^2}}{y}dx$$ Show that the path of shortest hyperbolic length ...
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vote
2answers
48 views
Show that the infimum of a functional is zero, but this infimum is never achieved.
Show that the infimum of the integrals $$\int_0^1(y'^2(x)-1)^2dx$$ among all $y(x)\in C^2[0,1]$ such that $y(0)=y(1)=0$, is zero, but is not achieved by any function in this set. What I've worked on: ...
2
votes
3answers
178 views
Procedure for Gâteaux Derivative with functionals
Not after an answer, just the method/procedure as I'm stumped...
We have the functionals:
$$ T[y] = \int_2^3 \left( 3\left| \frac{dy}{dx}\right|^2 - 8y \right)dx $$
$$ S[y] = \cosh(T[y]) $$
Now, to ...
2
votes
2answers
43 views
In calculus of variation: why are minimizing sequences bounded?
Assume the usual variational setting: Let $\mathcal{A} \subset W^{1,q}$ be the set of admissible functions and \begin{equation} I: \mathcal{A} \to \mathbb{R} \end{equation} the functional that needs ...
0
votes
1answer
106 views
Weak Minimizer of a Functional
I showed that $u(x) = \frac{x^2}{2}$ is a potential minimizer for the functional $\int_0^2 \frac{n}{2}u'(x)^2-nu(x) \, dx$ in $C^2[0,2]$ with $u(0) = 0$ and $u(2)=2$ where $n$ is a positive constant ...
