Questions on the calculus of variations, a subfield of calculus that deals with the optimization of functionals.
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What is a differentiable functional?
I saw in the article Alt, H. M. and Caffarelli, L. A. Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math., 325, (1981), 105–144. That the functional
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116 views
Prove that a flat shape minimizes a functional
The following functional arises in an information theoretic problem that I work on currently.
$$I(G(\omega)) = \int_{-\kappa\pi}^{\kappa\pi} \frac{A}{G(\omega)+A}d\omega-\frac{\left| ...
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142 views
Pre-requisites for the Calculus of Variations
I'm interested in working through the book : "Calculus of Variations" by Gelfand and Fomin. However, I lack the pre-requisites to do so (I'm familiar with linear algebra and one-variable calculus ...
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99 views
Who came up with the Euler-Lagrange equation first?
Could someone explain who came up with the specific equation first?
http://en.wikipedia.org/wiki/Euler-Lagrange
makes it sound like Lagrange got it first, in 1755, then sent it to Euler.
but:
...
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1answer
118 views
variation problem of constrained area and minimized distance
$$c=\int_{x_1}^{x_2}f_{gr}(x)\;dx$$
The integral is a time-like curve between $x_1$ and $x_2$ and at imagine fgf(x1) is a lower left corner of the rectangle and fgf(x2) is the upper right corner and ...
3
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1answer
970 views
Simple simple Euler Lagrange Equation
Just starting a course on Lagrangian Mechanics and I'm just wondering what about the Euler-Lagrange equation, and more specifically what I'm meant to be trying to do ..
One of the questions from my ...
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2answers
77 views
Am I allowed to move around an operator like this?
Can I take this product:
$$\frac{dL}{dt}\frac{d L}{d \dot{x}}$$
And factor out one of the $L$'s to get:
$$L\frac{d}{dt} \left( \frac{d L}{d \dot{x}}\right)$$
Where the operator $\frac{d}{dt}$ now ...
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1answer
122 views
Why can't we construct a counter-example to the Fundamental Lemma of the Calculus of Variations?
"The fundamental lemma of the calculus of variations states that if the definite integral of the product of a continuous function $f(x)$ and $h(x)$ is zero, for all continuous functions $h(x)$ that ...
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1answer
119 views
Find function to make maximum value
Let ${f : [0, 1] \rightarrow [-1, 1] }$ is a continuous function such that ${ \int_{0}^{1} x f \left(x\right) dx =0}$
Find $f(x)$ such that ${ \int_{0}^{1} \left(x ^{2 } + \frac{1}{4} \right) f ...
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138 views
Closed Geodesics as minimisers of action functional
Suppose I have a Riemannian surface $(M,g)$. It's clear that closed geodesics are critical points of the length functional $l(\gamma)=\int\left|\gamma(t)^{\prime}\right|_{g(\gamma(t))}dt$ over the ...
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2answers
231 views
minimal surface of revolution when endpoints on x-axis?
What is the formula for the planar curve through $(\pm a,0)$ of fixed length $l$ which has minimal-area surface of revolution when rotated about the x-axis?
I get the area of the surface to be ...
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1answer
115 views
Finding the Euler-Lagrange operator $\mathcal M$ of a functional $\mathcal F$
I'd appreciate some help with the following problem:
Let $F = F(x, \{p_\alpha\}_{|\alpha|\le m})$ be a smooth function of the variables $x\in \overline \Omega$, and $p_\alpha \in \mathbb R, ...
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78 views
First variation and the usual derivative
Sometimes I see that when you want to calculate the first variation of a functional $E$, instead of calculating $\frac{d}{ds}E(u + sv)\bigg|_{s=0}$ people just calculate $\frac{d}{ds}E(u)$ especially ...
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51 views
First Weighted Eigenvalue of the Laplacian
Let $\Omega$ be a ball centered in the origin and let $\lambda_1(\Omega)$ be the first (or lowest) eigenvalue of the Dirichlet Laplacian in $\Omega$:
$$\lambda_1 (\Omega) =\min_{u\in H_0^1 (\Omega),\ ...
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1answer
190 views
An elementary (?) minimization problem
This morning, in Italy, there was the national exam of mathematics for students of high schools. One of the exercises asked to solve Heron's problem: given a straight line and two points lying on the ...
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107 views
Problem of finding strong maxima or minima of a functional
I have got this problem in exam where I have to to check for strong maxima (or minima) or weak maxima(or minima) of the functional given by
$\int_{0}^{1} (1+x)(y^')^2 dx ~~~~~ y(0) = 0, ~~ y(1) = ...
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1answer
75 views
how to solve differential equation $y^4 = k^2 (y^2 + y'^2\csc^2\alpha)$?
What's the solution of the differential equation $y^4 = k^2 (y^2 + y'^2\csc^2\alpha)$, where $y$ is a function of $x$ and $\alpha$ is a constant?
Polynomial solutions don't seem to work, because the ...
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2answers
108 views
Intuition of 'Transversality conditions' needed
I have read different matrials on Calculus of Variations, but I still do not grasp the intuition of transversality condition. From textbooks, I can only roughly get an idea that with a transversality ...
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59 views
$u''+\frac{4}{x+1}u'+\frac{2}{\left(x+1\right)^{2}}u=0$ variational solution
This is a concept solution scheme derived from a particular example that I have not been able to generalise sufficiently. The objective is to find a particular solution to a certain second-order ...
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1answer
77 views
Lagrange multipliers — What have I done wrong?
I am trying to find the stationary points of the potential $U(x,y)=x^2+y^2$ with constraint $x^2-2y^2=1$
So I set the Augmented potential $U^*=x^2+y^2+m(x^2-2y^2)$ where $m$ is the Lagrange ...
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1answer
77 views
Volume integral and Variations
Suppose I wish to find the Euler-Lagrange equation for an integral $\int_V f(u,\mathop{\mathrm{grad}} u)\,dV$ where $V$ is a volume given by some equation, for example say $x^2+y^2+z^2\le 1$, and ...
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1answer
110 views
Could someone please explain what this question is asking?
I have some trouble understanding the following question:
Suppose we have 1st fundamental form $E \, dx^2+2F \, dx \, dy+G \, dy^2$ and we are given that for any $u,v$, the curve given by $x=u, y=v$ ...
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174 views
Is there a fundamental misunderstanding here or have I made an algebraic slip?
Is there a fundamental misunderstanding here or have I made an algebraic slip?
I have a Riemannian metric of the form $ds^2={du^2+dv^2\over 1-u^2-v^2}$ on an open disc and I want to prove that radial ...
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203 views
Simple form of Euler-Lagrange equations in cylindrical polars for path of light?
What are the Euler-Lagrange equations in cylindrical coordinates $(r,\theta,z)$ for light moving at speed $v(r,\theta)$, where $r$ and $\theta$ depend on $z$?
I.e. for the problem of minimising the ...
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0answers
196 views
Constraint of a Lagrange multiplier
My question concerns Lagrange multipliers and the possibility to impose constraints on the multipliers themselves. I have a Stokes flow which is solved using the Finite Element Method on a domain ...
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1answer
98 views
Find an upper bound for lowest eigenvalue using calculus of variations.
So I'm doing a little calculus of variations on an eigenvalue problem. The goal of this is to find an upper bound for the $\omega_0$ as follows:
$\omega_0^2 \leq ...
3
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0answers
86 views
Nonlinear BVP pde and variational inequality
Suppose $f \in L^2(\Omega)$ where $\Omega$ is bounded. The problem: for $a \in \mathbb{R}$ find $u_a \in H^1_0(\Omega)$ s.t
$$-\Delta u_a + \frac{m(u_a)}{a} = f$$
where $m(r) = \begin{cases} r ...
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0answers
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A “bounded” constraint in a variational problem
Is there any standard approach to solve the following kind of variational problem?
Maximize $F=\int_0 ^1 L(x,y,y')dx $ subject to the constraint $|\int_0 ^1 M(x,y,y')dx| \lt k$ where $y$ ...
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141 views
Finding a force function from bodies in equilibrium
(This is an edited version of the original question, since I'm starting a bounty)
I'm trying to find a function $y$ from given data. Reverse optimization, so to speak.
Say we have two ...
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1answer
115 views
How do I obtain an appropriate energy functional from the weak formulation of a partial differential equation?
I'm reading a textbook example on the finite element method:
$\nabla^T[D(x,y,z)\nabla u] - a(x,y,z)u + f = 0 $ in R
$\partial R= \partial R_1 \bigcup \partial R_2$, $\partial R_1 \bigcap ...
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1answer
92 views
Equation for stationary values of $px^2+qy^2+rz^2$ given sphere and plane constraints
Consider stationary points of the function $V=px^2+qy^2+rz^2$ subject to the constraints $x^2+y^2+z^2=1$ and $lx+my+nz=0$, where $l,m,n$ not all zero and $p,q,r$ not all equal. How can we show that ...
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73 views
Extremizing an Integral under a cyclic condition
Let $h$ be a nonnegative, smooth and convex function on $[0,1]$ and let $f(x,y):[0,1]\times[0,1]\rightarrow[0,1]$ with $f(x,y)=f(y,x)$ and $f$ continuous. Suppose I fix $r>0$ and demand that
$$
...
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1answer
70 views
Simplifying the search for a geodesic
How can calculations for the geodesics on the surface $U=\{(x,y,z): c(x^2+y^2)-z^2=0, z>0\}$ be simplified by noting that is locally Euclidean? I can see that the property means that when we open ...
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1answer
65 views
Existence theorems for problems with free endpoints?
It is well known that the problem of minimizing
$$ J[y] = \int_{0}^{1} \sqrt{y(x)^2 + \dot{y}(x)^2} dx $$
with $y \in C^2[0,1]$ and $y(0) = 1$ and $y(1) = 0$ has no solutions. However, if we remove ...
4
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1answer
165 views
Online tutorial requested: functional derivatives
I am taking a course on Quantum Field Theory where we work alot with the functional derivative.
Does anyone know of a good, free online PDF tutorial with some examples?
Cheers!
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78 views
Question about ellipses and calculus of variations
I don't know much about calculus of variation, but I think it applies to a problem I've come across. If you have a closed loop in a 2 dimensional space defined by some parametric equation r(t), is ...
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2answers
100 views
How to solve $\max_{f}\int_{0}^{\infty}f\left(x\right)dx$ subject to $\int_{0}^{\infty}xf\left(x\right)dx=x_{0}$?
How to find $\max_{f}\int_{0}^{\infty}f\left(x\right)dx$ subject to $\int_{0}^{\infty}xf\left(x\right)dx=x_{0}$,
where $f$ is a function and $x_{0}$ is a constant?
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1answer
217 views
Euler lagrange assumptions
I have a question related to these two posts:
(1) Euler-Lagrange, Gradient Descent, Heat Equation and Image Denoising
and
(2) When the Euler Lagrange equation simplifies to zero
Background
...
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1answer
69 views
Lines through the origin and Euler-Lagrange
What form should the minimal-length curve to $\int \sqrt{dr^2 +r^2d\theta^2\over{1-r^2}}$ take? I think I can use the Euler-Lagrange equations. So write the integral as $\int\sqrt{({dr\over ...
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1answer
93 views
examples of functions whose arc-length from the origin is given by their derivative
I'm looking for functions $y:\mathbb{R}\rightarrow\mathbb{R}$ such that
$$\int_{0}^{a} \sqrt{1+\left(\frac{dy}{dx}\right)^{2}} dx = \frac{dy}{dx}\Bigg|_{a}$$
(this kind of feels like a ...
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1answer
260 views
When the Euler Lagrange equation simplifies to zero
My question is rather simple, and I'm sure I'm missing something simple, and yet...
I'm trying to calculate the Euler Lagrange Equations for the example function here:
...
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1answer
70 views
Infimum length of curves
Let the unit disc $\{(x,y): r^2=x^2+y^2<1\}\subset\mathbb R^2$ be equipped with the Riemannian metric $dx^2 +dy^2\over 1-(x^2+y^2)$. Why does it follow that the shortest/infimum length of curves ...
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1answer
189 views
Monge Ampere and Calculus
I am learning about mass transportation theory and the Monge-Ampere equation, to transport a function $f$ toward $g$ by a change of variable $T$. In particular, in order to solve for :
$$ \min \int ...
4
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1answer
374 views
Polar coordinates, line integrals, and the Beltrami Identity
Imagine you are walking along the xy-plane. There is a landmark at the origin of the plane which distorts time at every point on the plane, such that the distortion is a function of the distance ...
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2answers
228 views
Euler Lagrange sufficient minimization condition
Suppose $g \in C^1([a,b]\times\mathbb{R}\times\mathbb{R})$. Let $S = \{ f \in C^1([a,b]): f(a)=a_0, f(b)=b_0\}$. I am trying to show that if $f \in S$ satisfies
$$
\frac{d}{dx}g_z(x,f(x),f'(x))= ...
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96 views
Positive rotational symmetric solution for p-Laplacian
I have the the following problem and I just can't get my head around how to solve it.
Be $1<p<n$ and $q=\frac{np}{n-p}$, $u\in\mathcal{C}_{n,p}=\{f\in W^{1,p}_{loc}: ...
1
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2answers
307 views
Lagrange multiplier
Suppose $(M,g)$ is a Riemannian manifold. Let $E(u)=\displaystyle\int_MuP(u) dV_g$, where $P$ is a linear self-adjoint operator, and $dV_g$ is the volume form with respect to $g$. Given a smooth ...
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2answers
271 views
Can there be a cubical bubble?
Although not physically perfect, a reasonable mathematical model for a bubble's shape is that it minimizes surface area subject to fixed volume.
A single floating bubble is usually a sphere, but ...
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1answer
70 views
Solve $I[y]=\int_{x_0}^{x_1}y^{-\frac{1}{2}}(1+(y')^2)^\frac{1}{2} \mathrm dx$ parametrically
If $$I[y]=\int_{x_0}^{x_1}F(x,y,y') \mathrm dx$$
Where $$F=y^{-\frac{1}{2}}(1+(y')^2)^\frac{1}{2}$$
Then I have shown the Euler-Lagrange equation implies that
$$y(1+(y')^2)=2a$$
For some ...
2
votes
1answer
163 views
Find the extremals of $I[y]=\int_0^1(y')^2 \mathrm dt+\{y(1)\}^2$
Could anyone help me find the extremals of
$$I[y]=\int_0^1(y')^2 \mathrm dx+\{y(1)\}^2$$ subject to $y(0)=1$
Most crucially I can't work out how to find the boundary $x=1$. I'm trying to go back ...
