Questions on the calculus of variations, a subfield of calculus that deals with the optimization of functionals.
1
vote
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
164 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 ...
1
vote
0answers
161 views
Deriving an expression for minimum arc length along a 3D surface between any two points.
Consider a 3D surface, defined by the function $z = f(x, y)$. Assuming the surface is differentiable (no kinks), is there a function that expresses the minimum arc length traced along the surface ...
1
vote
0answers
194 views
Book Recommendation Needed: Gradient Descent, Euler-Lagrange
On a lecture note I read about Calculus of Variations
faculty.uml.edu/cbyrne/cov.pdf
the author talks about Euler-Lagrange equation, then continues to say "unfortunately, many times a closed form ...
1
vote
1answer
107 views
How to use Euler-Lagrange equation when obj fn integrated over two parameters?
If I want to find the minimizing function $f(t)$ over a single parameter, like time, then I take the integrand of
$$\int_{t}L(t,f(t),f'(t))\:\:\:\:dt$$
and substitute it into the Euler-Lagrange ...
3
votes
0answers
65 views
what is the domain of the Lagrangian of a surface embedding?
If we view our Lagrangian particle mechanics geometrically, then we describe a particle trajectory as a map from R to a manifold, and the Lagrangian $L(x,\dot{x})$ as a function on the tangent bundle ...
1
vote
0answers
141 views
How does a geodesic equation on an n-manifold deal with singularities?
My general premise is that I want to investigate the transformations between two distinct sets of vertices on n-dimensional manifolds by:
Minimalizing the change in the fundamental shape of the ...
36
votes
1answer
1k views
What's the largest possible volume of a taco, and how do I make one that big?
Let $f$ be a continuous, even, positive function over some interval $I=[-a,a]$ such that the total arc length of $f$ over $I$ is at least $2$, $f(0)=0$, and $f$ is increasing on $(0,a)$. View the ...
2
votes
0answers
73 views
Extension of Uncertainty Relations to a specific potential in Schrödinger Equation
Given some $\|\psi \|$ $\in$ $L^2 (\mathbb R^n) $ such that $\| \psi \|_2 =1$ and a function (potential) $V: \mathbb R^n \rightarrow \mathbb R$. The Schorödinger equation tells us that $-\triangle ...
1
vote
2answers
147 views
Weak lower semicontinuity of a functional on Hilbert space?
Let $H:=\left\{u\in L^2(R^N):\nabla u \in L^2(R^N)\right\}$ and a functional $$f(u)=\int_{R^N} |\nabla u|^2dx+\left(\int_{R^N} |\nabla u|^2dx\right)^2.$$
If $\{u_n\}\subset H$ is a sequence such that ...
1
vote
0answers
89 views
Complicated “functional integral”
I came across the following "functional" at work:
$$
\Pi [b]=\iint_0^{\lambda b(v,\lambda)} vf(v,\lambda) \; dv \; d\lambda
$$
it's part of an optimization problem that tries to find $b$, subject ...
0
votes
2answers
305 views
Reference request: Calculus of Variations “cheat sheet”
I would appreciate any suggestions for "cheat sheets" (summary sheets) on the calculus of variations/ variational calculus in particular on the Euler -Lagrange equation, Lagrange multipliers, Legendre ...
5
votes
1answer
353 views
Shortest path on hyperboloid
On the sphere $S^2$, the shortest path between two points is the great circle path. How about $H^2$, the hyperboloid $x^2+y^2-z^2=-1, z\ge 1$, with the Euclidean distance? Is there a formula for the ...
13
votes
1answer
283 views
Hilbert's 19th problem: Why do we care?
Hilbert's 19th problem asks:
Are the solutions of regular problems in the calculus of variations always necessarily analytic?
This was proven to be true (through the work of Sergei Bernstein, ...
1
vote
0answers
84 views
Legendre test re First Variation
The Legendre test (as mentioned in An Introduction to the Calculus of Variations by Charles Fox, requires that the sign of $\partial^2 F \over\partial y'^2$ is constant throughout the range of ...
1
vote
0answers
59 views
Paths of minimum time
I am reading An Introduction to the Calculus of Variations by Charles Fox and would be grateful if someone could explain the following bits to me.
1) Legendre test: one of the conditions stated is ...
1
vote
1answer
186 views
Isoperimetric problem in the calculus of variations
I'm trying to solve the following isoperimetric problem:
A plane curve has length $l$ and end points at $(0, 0)$ and $(a, 0)$ on the positive $x$ axis. Show that the area $A$ under this curve is ...
3
votes
1answer
202 views
Does a maximum entropy probability distribution with KL-divergence constraint not exist?
In my earlier question I asked about a technical aspect of solving a system of equations arising from looking for an entropy-maximizing distribution $p(x)$ continuous on $\mathbb{R}$ and constrained ...
2
votes
1answer
364 views
Troublesome functional derivative: second term of Euler-Lagrange equation
I am attempting to calculate the functional derivative of a functional
$$E[\rho] = \int G(\rho(\mathbf{r}),\nabla\rho(\mathbf{r}),\mathbf{r})d\mathbf{r},$$
where
...
4
votes
0answers
161 views
Gradient flow of a surface
I found the following definition in a book (S. Osher, R. Fedkiw, "Level Set Methods and Dynamic Implicit Surfaces", p. 140):
[the context is reconstruction of surfaces from unorganized point sets]
...
3
votes
3answers
259 views
Treacherous Euler-Lagrange equation
If I have an Euler-Lagrange equation:
$(y')^2 = 2 (1-\cos(y))$ where $y$ is a function of $x$ subjected to boundary conditions $y(x) \to 0$ as $x \to -\infty$ and $y(x) \to 2\pi$ as $x \to ...
1
vote
1answer
169 views
Wikipedia Article — Legendre Transform
I was reading the wiki article on Legendre Transform. I would be grateful if someone could explain the section at http://en.wikipedia.org/wiki/Legendre_transformation#Examples
ie how they arrived at ...
1
vote
1answer
154 views
Question on the catenary
The catenary minimizes the potential energy of a cable and has equation $y - y_0 = A \cosh (\frac{x-x_0}{A})$. It is physically intuitive that the catenary is unique, but is there a mathematical ...
3
votes
1answer
142 views
Help deriving geodesic of $S^2$ by considering small deviations
For $s(t)$ the geodesic confined to the surface of a (3D) sphere, how does one get $\|\dot{s}\|^2 s + \ddot{s} = 0$ by setting $\frac{d}{d\delta} \left( \int \|\frac{d}{dt} \frac{s(t)+\delta ...
6
votes
2answers
208 views
Symmetry of Solution to Classical 3-Dimensional Isoperimetric Problem
A while ago I attempted to solve the classical isoperimetric problem in 3-dimensions, namely "Find the surface that has the smallest surface area for a given volume".
At that time for me to write ...
1
vote
1answer
133 views
The uniqueness of the brachistochrone
How does one show the uniqueness of the solution to the brachistochrone problem? Doesn't the fact that the solution is of the form
$x=a-c(2t+\sin2t)$ and $y=c(1+\cos2t)$
naturally guarantee ...
4
votes
2answers
575 views
Geodesics of a Sphere in Cartesian Coordinates
I want to minimize $I = \int |\dot{x}|^2 dt$
subject to the constraint $|x|^2=1$ (sphere)
which gives an Euler equation of $\lambda x - \ddot{x} = 0$.
I have to show that the Euler equation is ...
-1
votes
2answers
142 views
Calculus of variations question from Darcogona
I asked the question in the next forum, the 4th post, hopefully someone can help me with this here or there: https://nrich.maths.org/discus/messages/7601/151442.html?1310911861
Thanks in advance.
4
votes
1answer
473 views
Euler-Lagrange, Gradient Descent, Heat Equation and Image Denoising
I have a question about Euler-lagrange equation which you can check this file. http://www.bekirdizdaroglu.com/ceng/Downloads/ISCE10.pdf, specifically in page 6,equation 8 , not equation 9...
There ...
2
votes
2answers
81 views
An extremal property of the distance function
Let $\Omega \subseteq \mathbb{R}^N$ be open and bounded, let $\mathcal{I}:C(\overline{\Omega}) \ni u\mapsto \int_\Omega u(x)\ \text{d} x \in \mathbb{R}$ and set:
$$\phi(x):=\text{dist} (x,\partial ...
4
votes
1answer
143 views
Why weak formulations in numerical mathematics?
Regard the Poisson equation on the domain $\Omega = [-1,1]^n$ with $f \in H^{-1}$
$- \triangle u = f$
with homogenous Neumann boundary conditions. From standard regularity theory we know $u \in ...
3
votes
0answers
113 views
Calculus of Variations: Contains an integral of my goal function
OK, using the calculus of variations, I want to find a function $f$ that maximizes:
$$J = \int_0^n L(x,f(x)) \text{d}x$$.
But $L$ has multiple integrals in it (for example, $\displaystyle \int_0^n y ...
15
votes
3answers
452 views
What's the minimum of $\int_0^1 f(x)^2 \: dx$, subject to $\int_0^1 f(x) \: dx = 0, \int_0^1 x f(x) \: dx = 1$?
The question is as in the title: what's the minimum of $\int_0^1 f(x)^2 \: dx$, subject to $\int_0^1 f(x) \: dx = 0, \int_0^1 x f(x) \: dx = 1$? (Assume suitable smoothness conditions.)
A problem in ...
6
votes
5answers
733 views
Introductory text for calculus of variations
I am currently working on problems that require familiarity with calculus of variations. I am fairly new to this field. Please suggest a good introductory book for the same that could help me pick up ...
3
votes
1answer
243 views
minimizing the norm of a curl over a domain
According to my computations, the function which minimizes $\int_\Omega \|\operatorname{curl} f\|^2~dx$ should satisfy $\operatorname{curl}(\operatorname{curl}(f)) = 0$ everywhere on $\Omega$, ...
3
votes
1answer
178 views
Is it possible to combine the Euler-Lagrange equations with the method of Lagrange multipliers?
In particular, say we seek a sufficiently smooth function $ u : [a,b] \to \mathbb{R} $ such that the solution $x$ to the differential equation with given initial conditions
$$ G(x, x', \dots, ...
2
votes
2answers
701 views
Partial derivative with respect to a function
Let $p$ be a function of the form $\mathbb{R}^2 \to \mathbb{R}$.
How do i find the derivatives of the following expressions with respect to p(x,y) :
a. $\int_\mathbb{R^2} p$
b. $ \dfrac{\partial ...
1
vote
1answer
212 views
Maximizing a function by finding derivative
I want to find the value of $\vec{p}$, $p_s$, $p_t$ each of which is a function of the form $f:\mathbb{R}^2 \to \mathbb{R}$ that maximize the following function :
$$\begin{align}
\int_\mathbb{R^2} ...
0
votes
0answers
78 views
Variability of a curve with Vapnik-Chervonenkis dimension 4
Say I have 10,000 data in 2-D and I want to fit a curve to them. There are many functional forms this curve could take -- polynomial, B-spline, trigonometric, and so on. I've decided that I only want ...
3
votes
1answer
201 views
Functional derivative of $\int \left( \frac{df^2 }{d^2 x} \right)^2 dx$
According to page 7 of the PDF document
$$
\frac{\delta}{\delta f} \int \left( \frac{df^2 }{d^2 x} \right)^2 dx = \int \frac{df^4}{d^4 x} dx
$$
I would like help proving this statement.
Although ...
2
votes
2answers
326 views
Divergence Theorem, Laplacian, Energy Minimization
I am trying to understand a proof for critical points of certain energy functions being harmonic functions. It goes as follows: For a function $u(x_1,..,x_n)$, a functional E(u) is defined as
$E(u) ...
5
votes
2answers
953 views
Euler-Lagrange, Gradient Descent, Heat Equation and Image Denoising
For an image denoising problem, the author has a functional $E$ defined
$$E(u) = \iint_\Omega F \;\mathrm d\Omega$$
which he wants to minimize. $F$ is defined as
$$F = \|\nabla u \|^2 = u_x^2 + ...
1
vote
1answer
151 views
Euler-Lagrange expression uniformly non-negative
If you form the Euler-Lagrange equation for some calculus of variations problem in x, f(x) and f'(x), and the resulting expression is always non-negative over the domain of x (because the expression ...
4
votes
5answers
242 views
Function that maximizes a function
Let's say we have a real, continuous,
positive function f(x) for which we
define the quantity:
$$\pi(f,a) = \frac{\int_0^a f(x)
dx}{\int_0^a \sqrt{1+\left(\frac{df(x)}{dx}
\right)^2 ...
8
votes
1answer
676 views
Calculus of Variations and Lagrange Multipliers
A general problem for the Calculus of Variations asks us to minimize the value of a functional $A[f]$, where $f$ is usually a differentiable function defined on $\mathbb{R}^n$.
What if, however, the ...
8
votes
3answers
454 views
Supremum length of space curves contained in the open unit ball having always less than unity curvature
I am in the process of proving that if a space curve (in R^3) has infinite length and the curvature tends towards 0 as the natural parameter s tends to infinity, the curve must be unbounded - i.e. not ...
2
votes
1answer
314 views
Quadratic minimization in a Hilbert space
If $A$ is a positive definite matrix, then the solution to the minimization problem $(1/2)x^TAx - b^Tx$ is given by $A^{-1}b$. I'm interested in the generalization of this to a Hilbert space. What ...
12
votes
7answers
2k views
Why Circle encloses largest Area?
In this wikipedia, article http://en.wikipedia.org/wiki/Circle#Area_enclosed its stated that the circle is the closed curve which has the maximum area for a given arc length. First, of all i would ...
