Questions on the calculus of variations, a subfield of calculus that deals with the optimization of functionals.

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137 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 ...
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118 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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178 views

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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161 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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2answers
262 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
135 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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87 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
77 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
89 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 ...
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257 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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99 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
111 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
461 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
81 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
110 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
422 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
87 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
245 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 ...
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1answer
460 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
342 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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137 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}: ...
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422 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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372 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
82 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 ...
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1answer
253 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 ...
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241 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 ...
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287 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 ...
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1answer
215 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 ...
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73 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 ...
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180 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 ...
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1answer
2k 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 ...
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81 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 ...
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240 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 ...
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1answer
124 views

Complicated “functional integral”

I came across the following "functional" at work: $$ \Pi [b]=\int_0^\infty\int_0^{\lambda b(v,\lambda)} vf(v,\lambda) \; dv \; d\lambda $$ it's part of an optimization problem that tries to find ...
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535 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 ...
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744 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 ...
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441 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, ...
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173 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 ...
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77 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 ...
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1answer
495 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 ...
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334 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 ...
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1answer
535 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 ...
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441 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] ...
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358 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 ...
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1answer
292 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 ...
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314 views

It is physically intuitive that the catenary is unique?

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 ...
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153 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 ...
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3answers
264 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 ...
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1answer
171 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 ...
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873 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 ...