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

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How to find a function which maximizes a stochastic process containing sum?

Let $X=\lbrace X_t : t\geq 0\rbrace$ denote a Lévy process with initial value $X_0=0$. Let the process be sampled equally in time ($t_n-t_{n-1}=const.$). I am looking for the ...
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Weierstrass conditions, what does strong mean, and are both conditions required?

I have the Weierstrass condition: In order that the extremal $\bf{C^*}: x = x^*(t)$ give a strong local minimum to $\bf{J[x]}$ it is sufficient that: # ...
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36 views

Counterexample for existence of a minimiser in a variational problem

I'm trying to find an example of a minimisation problem of the form $$ \inf \{ J(u) := \int_{\Omega} f(x)|u(x)| + |\nabla u(x)|^2:\, u \in H^1, \, \int u = 1\}$$ with $\Omega$ an open and bounded ...
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56 views

Strongly minimizing curve optimisation with Weierstrass condition

No idea where to start on this one: Find the strongly minimizing curve and value of $J_{min}$ for cases: $$\int_1^2 (t^2\dot{x}^2 + 2x^2) dt$$ where $x(1)=0,x(2)=7$ Using the Weierstrass ...
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80 views

how to solve the system of differential equations for this particle?

I'm trying to solve this problem A particle of mass m moves under the action of gravity on the inner surface of a paraboloid of revolution $x^2+y^2=az$ which assumed frictionless. Obtain the ...
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problem of calculus of variations

A necessary and sufficient condition for the calculus of variations problem $\delta\int_{a}^{b} L(x(t), \frac{dx}{dt}) dt = 0$ be independent of the choice of parametric representation of the curve ...
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42 views

Inverse optimization problem

This may seem like a weird question, but it's something which has been intriguing me for quite a while. In the Calculus of Variations we are told to find the extrema of a functional defined over a ...
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1answer
71 views

Circular variation with repetition

I would like to know formula for circular variation with repetition. What I mean is : You have round table with n-spots. On every spot there can be number from 1 to k. So for n = 4 and k = 3 ...
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Change of variables in Lagrangian

Question Let $\psi : [t_0, t_1] \to \mathbb{R}$ be a smooth function such that for $t \in[t_0, t_1], \dot{\psi(t)} > 0$ and also so that $\psi(t_0) = x_0$ and $\psi(t_1) = x_1$. Using the ...
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Volterra derivative of utility function in economics

I have difficulties concerning the calculus of first variation (which is also nominated as Volterra derivative I suppose.) Except the explanation on wikipedia, I did not see any useful explanation on ...
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Explanation as to why we treat position and velocity as independent variables in the lagrangian?

Although having studied calculus of variations and lagrangian mechanics, something I've never felt that I've fully justified in my mind is why the lagrangian is a function of position and velocity? ...
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Problems on calculus of variations

I'm reading a paper in which it gives the following Lagrangian $$L[u,\rho,\phi]=L_0[u,\rho]+\phi(x)(\partial_t\rho+\nabla\cdot(\rho u))$$ where $L_0$ is part of Lagrangian and $\phi(x)$ is Lagrange ...
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97 views

Lagrangian equivalence up to total time derivative: dependence on higher derivatives

I recently encountered the problem Show that the Euler-Lagrange equations of motion for $L_1$ and $L_2$ are the same when $$L_2(\ddot{q},\dot{q},q,t) = L_1(\dot{q},q,t) + \frac{d}{dt} ...
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76 views

Euler-Lagrange equation notation: $\delta$ instead of $\partial$

I have seen the equation written as: $$\frac{\delta L}{\delta q} - \frac{d}{dx} (\frac{\delta L}{\delta \frac{dq}{dx}}) = 0$$ Here, "variation of $L$ divided by variation of $q$ or $\frac{dq}{dx}$" ...
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Is every set a subset of a vector space?

I was taught that a functional is a map from a subset (not subspace) of a vector space into the reals, $F: D\subset V \to \mathbb{R}$. I know there are other definitions, but is there any reason to ...
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41 views

Euler-Lagrange equation: Differentiation with respect to x

I got stuck in my lecture notes after a supposed differentiation of the Euler-Lagrange equation: $$\dfrac{\partial f}{\partial y}-\dfrac{d}{dx} \left( \dfrac{\partial f}{\partial y'}\right) = 0$$ ...
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Formulation of variational problem

I've been stuck on this problem for a long while. I'd be grateful for your help. The problem A variational problem is given by the functional: $$I[y]=\int_a^b{F(x,y,y',y'')dx}+[y'(a)]^2$$ Where $F$ ...
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44 views

Does the square root function change the variations of a function?

If I have $$f(x) =\sqrt{g(x)}$$ Will the variations of $f(x)$ be the same than the variations of $g(x)$ ?
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68 views

Strong and weak extrema

I am confused about the "strength" of the two definitions. The definitions I use are the following: Let $y$ be a function defined on the set $M$. Neighborhood (0. order) of the function $y$ is the ...
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38 views

Proving isoperimetric inequality using calculus of variations

I was trying to prove isoperimetric inequality, which states that for any simple closed curve of length $l$, the area that it encloses is $\leq \frac{l^2}{4\pi}$. I wanted to use calculus of ...
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33 views

Finding the extremal curve satisfying a variable endpoint

Below is a question I am trying to solve, and my attempt. $\int_0^T \frac{\dot{x}^2}{t^3} \mathrm{d} t$, where $x(0)=1 $ and $x(T)$ lies on the curve Transversal condition: $$f-(\dot{c} ...
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Why does the arc-length formula have form $\int_a^b\left|\left|\frac{d\vec{f}(t)}{dt}\right|\right|_2dt$ for C1 curves?

This discussion focuses on $\mathcal{C}^1$ curve on $\mathbb{R}^n$. But feel free to talk about the case where we only have a continuous curve or the scenario with a manifold with a metric in general. ...
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variation of functional

A little confused about finding the variation of the functional J = $\int_{t0}^{tf}(e^{x_1(t)+x_2(t)})dt$ When I perturb and find the increment, I get: $\Delta J = \int_{t0}^{tf} (e^{x_1(t) + ...
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Given a point $A$, describe those points to which a catenary cannot be drawn from $A$.

Background An elementary problem in the calculus of variations shows that among all curves joining two points $A$, and $B$ in the first quadrant, the one which generates the surface of minimum area ...
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56 views

Are all definite integrals considered functionals?

In my Optimization class, we are messing around with some Calculus of Variations in an effort to find functions which minimize functionals. In these cases, the spaces we're working with are spaces ...
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66 views

A derivation of Euler–Lagrange equations with a general metric

$g_{\mu \nu}(x)$ is a metric of a space and $X^{\mu}(\lambda)$ a curve with $\lambda$ a parameter that varies monotonically along the curve: $$0 = \delta \int d\lambda L = \int d\lambda \delta L = \\ ...
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34 views

Prove that solution of a variational problem exists

Let $X$ is a Hilbert Spaces We define two operators $$a:X\times X\rightarrow\mathbb R$$ and $$b:X\rightarrow\mathbb R$$ where $a$ is a symmetric, bounded, strongly positive operator, and $b$ is a ...
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1answer
49 views

Finding extremal of a fixed end point problem. Optimisation

I want to find the extremal of the fix-end point problem $\int_1^2 \frac{\dot{x}^2}{t^3}$ with $x(1)=2,x(2)=17$ First I check the euler-lagrange equation is equal to $0$. We have: ...
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Local minima: Sufficient conditions. Comparison of Calculus verses Calculus of Variations

My lecturer has written: Let $y=x^*+\epsilon \eta$ where $x^*,\eta,y\in \mathbb{R}^2$ $0\leq f(y) - f(x^*) = \epsilon V_1 + \epsilon^2 V_2 + O(\epsilon^3)$ $V_1 = \nabla f(x^*)\eta$ $V_2 = ...
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Constrained optimization minima and maxima and non-degeneracy answer check

Find the critical points of $$\def\1{x_1}\def\2{x_2}\def\3{x_3}\def\f{f(\1,\2,\3)}\def\l{\lambda}$$ $$\f=\1\2+\2\3+\3\1$$ subject to constraint $\1+\2+\3=1$ First I will construct the Lagrangian: $$L ...
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66 views

How to calculate this functional derivative?

How can I calculate the functional derivative of this functional? $$F[x](t) = \int_{0}^{t}x(t_1)a(t_1)\left \{ \int_{0}^{t_1}x(t_2)b(t_2) \,dt_2\right \} dt_1 .$$ Where $a(t)$ and $b(t)$ are real ...
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38 views

Confusion with Euler-Lagrange Derivation

This is mostly a re-hash of this thread, but it did not receive an adequate answer. In the derivation that I am reading, there is one step that is not justified. Perhaps obvious, but it is not clear ...
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40 views

isoperimetric problem:how to solve the given question

Determine $y(x)$ for which $\int_{0}^{1} x^{2} + y^{'2}dx$ is stationary, subject to $\int_{0}^{1}y^2=2$, $y(0) = 0$, $ y(1) = 0$. how to solve it? I tried it: $f=x^{2} + y^{'2}$ and $g=y^2$ then ...
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60 views

extremal problem-how to check istrong minima,maxima condition

The functional $I[y(x)]=\int_{0}^{2}(xy^{'}+y^{'2})dx$,y(0)=1,y(2)=0 possess a.strong minima b.strong maxima c.strong maxima but not weak minima d.weak maxima but not strong minima How do we show ...
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387 views

Rayleigh-Ritz-method-how to solve the given problem

how to solve this: An approximate solution of the problem $y^"-y^{'}+4x\epsilon^x =0$, $y^{'}(0)-y(0)=1$,$y^{'}(1)+y(1)=-\epsilon$ is: here we have to calculate the value of y(x)? what i did is: ...
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how to find the optimal function with lagged cost? (calculus of variations)

I need to find the function $b( )$ that maximizes this guy ($c()$ and $\beta()$ are functions too, and $c()$ is convex): $$\int_{0}^{T} \! e^{-\delta v}\beta(v) \left[\int_{0}^{v} b(s) \; ds - ...
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Open problems in variational analysis/PDEs

I wasn't sure whether this question was more appropriate for StackExchange or Overflow, but in any case I would really appreciate it if any active researchers in the field responded. I'm a PhD ...
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57 views

Fréchet normal cone

Given $x\in \Omega(\subset X)$ (X: Banach space) and $\varepsilon\geq 0$, the set of $\varepsilon-$normals to $\Omega$ at $x$ by \begin{align} \widehat N_\varepsilon(x;\Omega):=\left\{x^*\in X^*\mid ...
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green function, functional derivative

I am trying to find ${\delta F}/{\delta u}$ for the functional: $F[u]=\int u(x)\int G(x,y)u(y)dy dx $ G is green function for laplace operator. is there Euler-Lagrange version for double intrgral? ...
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29 views

Maximization of ratio of two functionals

I am trying to find a function prescribed in polar coordinates $r = f(\theta)$ that maximizes the following quantity $$\frac{\int_0^{2\pi}r^3\cos\theta\, d\theta}{\int_0^{2\pi}r^4\, d\theta}$$ ...
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Least surface of volume with constraints

We know that in 2D/3D the shape with the least surface of a certain volume is a circle/sphere (e.g. soap bubbles). Now Imagine we have a flat surface (tabletop) that can be used as part of the surface ...
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Minimum travel time of a fuel-less train: brachistochrone problem

It is suggested that a rail network should include a frictionless tunnel where fuel-less trains run under gravity. The trains are released from rest at the point of departure and run freely until ...
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1answer
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Maximizing the uniformity of density function subject to moment constraints

Background I want to find a probability measure for a continuous random variable, subject to moment constraints, that is maximally "uniform", as defined below: Definition: Maximally Uniform ...
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1answer
407 views

Using bordered Hessian matrix to determine non-degeneracy and type of constrained extremum

I have the following problem: $\def\f{f(x_1,x_2,x_3)}\def\1{x_1}\def\2{x_2}\def\3{x_3}\def\n{\nabla}\def\g{g(x_1,x_2,x_3)}\def\l{\lambda}\def\q{\begin{pmatrix}}\def\p{\end{pmatrix}}$ Find the ...
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42 views

Local minimum of the function:

Find the local minimum of the function: $$\def\f{f(x_1,x_2,x_3)}\def\1{x_1}\def\2{x_2}\def\3{x_3}\def\n{\nabla}$$ $$\f=\1^2-2\1\2+2\2^2+\3^2 \text{ in } \mathbb{R}^3$$ $\n\f=(2\1-2\2,-2\1+4\2,2\3) ...
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36 views

Energy functional in Geodesic Active Contours

I have read some papers about Geometric active contours of the author C.Gout and Le Guyader [1] Segmentation under geometrical conditions using geodesic active contours and interpolation using level ...
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222 views

Minimize Energy in Image processing - Geodesic active contours

I've read some papers in Geodesic active contours (Image processing), which use the minimization of an Energy, consist of Internal Energy and External energy, for example, in the paper of Kass (Snake: ...
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Minimum surface attaching two parallel (non-planar) circles.

While studying for a comprehensive exam, I found this old problem: Consider two parallel coaxial wire circles, not in the same plane, to be connected by a surface of minimum area that is ...
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Prove that the maximizing point configuration on the unit circle for a Vandermonde like functional is a picket fence

For $\lambda_i \in S^1 \subset \mathbb{C}$, consider the functional $H(\{\lambda_1, \ldots, \lambda_n\}):= \sum_{j < k} | \lambda_j - \lambda_k | $. I want to show that $H$ is globally maximized by ...
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72 views

Calculus of variations: big-O notation?

I have a formula in my text-book $$y(x+C) = y(x) + \frac{dy}{dx}C + O(C^2)$$ Can someone explain this formula?