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

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Proving a Sobolev-Type inequality (also it is related to variational problem)

This is question 8.23 part $4$ from H. Brezis Functional analysis I already have that for any $f\in L^p(I)$, $p>1$ and $I=(0,1)$ there exists a unique $u\in H_0^1(I)$ satisfying ...
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5 views

Epi-convergence and normal cones

I have a series of lower semi continuous, eventually level bounded and proper functions $ f^\nu(p)$ that epi-converges to $f(p)$. In this context, it is known from e.g., [7.33, Variational analyis, ...
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8 views

Euler Lagrange variational problem with $n$ independent variables and up to the hessian term

I'm trying to evaluate Euler Lagrange equation from the following relation: $$ F[f(\vec{r})]=\int_{\vec{r_1}}^{\vec{r_2}} d^n r J[f(\vec{r}),\nabla f (\vec{r}),H f(\vec{r}) ] $$ where $H$ is the ...
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1answer
15 views

Existence of minimum in $H^{1,2}(\Omega)$

I am considering a functional $$\mu(\Omega) = \min \{ u \in H^{1,2}(\Omega), \frac{\alpha \int_{\partial \Omega} u^2 ds + \int_{\Omega} |\nabla u|^2}{\int_{\Omega} u^2 dx} \}$$ I want to show the ...
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8 views

Supremum of $\phi[x]=\int_{0}^{\frac{3\pi}{2}} x(t)^2-4x(t)\cos t-(x'(t))^2 \;dt$

Find supremum of $\displaystyle \phi[x]=\int_{0}^{\frac{3\pi}{2}}x(t)^2-4x(t)\cos t-(x'(t))^2 \;dt$, where $x \in C^{1}[0,\frac{3\pi}{2}]$, $x(0)=0$ and $x(\frac{3\pi}{2})=-\frac{3\pi}{2}$. Using ...
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1answer
17 views

Is $xyz=0$ a joint variation

Is $xyz=0$ a joint variation I know that a joint variation is $\dfrac{x}{yz} = k$ I just want to know if $k$ is allowed to be zero
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69 views

Time-optimal control to the origin for two first order ODES - But wait, the node is unstable? Hard-mode active!

I want to find the time optimal control to the origin of the system: $$\dot{x}_1 = 3x_1+ x_2$$ $$\dot{x}_2 = 4x_1 + 3x_2 + u$$ where $|u|\leq 1$ I ran straight into the problem full strength, hit it ...
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1answer
29 views

Infimum and supremum of $\int_{0}^{1} e^{x(t)}(x'(t))^{2} \; dt$

Find infimum and supremum of $$\phi[x]=\int_{0}^{1} e^{x(t)}(x'(t))^{2} \; dt$$ where $x \in C^{1}[0,1]$ and $x(0)=0$ and $x(1)=\log 4$. It's easy to show that $\sup \phi[x]=\infty$, but what about ...
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1answer
20 views

What'd the author do here? (Euler-Lagrange equation)

I was reading the section of calculus of variations in Taylor's Classical mechanics and he went over some examples. The first being: When he reaches the portion $\frac{d}{dx}\frac{\partial ...
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12 views

Sobolev spaces of maps between manifolds and the Palais-Smale Condition

I'm currently reading some papers by Uhlenbeck on harmonic maps. She mentions the following facts: Let $M^m$ and $N^n$ be compact Riemannian manifolds, $N$ embedded isometrically into Euclidean ...
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13 views

Three dimensional plate model

Does anyone know of a good book or paper where the natural boundary conditions for the three dimensional plate model with simply supported edges are derived? I think that the bending moments should ...
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0answers
14 views

The variational bicomplex with dependent fields

I would like to understand a certain approach to variational problems that I've seen in the physics literature. In particular, I'd like to express it in terms of the variational bicomplex. However, ...
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1answer
36 views

Determining the Euler-Lagrange equations for a minimizataion problem

I'm working on a problem in computer vision and I've ended up trying to minimize the functional $$\int \left[\lambda(S''(x))^2 + (f(x) - S(x))^2 \sum_k \delta (x - x_k)\right]dx$$ where $\lambda$ is ...
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1answer
24 views

Deriving a high ordered Euler-Lagrange equation.

I've been able to derive the Euler-Lagrange equation for $$\int_a^b F(x,y,y')dx$$ relatively easily by using the total derivative and integration by parts. However, I was unable to apply the same ...
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1answer
29 views

Elementary calculus equation

If I have $L = y^2(1-y')^2$ are the following partial derivatives correct? Wolfram Alpha tells me otherwise... $$\frac{\partial L}{\partial y} = 2y - 4yy' + 2y{y'}^2$$ $$\frac{\partial L}{\partial ...
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1answer
23 views

Prove that a product of functions of bounded variation is a function of bounded variation

We consider functions defined on an interval $[a,b]$. I have to prove that a product of functions of bounded variation is a function of bounded variation. I have to also show that this isn't true for ...
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1answer
41 views

Functional derivatives in (Physics) Field Theory

The functional or variational derivative as defined in several places like Wikipedia seems to be defined as a functional, $L$ that takes a single input function, say $f(x)$ and then we define a ...
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39 views

Time-optimal control - Coupled system of equations, control to the origin

I want to find the time-optimal control to the origin $\underline 0$ for the following: $\dot{x}_1=-3x_1 + x_2$ and $\dot{x}_2 = x_1 - 3x_2 + u$, $|u|\leq 1$ How do I go about doing this. I ...
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44 views

Elastica - numerical check

Following on from rmhleo's fantastic answer here, where he states that the deformation of an ideally elastic circle is a problem of the calculus of variations which may be solved with an ODE of the ...
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20 views

Assumptions in Noether's theorem

Noether's theorem states conserved quantities exist when Lagrangian admits continuous symmetry. In the derivation of Noether's theorem here, it is assumed that Euler-Lagrange equation is satisfied, as ...
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26 views

Converting partial DE to integral Equation [closed]

Can anybody help me solving the below problem: What would be the functional corresponding to the following problem: $$ \frac{\partial ^{2}u}{\partial x^{2}}+ \frac{\partial ^{2}u}{\partial y^{2}} = ...
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1answer
27 views

Integral of homogeneous partial differential equation

From the book "Radio Occultations Using Earth Satellites" by William G. Melbourne: From Calculus of Variations a necessary condition for stationarity is that the ray at all points must satisfy ...
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40 views

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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0answers
13 views

stationary stokes problem - inf-sup

I want to show that $\inf_v \sup_{p} \int_{\Omega} \vert \nabla v \vert^2 + p \nabla \cdot v \, dx$ (where $v \in W_2^1(\Omega)$ and $p \in L^2(\Omega)$) is equivalent to the minimization of the ...
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0answers
11 views

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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1answer
23 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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1answer
41 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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1answer
67 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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34 views

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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1answer
27 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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12 views

First Variation of Length in the Euclidean Case

I'm reading a paper that says "the first variation of length can be viewed as the scalar product between the direction of variation and the inward pointing unit vector" in the context of straight ...
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1answer
50 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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9 views

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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1answer
50 views

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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1answer
29 views

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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15 views

Extrema of a functional simple question

I'm working out extrema of this functional $$ \int_a^b (y+yy'+y'+\frac12(y')^2) dx$$ Where $y$ is a function of $x$. Since it does not (explicitly) depend on x, the solution should the the solution ...
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1answer
66 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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1answer
45 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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2answers
32 views

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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2answers
29 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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21 views

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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2answers
37 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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1answer
15 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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1answer
21 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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1answer
20 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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44 views

Denominator of a function

I have a function $S(x,y)$ which satisfies the following PDE $$\frac{\partial S(x,y)}{\partial y}=-H\left(x,\frac{\partial S(x,y)}{\partial x}\right)$$ where the known function ...
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1answer
54 views

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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1answer
21 views

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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0answers
31 views

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 ...