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

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12
votes
3answers
99 views

Show $\inf_f\int_0^1|f'(x)-f(x)|dx=1/e$ for continuously differentiable functions with $f(0)=0$, $f(1)=1$.

Let $C$ be the class of all real-valued continuously differentiable functions $f$ on the interval $[0,1]$ with $f(0)=0$ and $f(1)=1$. How to show that $$\inf_{f\in ...
3
votes
1answer
57 views

Relation between Fourier components of a positive function

Here's a problem that has recently come up in my physics research: Let f be a function on [0, 2 $\pi$], which yields positive real numbers. Let the integral of $\int_0^{2\pi}f(x)= 1$. (Just for the ...
0
votes
1answer
27 views

I want to find Euler-Lagrange equation for the given functional.

I want to find Euler-Lagrange equation for the following: $$J(u) = \int \left( \frac{\psi(x) u + \dot{u}}{\psi(x)u - \dot{u}} \right)dx, \text{where} \ \psi(x) \ \text{is an explicit function of} \ ...
1
vote
1answer
42 views

How do first integrals help you solve differential equations?

I am reading about Euler-Lagrange equations and this particular section is a little unclear. Consider the differential equation $$\begin{bmatrix} \dot{x}\\ \dot{y} \end{bmatrix} = \begin{bmatrix} ...
-1
votes
0answers
25 views

how to numerically solve a problem in calculus of variations?

I am asked to solve a problem in calculus of variations via numerical methods as below in 2 states: I have no idea what to do? can any one help me urgently? or offer me some tutorials, books, or ...
2
votes
0answers
46 views

First variation of the Ginzburg-Landau functional

Suppose the Ginzburg-Landau free energy functional $‎E[u]=\frac{1}{2}\int_{\Omega}\{\vert\nabla u^2\vert+\frac{1}{2k^2} (\vert u \vert^2 -1)^2 \} \, dx‎$.Where $\Omega \subset R^2$ is an annular ...
1
vote
0answers
18 views

Functional derivative of a repeated integral

For a given function $f$, the functional derivative of the functional $\mathcal{F}[\rho]=\int f(x,\rho(x))\,dx$ is well-known to be $\frac{\delta}{\delta \rho(x)}\mathcal{F}[\rho]=\frac{\partial ...
1
vote
0answers
19 views

Determine whether it's min or max of functional.

so I have such functional: $$\phi(y)=\int\limits_0^1 (y^2+2y'^2+y''^2)dx, \ \ y(0)=y(1)=0, \ y'(0)=1, \ y'(1) = -\sinh1.$$ By using Euler-Lagrange formula, I get $$y^{IV} - 2y'' + y = 0$$ After ...
0
votes
0answers
19 views

Write an equation for variations

Write an equation for variations in relation to the parameter: $$ \frac{dx}{dt}=x+ \mu y^2$$ $$\frac{dy}{dt}=x+y$$ with initial condition x(0)=1, y(0)=0 in the point $ \mu=0 $
1
vote
2answers
29 views

Find the first-variational curve which corresponds to the functional $\int_{-1}^1 t^2 \dot{x}^2 dt$ when $x(-1) = -1$ and $x(1) = 1$.

Find the first-variational curve which corresponds to the functional $$\int_{-1}^1 t^2 \dot{x}^2 dt$$ when $x(-1) = -1$ and $x(1) = 1$. Here is what I did: \begin{align} \delta J(x)(h) &= ...
0
votes
0answers
15 views

Rayleigh-Ritz method to solve the P.D.E.

How we take an approximate solution from boundary conditions to find the solution of a partial differential equation by Rayleigh-Ritz method?
1
vote
1answer
25 views

Does $x^*(t) =(\frac{2 - e + e^2}{2 - 2e^2})e^t + (\frac{e - 3e^2}{2 - 2e^2})e^{-t} + \frac{1}{2}te^{-t}$ contain corner points?

I want to know if $x^*(t) =(\frac{2 - e + e^2}{2 - 2e^2})e^t + (\frac{e - 3e^2}{2 - 2e^2})e^{-t} + \frac{1}{2}te^{-t}$ can contain corner points. This $x^*(t)$ is the solution to the differential ...
2
votes
2answers
27 views

Solution verification for finding an extremal under constraints

Find the extremal of $\int_0^1 \left[\dot{x}^2 + 2x\dot{x} + 2x\right] dt$ with $x(0)=0$, and $x(1)=\frac12$ subject to the constraint $\int_0^1 12tx dt=24$ Could anyone verify the anwer to this? I ...
1
vote
0answers
26 views

Proof of fundamental lemma of calculus of variation.

Suppose $\Omega$ is an open subset of $\mathbb{R}^n$ and let $L^1_\text{Loc}\Omega$ denote all locally integrable functions on $\Omega$ and $C^{\infty}_0\Omega$ for smooth functions whose support lie ...
1
vote
1answer
42 views

Select $x(t)$ such that it minimizes $J$, $J = \int_0^T \left[(x - e^{-t})^2 + (\dot{x} + 1)^2\right]dt$.

I am given two situations, $A$ and $B$. For $A$, the ideal $x(t)$ is $e^{-t}$ for $0 \le t \le T$. However, for $B$, the ideal $\frac{dx}{dt}$ is $-1$ for $0 \le t \le T$. I am to make a compromise ...
2
votes
0answers
39 views

Calculus of variations: time of travel between two points

I'm reading Calculus of Variations by Elsgolc. On the page 35 there is example number 7. Let me introduce the problem. We have a functional given by: $v(y(x)) = \int_{x_0}^{x_1}F(x,y,y')dx$ If $F$ ...
0
votes
1answer
20 views

Find the minimum of $J = \int_{x_a}^{x_b} [1 + (\frac{dy}{dx})^2]^\frac{1}{2}dx$ with respect to $y(x)$.

Find $y(x)$ such that the Euclidean distance between $(x_a, y(x_a))$ and $(x_b,y(x_b))$ is a minimum, i.e., find the minimum of $$J = \int_{x_a}^{x_b} \left[1 + ...
0
votes
1answer
24 views

problem with a functional derivative

I've the following problem with a functional derivative (I'm not a specialist). Let's start with something I know (hope!): given a functional $\mathcal{F}[\psi]$, say $$ \mathcal{F}[\psi] = ...
1
vote
1answer
20 views

Minimal surface between two non coaxial rings

I'm currently studying minimal surfaces using the Euler-Lagrange equation. I'm particularly interested in minimal surfaces between two circles. I have already examined the case of two coaxial ...
1
vote
0answers
17 views

If $V\subset H\subset V^*$ is a Gelfand triple, which is the natural inner product on $V^*$?

is there any natural way to define a inner product on $V^*$? First we could consider Riesz isomorphism $\mathfrak{R}:V\rightarrow V^*$, and define $\langle F, G\rangle_{V^*}:=\langle ...
0
votes
1answer
68 views

Related Rates Cylinder

a) Assuming even distribution of oil, calculate the volume in cubic meter oil slick when the radius is 1 km and the height is .23 meters B) AT the exact instant in part a, the radius is increasing at ...
2
votes
0answers
19 views

Non-differentiable variational calculus (Dido's problem)

I wonder what is the alternative to Euler-Lagrange equations when we have non-differentiability issues. I'll give an example: Dido's problem can be stated as: Find the figure bounded by a line ...
1
vote
0answers
45 views

Expanding in powers of $\epsilon$ and big O notation

I do not understand how to approach (D.1) equation Where did that big O notation come from?Is it using taylor series and linear approximation? Thanks in advance
3
votes
1answer
47 views

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 ...
0
votes
0answers
14 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, ...
0
votes
1answer
30 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 ...
1
vote
1answer
20 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 ...
0
votes
0answers
12 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 ...
0
votes
1answer
19 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
18
votes
2answers
211 views

Time-optimal control to the origin for two first order ODES - Trying to take control as we speak!

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 ...
0
votes
1answer
30 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 ...
1
vote
1answer
26 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 ...
0
votes
0answers
13 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 ...
0
votes
0answers
15 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 ...
1
vote
0answers
15 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, ...
0
votes
1answer
37 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 ...
1
vote
1answer
37 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 ...
0
votes
1answer
30 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 ...
0
votes
1answer
29 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 ...
3
votes
1answer
53 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 ...
3
votes
0answers
49 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 ...
1
vote
0answers
52 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 ...
0
votes
0answers
21 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 ...
-1
votes
0answers
31 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}} = ...
1
vote
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 ...
0
votes
0answers
45 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 ...
1
vote
0answers
13 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: # ...
1
vote
1answer
27 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 ...
1
vote
1answer
46 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 ...
1
vote
1answer
72 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 ...