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

learn more… | top users | synonyms

0
votes
0answers
25 views

second variation of a modified Dirichlet energy

Given: $E(h)=\int (\frac{1}{2}h_x^2-F(h))dx$ $f(h)=F'(h)$ and $f'(h) $is integrable $\int \phi dx=0$ (may not need this) Trying to show: $E(h+\epsilon\phi)=E(h)+\epsilon \int h_x \phi _x \, ...
0
votes
0answers
34 views

Euler Lagrange Equations for the following minimization problem

Say I have a curve C in the plane Using Euler-Lagrange I am trying to solve the following minimization problem: $$\int \int (E - R) \nabla C dx dy$$
0
votes
0answers
24 views

Euler Lagrange of the following equation.

I've worked the Euler -Lagrange for the following equation $$E(u(x,y),v(x,y)) = \int \int (I_x u + I_y v + I_t) + \alpha ((u_x^2 + u_y^2) + (v_x^2 + v_y^2))dx dy$$ and got the following equations: $$0 ...
0
votes
0answers
33 views

Using Euler-Lagrange to find the first variational curve of

I have been working on this optimization problem for days but I cannot figure out the right way to finish it off. I am reading from Optimization Theory by Pierre, and this is problem 3.3. Note that ...
2
votes
0answers
46 views

Doubt in the derivation of the field Euler-Lagrange equations

I'm looking at a derivation of the Euler-Lagrange equations in a field setting, and one step in the proof is continually eluding me. Let $\phi(\vec x,t)$ be a field and $\mathscr ...
12
votes
3answers
140 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
81 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
53 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
79 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
vote
0answers
24 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
21 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
20 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
42 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) &= ...
1
vote
1answer
27 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 ...
3
votes
2answers
42 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 ...
2
votes
0answers
98 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
48 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
58 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
27 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
34 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
33 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
25 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
145 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
45 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
67 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
54 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
20 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
43 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
26 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
20 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
28 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
221 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
43 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
36 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
25 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
20 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 ...
0
votes
1answer
55 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
80 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
31 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
57 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
80 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
57 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
54 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
32 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
39 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
31 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
53 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
16 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
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 ...
2
votes
1answer
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 ...