Questions on the calculus of variations, which deals with the optimization of functionals mostly defined on an infinite dimensional spaces.

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Intuition for second frechet derivative

I am now used to thinking of the first derivative of a map between vector spaces $f:V\to W$ in the "proper" Frechet sense, as being "the assignment to each point $v$ of $V$ of the linear map ...
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19 views

Determining whether the extremal problem has a weak minimum or strong minimum or both

The extremal of the functional $\int_{0}^{\alpha}{\left((y')^2 - y^2\right)dx}$ that passes through (0,0) and (${\alpha}$,0) has a weak minimum if ${\alpha}$ < $\pi$ strong minimum if ${\alpha}$ ...
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2answers
118 views

Calculus of variation with inequality constraints

I want to find the function $y$ which maximizes the functional $J[y] = \int_0^1 g(x) y(x) dx$ subject to $0 \leq y(x) \leq 1$ for all $x\in [0,1]$ and $\int_0^1 y(x) dx = k$ where $g$ is a strictly ...
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2answers
222 views

Why is the integral of $\|\nabla f\|^2$ called the energy of $f$?

Let $\Omega$ be a region in $\mathbb{R}^2$ with $f:\Omega \to \mathbb{R}$ a smooth function. Why is the quantity, $$ \tfrac{1}{2} \iint_{\Omega} \|\nabla f\|^2 $$ Called the "energy" of $f$? I am ...
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1answer
36 views

Calculus of variations with inequality and non-integral constraints

I have a question on solving an optimization problem with calculus of variations. I am attempting to maximize the functional $$ J[y] = \displaystyle\int_a^b F(x,y,y') \, \mathrm{d}x, \tag{1}$$ ...
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3answers
794 views

Find the extremal to the functional $J(y) = \int_{0}^{1} ((y')^2 -y)dx$ and discuss whether they provide a max/min

I am having a hard time getting my head around Functionals and Calculus of Variations, My question is: Given a functional and using the Euler-Lagrange equation to find an extremal, how do we show ...
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2answers
95 views

Proving that a sphere has a minimal surface to volume ratio using Calculus of Variations

I know the problem is traditionally solved via the isoperimetric inequality, but I was hoping to solve it by minimizing a surface of revolution subject to a volume constraint. The surface area of a ...
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1answer
35 views

Solution of a sublinear elliptic problem.

In a lecture notes, the author showed the problem $\tag{$P$}$ $\begin{cases} -\Delta u = |u|^{q-2}u \textrm{ in } \Omega, \\ u(x)= 0 \textrm{ in } \partial\Omega, \end{cases}$ where $\Omega ...
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1answer
56 views

Show that $\eta(z) \det P$ is a null Lagrangian

This is problem 8.7.4 from Evans' PDE book. Assume $\eta: \mathbb{R}^n \to \mathbb{R}$ is $C^1$. Show that $L(P,z,x) = \eta(z) \det P$ is a null Lagrangian. Here $P$ is a $n \times n$ matrix and $z ...
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2answers
54 views

Alternative proof of Fundamental Lemma of Variational Calculus?

I am confused by one of the proof in the Calculus of Variations by Gelfand and Fomin. On page 9, we have Lemma: If $\alpha(x)$ is continuous on $[a,b]$, and if $\int_a^b \alpha(x)h(x)=0$ for every ...
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0answers
12 views

Euler-Lagrange Single function of single variable with higher derivatives

Here is the page on Wikipedia: So it says the fixed boundary conditions for the function itself as well as for the first $n-1$ derivatives. You can fix the boundary points physically say $y(a)=a'$ ...
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0answers
15 views

Obscure first order approximation

I don't understand this first order approximation from Gelfand, Fomin "Calculus of Variation": $$ \int_{x_0}^{x_0 + \delta x_0} F(x, y + h, y'+h') dx \sim F(x,y,y')\big|_{x = x_0}\delta x_0$$ where ...
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1answer
59 views

How do I determine a cricital point of an area functional?

The orientated area $A(\gamma)$ of a regular closed plane curve $(\gamma, \tau)$ is defined as $$A(\gamma) :=\frac{1}{2}\int_{0}^\tau \det (\gamma,\gamma')$$ Now how can I determine the cricital ...
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1answer
25 views

When the Euler-Lagrange equation reduces to 0=0

I've gotten the functional $$\int_a^b(y^2+2xyy')dx$$ with Dirichlet boundary conditions. Applying the Euler-Lagrange equation I get: $$0=\frac{\partial f}{\partial y}-\frac{d}{dx}\frac{\partial ...
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0answers
46 views

$\int_a^b M(x)\eta''(x)dx=0$ for all $\eta$ implies $M(x) = c_0 + c_1x$ proof

Here's a lemma that my book proves: If $M(x)\in C[a,b]$ and $$\int_a^b M(x)\eta'(x)dx = 0$$ for all $\eta(x)\in C^1[a,b]$ such that $\eta(a)=\eta(b)=0$, then $$M(x)=c,$$ a constant, for all $x\in ...
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1answer
34 views

Weak solution in $\mathbb{R}^{N}$

I'm bit confusing about definition of weak solution. If I have the following problem: $\begin{cases} \tag{P} -\Delta u = f \textrm{ in } \Omega, \\ u = 0 \textrm{ in } \partial\Omega, \end{cases}$ ...
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1answer
30 views

Explanation of this proof of the Euler-Lagrange

(Please note that I am a physics student, so please try to avoid rigorous mathematics in the explanation.) I have the following proof for the Euler-Lagrange: Consider the integral $$I=\int ...
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1answer
25 views

Find critical points of the functional $I[y] = c \int_0^L y(y')^3 dx$

Find critical points of the functional $I[y] = c \int_0^L y(y')^3 dx$ with $y(0)=0$ and $y(L)=R$ Euler-Lagrange equation: I arrive at $(y')^3+3yy'y''=0$ and so solve $y'=0$ and ...
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0answers
29 views

Calculus of Variations Open ball constraint

I'm stuck on this question, Let $V$ be an open ball in $\mathbb R^3$ in $ { x,y,z \text{ st }x^2 +y^2 +z^2<1}$ I need to minimise the integral $$ \iiint_V (u_x^2+u_y^2+u_z^2) \, dx \, dy \, dz $$ ...
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2answers
73 views

How to find the minimum value of following integral?

Let $A$ be the set of twice differentiable functions on the interval $[0,1]$, and $$B=\{f \varepsilon A: f(0)=f(1)=0, f'(0)=2\}.$$ What is $${\rm Min}_{f\varepsilon B}\int_{0}^{1} (f''(x))^2 ...
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2answers
240 views

Intuition behind variational principle

Hofer-Zehnder, in section 1.5, proves that every Hamiltonian field on a strictly convex compact regular energy surface carries a periodic orbit. I have understood the proof. What I am wondering about ...
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1answer
38 views

Passage in a proof from Hofer-Zehnder

The proof I'm referring to is to the following theorem. Assume $S$ is a compact regular and strictly convex energy surface for the Hamiltonian field $X_H$ in $\mathbb{R}^{2n}$. Then $S$ carries a ...
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11 views

Natural boundary and natural transient conditions of $F(x, y, y', y'')$

In view of finding the extreme points of the integral $$I=\int _a ^b F(x,y,y',y'') dx$$ what natural conditions should be used if $y(a), y(b), y'(a), y'(b)$ are not preassigned? Also, if $F$ is ...
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1answer
36 views

Integral with variation (principle of least action)

I have \begin{equation} \delta S=\delta \int_{y_1}^{y_2}\bigr[p_x\frac{dx}{dy}+(-E)\frac{dt}{dy}-(-p_y)\bigr]dy \end{equation} with fixed values at the limits on x(y) and t(y), and the teacher asks me ...
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2answers
40 views

Find the curve connecting $(x_1,y_1)$ to $(x_2,y_2)$ that minimizes the surface area of the volume of revoluion

Given two points $(x_1,y_1)$ and $(x_2,y_2)$, find the curve $\gamma$ connecting them such that the surface area of the volume obtained when rotating the curve along the $x$-axis is minimized. First ...
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Uniqueness of solution to Helmholtz-style equation

Let $\Omega\subset \mathbb{R}^n$ be open, bounded and connected with $C^1$ boundary. Suppose $q\in L^\infty(\Omega)$ and that $q\geq C>0$ almost everywhere. Consider the following PDE problem: ...
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1answer
678 views

DuBois-Reymond Lemma

I know thats the following statement is true. $f,g$ are continuous function $[a,b]$.Suppose $\int\limits_a^bf(t)h(t)+g(t)h'(t) \, dt=0$ for every $h$ belonging to $C_0^{\infty}[a,b]$, then $g$ is ...
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1answer
30 views

Simplifying trick for extremizing functionals

If I have a functional $I[y] = \int{({y^\prime}^2- 1)^2}dx$ , since $f(x) = x^2$ is an increasing function for $ x > 0$, can I make the conclusion that $I[y]$ is extremized for at the same ...
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1answer
44 views

Calculating the time of a Brachistochrone

I derived the general equation of a Brachistochrone, which is a cycloid. $y=A(1-\cos\theta)$ $x=A(\theta-\sin\theta)$ I'm now trying to calculate the time needed to go from the origin to a point ...
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1answer
34 views

Generalization of the Beltrami identity to functionals with higher derivatives

Suppose that I have a functional $S[q]=\int_a^b L(t,q(t),q'(t))\,dt$. Such a functional is well-known to extremized by a choice of $q(t)$ satisfying the Euler-Lagrange equation ...
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1answer
53 views

4th order differential equation from Euler-Lagrange

I am trying to extremise the functional $\int{[y + \frac{1}{2}y^2 - \frac{1}{2}(y^{''})^2]}dy$ and so from Euler-Lagrange I get the differential equation $1 + y + y^{(4)} = 0$ and I have no idea how ...
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39 views

How to solve these simultaneous equations?

I'm doing questions from this page: http://tartarus.org/gareth/maths/tripos/IB/Variational_Principles.pdf and I'm doing Question 2013 1/I/6A The question asks to find the cylindrical cup of least ...
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19 views

Minimum surface area soap film variational principles

An axisymmetric soap film $y(x)$ is formed between two circular wires at $x = ±l$. The wires both have radius $r$. Show that the shape that minimises the surface area takes the form ...
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1answer
23 views

Reference about the space of closed curves in Riemann manifold

Some days ago, I listened a report about the width of Riemann manifold. I am interesting in the space of closed curves in Riemann manifold. It seemly has good topology and different construction.For ...
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3answers
52 views

Euler-Lagrange equation with higher derivatives in the functional

Find a function $\phi$ of class $C^2$ (first and second derivatives exist and are continuous) that minimice the functional: $I(\phi)= \int_0^1 \frac {\phi''(t)} {\phi(t)} dt$ and $\phi(0)=1$, ...
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What is meant by Lyapunov functional?

In the context of variational calculus, what is meant by 'Lyapunov functional'? Frankly speaking, in calculus of variation, we are searching for some $u(x)$ to put in $F(x,u(x),u'(x))$ in order to ...
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1answer
29 views

Computing the “Mean Value” of a Point Sample From an Arbitrary Manifold

A friend of mine noticed that taking the "mean" of two points on the circle isn't as easy as just computing the arithmetic mean of their arguments: If one point has argument $-3.13$ radians and one ...
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1answer
515 views

Determine the minimum and maximum values of an integral subject to end conditions

The question: determine the minimum and maximum values of the integral $$\int_0^1 yy'dx$$ subject to the conditions $y(0)=0$ and $y(1)=1$. There is no explicit y dependence, so our Euler-Lagrange ...
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Variational Calculus with Discrete Objective

I'm trying to infer a smooth, non-negative function from some given data ($\vec{m},\vec{\alpha},\vec{\beta}$). That is, I want to solve (I think) $$ \mathop{\arg\!\min}_{g \in ...
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Show that the graph of the function $z(x,y) = x\tan(y)$ is a minimal surface

Show that the graph of the function $z(x,y) = x\tan(y)$ is a minimal surface I'm really lost on how to do this question. I know we have to use the Euler equation to show this, but other than that ...
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Question about trace operator

From the general trace theorem we know for instance that if $f\in W^{1-\frac{1}{p},p}(\partial\Omega)$, then there exists a function $f\in W^{1,p}(\Omega)$ such that $f|_{\partial\Omega}=f$. But is it ...
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1answer
565 views

Sufficient conditions for functional extrema

In the calculus of variations, we can develop a sufficient condition for a functional $J: S \to \mathbb{R}$, $$J(y) = \int_a^b f(x,y,y') \, dx$$ to have a local maximum or minimum, where $S \subseteq ...
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1answer
91 views

Calculus of variations with double integral, inner integral's upper limit is outer variable of integration

I want to find the function $\gamma(t)$ that makes the following stationary. $S(t) = \int_0^t \gamma(t'') \Big(\beta\int_0^{t''}(1-\gamma(t'))dt' + e_0\Big) dt''$ Constraints: $\beta\gt0$, ...
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Prove Ladyzhenskaya- Babuska-Brezzi condition for Poisson problem with homogenoeus Dirichlet boundary condition

I'm considering the problem: \begin{equation} \label{eq:PM} \begin{cases} \mathbf{u} -\nabla p=0\quad \text{ in } \Omega\\ \mathrm{div} \mathbf{u}=-f \quad \text{ in } \Omega\\ p=0\quad \text{ in } ...
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1answer
33 views

Solving a 2-variable Second Order Linear Partial Differential Equation

Part 1: Initial Problem I am trying to solve the following partial differential equation. $$py + q= a\frac{\partial f}{\partial x} + by\frac{\partial f}{\partial y} + c\frac{\partial^2f}{\partial ...
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0answers
73 views

Green's first identity and the calculus of variations

UPDATE: I was able to solve this problem using iterative integration by parts. However, I still cannot find how Green's first identity would apply here. Suppose I had a multiple integral over ...
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90 views

Calculus of Variations with discontinuous Lagrangian

Consider the classical problem of extremizing a functional of the form $$S[x] = \int_a^b L\left(t,x,\dot{x}\right)\ dt.$$ In almost all cases of consideration, the integrand $L$ is considered to be a ...
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43 views

Calculation of Variational Derivative

Following is from Olver's book on Lie groups and Differential Equations: Define the variational problem: \begin{eqnarray*} \mathcal{L}= \int_\Omega L(x,u^{(n)}) \end{eqnarray*} where $u^{(n)}$ ...
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23 views

Can be the derivative the marginal variation of a function?

I know that derivative means in other words slope of the function. $$f(x) = x^2-4x+4$$ $$\frac{d}{dx} = 2x-4$$ So the slop of function at the point 3 is: $2*(3)-4=2$ Can be the derivative the ...
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16 views

Simply Supported Beam Deflection

I would appreciate confirmation on my method and answers for part a), also if anyone knows how to solve this problem that would be appreciated.