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

learn more… | top users | synonyms

2
votes
0answers
47 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 [...
0
votes
1answer
49 views

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

I'm bit confusing about definition of weak solution. If I have the following problem: $\begin{cases} \tag{P} -\Delta u = f \textrm{ in } \Omega, \\ u = 0 \textrm{ in } \partial\Omega, \end{cases}$ ...
1
vote
0answers
37 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 $$ \iiint_V (u_x^2+u_y^2+u_z^2) \, dx \, dy \, dz $$ ...
1
vote
1answer
34 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 $(y')^2+3yy''=0.$...
0
votes
1answer
34 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 F(x,y,y')....
0
votes
0answers
12 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 ...
0
votes
1answer
38 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 ...
1
vote
0answers
36 views

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: $$-...
0
votes
2answers
42 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 ...
2
votes
2answers
76 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 dx~?$...
0
votes
1answer
34 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 ...
0
votes
1answer
49 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 $(...
2
votes
1answer
41 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 $\dfrac{d}{dt}\dfrac{\...
3
votes
1answer
56 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 ...
0
votes
0answers
26 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 $$y(x) = ...
0
votes
0answers
43 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 ...
1
vote
0answers
16 views

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 ...
1
vote
3answers
63 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 $\phi(0)=1$, $\phi(...
0
votes
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 ...
0
votes
1answer
30 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 ...
1
vote
0answers
17 views

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 C^1((-1,1)\to(0,\infty))}...
0
votes
1answer
30 views

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 ...
1
vote
0answers
46 views

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 I'...
0
votes
0answers
17 views

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 } \...
1
vote
1answer
39 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 ...
1
vote
0answers
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 $u^{(n)}$ ...
3
votes
0answers
77 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 $p$-...
0
votes
0answers
19 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.
0
votes
0answers
14 views

Factoring Weierstrass Excess function

I have a Lagrangian $$ L(x,\dot{x}) = \dot{x}^2(1+\dot{x})^2 $$ For this I have the Weierstrass Excess function as $$ E(x,\dot{x},x') = x'^2(1+x')^2 - \dot{x}^2(1+\dot{x})^2 - (2\dot{x}+6\dot{x}^2+4\...
0
votes
0answers
24 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 ...
0
votes
2answers
22 views

Equilibria of the system Calculus of Variations

The harmonic oscillator is described by the action functional $$J[x] = \int_{t_0}^{t_1}\left(mx'^2 −\frac{1}{2}kx^2\right) dt$$ where $m$ is the mass and $k$ is the spring constant. a. Show that the ...
0
votes
0answers
50 views

Explanation of this integral identity in the proof of Wirtinger's inequality from Hardy-Littlewood-Polya

I report the following excerpt from the book "Inequalities" by Hardy-Littlewood-Polya, page 184, where Wirtinger inequality is proven using variational methods. I'm trying to understand what is the "...
1
vote
0answers
21 views

calculus of variation

Among all curves joining a given point (0, b) on the y-axis to a point on the x-axis and enclosing a given area S together with the x-axis, find the curve which generates the least area when rotated ...
1
vote
0answers
16 views

Time independent vs. time dependent lagrange multiplier

What are the differences between these two in applications? For example: $$max\sum_{t=0}^{\infty} \beta^t u(c_t)$$$$s.t.f(c_t,c_{t+1},x_t,x_{t+1})=0$$ What are the differences between: $$L=\sum_{t=0}^{...
0
votes
0answers
33 views

The definition of the First Variation - Calculus of Variation

I have the following definition of the functional derivative $ \frac{\delta S}{\delta\gamma}$, where $S$ is my functional and $\gamma$ is a curve: $$\tag{1} \int^B_A \frac{\delta S}{\delta\gamma} h(x)...
0
votes
0answers
32 views

One dimensional obstacle problem - how to determine coincidence set

I was wondering whether someone could comment on my line of reasoning and, if possible, point me to some relevant literature etc. In general any help will be much appreciated! Suppose $\Omega = (0,1)$...
1
vote
1answer
30 views

$J[y]=\int_a^bF(x,y,y')dx$ with constraint and free boundary

Suppose the variation problem $$J[y]=\int_a^bF(x,y,y')dx$$ with free boundary and constraint $\int_a^bG(x,y,y')=l$, how can formulate the corresponding Euler-Lagrange equation? For fixed boundary ...
0
votes
1answer
39 views

Why the need of Sobolev spaces in this proof of isoperimetric inequality?

I was reading the chapter about isoperimetric inequalities in DaCorogna's book "Introduction to The Calculus of Variations". The isoperimetric inequality is proved to be equivalent to Wirtinger ...
2
votes
0answers
42 views

Where does Jacobi's accessory equation come from?

I'm reading Charles Fox's An Introduction to the Calculus of Variations and in section 2.4 he just suddenly introduces Jacobi's accessory equation and I don't understand where it's coming from. ...
0
votes
1answer
17 views

Quasiconvexity (in the sense of Morrey) implies Rank-One convexity

I am trying to understand why Quasiconvexity implies Rank-One convexity. In a standard proof of this fact a sequence of functions is constructed, which converges weakly to zero in $W^{1,p}.$ in ...
0
votes
0answers
24 views

Making some mistake in finding the solution to the catenary

I seem to be getting a little bit different solution to the catenary problem. Maybe I'm just not seeing how its equivalent to the regular solution. Here's what I've got Starting with $$U=\lambda g \...
0
votes
1answer
33 views

isoperimetric problem in variational methods

I have a problem that requires to find the smallest perimeter of a curve that goes trough points a,b and encloses an area A between the curve and x axis. I used the calculus of variations approach. $...
1
vote
1answer
59 views

Isoperimetric problem with constraint

Solve the isoperimentric problem to minimize $$\int_0^1 y(x)\,{\rm d}x$$ subject to the constraint $$\int_0^1 \sqrt{1 + y'(x)^2}\,{\rm d}x = \frac{\pi}{2}$$ with $y(0) = 0$ and $y(1) = 0$. My Work: ...
2
votes
1answer
76 views

Why is finding a closed form of a geodesic so difficult?

What is it about geodesics (does this term apply in $R^n$ or just $R^3$?) that makes finding a closed form of the curve so difficult? My thoughts: The minimal curve may not be unique which ...
0
votes
0answers
21 views

Exact Galerkin solution

Let's consider the following linear variational problem (a < b) : and its finite element Galerkin discretization, based on a trial and test space on some uniform mesh M of [a, b]. the space ...
1
vote
1answer
33 views

From Euler-Lagrange equation to time-dependent problem

I am reading this pdf which is about an image denoise model. Essentially, we want to find a function $u$ such that $u$ minimize the following functional: $$F(u) = \lambda \int_\Omega |f-u|^2 \,dx\,dy +...
1
vote
1answer
42 views

Is This Linear Functional Bounded?

In the space of $L^1[0,1]$, is the following linear functional $I:L^1[0,1]\rightarrow \mathbb{R}$: $$ I(f) = \int_0^1 x^3f(x)dx $$ bounded on this section of the norm ball $\{f:\|f\|_1 = 1 \text{ ...
1
vote
1answer
24 views

calculus of variations - connecting functionals in different regions.

This is a seemingly small but annoying problem I can't seem to find an answer to. I have a problem of the form $$J(y) = \int^{x_1}_{0} F(x,y,y') \,ds + \int^{x_2}_{x_1} G(x,y,y') \,ds$$ where the ...
2
votes
0answers
23 views

Lifting the Einstein-Hilbert action into the frame bundle

If we have a four dimensional real spacetime $(M,g)$, with $g$ being a $(-+++)$ signature Lorentz-metric, and $\{\theta^0,\theta^1,\theta^2,\theta^3\}$ is a local orthornormal coframe defined in some ...
0
votes
1answer
29 views

Proving a trajectory is critical for an action functional

Suppose a particle with mass $m$ moves freely in $\mathbb{R}^{n}$ under the influence of a conservative force field with potential $\Phi\in C^{2}([t_{1},t_{2}]\times\mathbb{R}^{n})$; that is, its ...