# Tagged Questions

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

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### Variation of a d'alambertian operator

Let $M$ be a pseudo Riemannian manyfold, $H$ be function of a scalar curvature $R$. Assume that variation of the metric tensor and it's first derivatives is zero on the border $\partial M$. Which ...
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### Maximise the integral w.r.t. probability measure.

Let $(Z_t)_{0\leq t\leq T}$ be a stochastic process. Then $Z_T$ is a r.v. and $F_{Z_T}$ a corresponding cdf. Suppose $\mathbb{E}[|e^{Z_t}|]<\infty$ for all $t\geq0$. Also \mathbb{...
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### Maximization of the Expectation of a function

Recently I was thinking in general on how to maximize the expectation of a function (not necessarily a utility function, but apparently this is a common case). To give an idea of the kind of problem, ...
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The Legendre transform, or transformation, seems to have many properties which are useful in different fields. For example: It switches between Lagrangian and Hamiltonian formalism in mechanics / ...
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### Movable end points Calculus of Variation.

Given problem is $J[y] =\int_{0}^{x_1}y'^2dx$ with $y(0)=0$ and $y(x_1)=-x_1-1$. After solving Euler Lagrange equation I got $y=Ax+B$ . And using first boundry conditon I got $y=Ax$ We have ...
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### Is there a reason for different nomenclature on Calculus of Variations?

While sightseeing aspects of Calculus of Variations, the following fact elludes me: there is a plethora of new definitions which seem redundant to me. This phenomenom happens, of course, with other ...
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### Finding minimal area of a cylindrically symmetric surface

I am told that I have a cylindrically symmetric surface that is bounded between two circles $r=a$ at $z=\pm b$. I'm meant to use the Euler-Lagrange equation, so I'm trying to a functional for the ...
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### Functional integration and Feynman path integrals in wolfram alpha

Is it possible to do Feynman path integrals in wolfram alpha? Say for a free quantum mechanical particle. The reason I am interested in this is because I would like to see how it arrives at the ...
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### Brachistochrone problem with extra derivative condition

I was given for HW in classical mechanics the brachistochrone problem but they also mention their that besides the given end points they demand that at the final point: $y'(x_f)=0$. Now, since I didn'...
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### Alternative to Arnold's mathematical methods

I have difficulties understanding Arnold's book of mathematical methods of classical mechanics. Yet I should get some familiarity with the subjects found at chapters 3,4,7,8 before next semester to ...
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### Maximum over Probabilistic Distribution Functions Space

Suppose $P$ is the set of functions where $p\in P: R^{+2}\to R^+$ and $p(t,s)$ is differentiable in $t$. $\forall t, p(t,\cdot)$ is a probability distribution on the positive axis $s\in [0,\infty)$, i....
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### Difficulty in solving calculus of variations problem.

I am solving a problem on calculus of variation in which $F(x,y,y')$ is given as $F(x,y,y')=e^yy'^2$ After solving Euler equation I got this $2y'' +2y'-y'^2=0$. I don't know how to proceed further. ...
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### textbook on calculus of variation which focuses on the following topics

I need a textbook (or set of online lecture notes) on calculus of variation which focuses on the following topics "Variation of a functional, Euler-Lagrange equation, Necessary and sufficient ...
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### Calculate $\Delta J$ for a functional

$J(y)=\int_{0}^{1} (x^2-y^2+(y')^2)dx$ $y(x)=x, h(x)=x^2$ I need to calculate $\Delta J$ and I am given this from the answer key: $\Delta J = J(y + \epsilon h)-J(y) = J(x + \epsilon x^2) - J(x)$ ...
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### Continuous representative for functions in $W^{1,2}(\mathbb{R})$

I want to prove that $K(x,y) = \frac{1}{2}e^{-|x-y|}$ is a reproducing kernel for $W^{1,2}(\mathbb{R})$ and as a hint I have given that for $f\in W^{1,2}(\mathbb{R})$ I should use its continuous ...
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### Feynman problem on action

It is very weird for me that a newbie can ask a new (may be silly, sorry...) question but must have 50 reputation to comment. When I see a good question like this but have no answer what I have to ...
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### Weak derivative of $\int_0^x g(t)\,dt$ is $g$ on $(0,1)$

I'm working on a functional-analysis problem set, and the question is: Let $g\in L^1(0,1)$, and define $f(x) = \int_0^x g(t)\,dt$. Show that $f\in W^{1,1}(0,1)$ and that the weak derivative of $f$ ...
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### A homogeneous double integral equation

I've happened to stumble upon an interesting double integral equation: $$0=\int_0^\ell\int_0^\ell f(s,t)\mu(t)\mu^\prime(s)\,ds\,dt$$ Here $f$ and $\mu$ are at least continuous (if you want higher ...
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### Minimizing distance between two curves. Can the Calculus of Variations be used?

Given two curves, one might want to find the minimum distance between two points. It is fairly straightforward to find minimums of the function $$(x_1-x_2)^2+(y_1-y_2)^2$$ which corresponds to the ...
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### Energy functional and Euler Lagrange equation

We know that for potential energy functional, its derivative is called the Euler Lagrange equation and physically, it means that at the given point there is a force balance. Now if the energy ...
### Time it takes a particle of $2x^2+3y^2+4z^2=9$ to reach the xy plane using derivatives.
Consider in $\mathbb{R}^3$ the surface $2x^2+3y^2+4z^2=9$. Suppose a particle leaves the point $(1, 1, 1)$ located in the surface along the normal at that point to the $xy$ plane at a speed of one ...
Say I have some function $$\mu:[0,\ell]\to\mathbb{R}^+,$$ and a bijective function $$\phi_\varepsilon:[0,\ell]\to[0,\ell].$$ Let $s$ represent the variable in $[0,\ell]$ and say that \$\varepsilon\in(...