# Questions tagged [variational-analysis]

Variational analysis is a branch of mathematics that extends the methods arising from the classic calculus of variations and convex analysis to more general problems of optimization theory, including topics in set-valued analysis, e.g. generalized derivatives.

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### Producing term with Lie derivative from variational principle

Is it possible to obtain variational principle (Lagrangian) such that the equations of motions contain Lie derivative? For example, if $g$ is a standard metric tensor in Euclidean space $E^3$ and we ...
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### Explicit Solution to Euler Lagrange Equation of Shortest Distance between two Points

So if $f(t)=(x(t),y(t))$ with $f(0)=a$ and $f(1)=b$, I should minimize $L(f)=\int_0^1{|\dot{f(t)}|}dt$. I get a jumble of equations when solving the Euler Lagrange equations with respect to $t$. Would ...
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### Minimal divergence-free projection onto unit vector field

I'm attempting to use a variational method to find a critical vector field. I want to find the local components of a given vector field, $\bf{v}$, that are as "close" to $\bf{v}$ as possible, but are ...
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I'm wondering if the Pohozaev identity: n\int_\Omega\int_0^{u(x)}f(t)\operatorname{d}t\operatorname{d}x-\frac{n-2}{2}\int_\Omega u(x)f(u(x))\operatorname{d}x=\frac{1}{2}\int_{\partial\Omega}\left|\... 1answer 380 views ### Confused by Kullback-Leibler on conditional probability distributions I understand the Kullback-Leibler divergence well enough when it comes to a probability distribution over a single variable. However, I'm currently trying to teach myself variational methods and the ... 0answers 51 views ### Error is the Projection on the Noise - Does the intuition carry over from least squares? My question concerns a generalization of the following: Background It is a well known that the least squares problem (on some Hilbert space) \begin{align*} \|AX-Y \|_2^2 \to \min_{A}! \end{align*} ... 1answer 60 views ### How to simplify the Euler-Lagrange equation of Brachistochrone in this way? I already know that in the Brachistochrone problem, we have Euler-Lagrange equation:\frac{1}{2y}\sqrt{\frac{1+y'^2}{y}}+\frac{d}{dx}[\frac{y'}{\sqrt{y(1+y'^2)}}]=0$$To solve this equation, we ... 0answers 20 views ### About the equivalence of two definitions of topological linking If E is a Banach space, denote with I the identity map of E, denote the space of continuous functions from E to E with \operatorname{C}(E,E) and denote the space of homeomorphism from E ... 0answers 31 views ### Minimizing a nonlinear functional over finite elements I am looking for a reference (an url, a book, or a paper) that could help me in discretization and minimization of the following cost function J\left(\omega\right), over 3D tetrahedrons (finite ... 0answers 85 views ### Variational formulation of a parabolic equation I have the following problem$$ \dfrac{\partial u}{\partial t} - \Delta u + F(u)= f(x,t) $$where u \in L^2([0,T[;H^1(\mathbb{R}^n)) with \partial_t u \in L^2([0,T[;(H^1(\mathbb{R}^n))') I want ... 0answers 12 views ### Bounded-length path that maximizes work in non-conservative field Given a "sufficiently well behaved", but not necessarily conservative, vector field {\bf E} in \mathbb{R}^3 defined in a bounded domain \Omega, what is the maximum value of$$ \int_C {\bf E} \...
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The task is to find the sequence of functions $\lbrace f_n \rbrace _{n\geqslant1} \subset \mathbb{K}\mathbb{C}^{1} \left[ 0;1 \right]$ (where $\mathbb{K}\mathbb{C}^{1} \left[ 0;1 \right]$ is a set of ...
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### Beam equation is neccesary condition for minimum of specific functional.

The beam equation is given by: $Mu^{(4)}(x)+Nu''(x) = f(x) \; x \in [0,L]$ This is supposed to represent the bending of a beam of length $L$ when a sectional mass density $f(x) \ge 0$ acts on the ...
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### Finding stationary point of the functional

Find the stationary point of the functional $$J[y]=\int \left( x^2y'^2+2y^2 \right) dx$$ where $y(0)=0, y(1)=2.$ My Solution: E-L equation: $x^2y''+2xy'-2y=0.$ This is also Cauchy-Euler equation. ...
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### Relation between two types of pseudomonotone function

As far as I know, there are two different definitions for pseudomonotone function. The first definition was introduced by H.Brezis in 1968: Pseudomonotone function in the sense of Brezis: Let ...
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### Calculus of Variation - Minimize a Functional

I am struggling with a basic question from a homework assignment regarding Calculus of Variation. I would love if somebody could get me started in the right direction, as I am pretty lost and the ...
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### Numerical compuation of functional derivative

I was wondering if it is possible to numerically compute $$\frac{\delta L}{\delta f}$$ where $L$ is a functional of the function $f$ known only in a discrete number of point by numerical computation ...
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### Legendre Transforms

I want to show that the Legendre Transform of a function $g(x) = kf(x-x_0) + m$ for $m$, $k$ constants is equal to $g^*(p) = kf\left(\frac{p}{k}\right)+p \cdot x_0+m$ where $f^*$ is the Legendre ...
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### showing a different form of the Euler Lagrange Equation provided $f\in{C^2}$ and $y'\ne{0}$

Suppose that $f(x,y(x),y'(x))$ is s.t $f\in{C^2}$ and $y'(x)\ne{0}.$ I am trying to show that the Euler-Lagrange equation $\frac{\partial f}{\partial y}-\frac{d}{dx}\frac{\partial f}{\partial y'}=0$ ...
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### Is this function differentiable

Define the following function \begin{eqnarray} f(Q_p) &=& \max \quad x^T(Q-Q_p)x +\mbox{trace}(Q_p X) \\ &&s.t. \quad (x, X) \in \mathcal{P} \end{eqnarray} where $\mathcal{P}$ is ...
### Given a smooth family of strictly convex norms in $\mathbb{R}^n$ ($n\geq 2$), what can we say about the family of dual norms?
Let $U\subset{\mathbb{R}^n}$ be an open set (may has compact closure) and $F:U\times\mathbb{R}^n\rightarrow\mathbb{R}$ such that $F(p,\cdot):\mathbb{R}^n\rightarrow\mathbb{R}$ is, for each $p\in{U}$, ...
Fox states in Section 2.4, pg. 38, that "Anticipating this result, it follows that even if $u(x)$ vanishes at either or both of the values $x=a$ and $x=b$, both $t^2(a)/u(a)$ and $t^2(b)/u(b)$ still ...