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.
Questions tagged [variational-analysis]
322 questions
2
votes
1 answer
Variational principal question regarding functions that have a minimum at the origin under a restriction.
I'm going over some old assignments from a couple terms ago and have come across a problem from my variational principals module.
I looked at the function in the hint and noticed that for some points $f(x,y)<0$ but $f(0,0)=0.$ So my issue is in my…
kam
- 1,255
2
votes
0 answers
variation method
Define $ F[\phi]=\int dr \{V(\phi)+\frac{1}{2} \kappa \vert \nabla \phi \vert^2 \}$, then how can we deduce that$ \frac{\delta F}{\delta \phi}= \frac{d V}{d \phi}-\kappa \nabla^2 \phi $?
I feel confused on some basic points, so details will be very…
Yidong Luo
- 411
- 2
- 12
1
vote
0 answers
Prove that $u(x_i)=u_h(x_i)$ in variational discretization
Suppose $f \in L^2(0,1)$ and $u \in H^1_0$ is the solution to the following problem:
\begin{equation}
\int_0^1 u'(x)v'(x) dx=\int_0^1 f(x)v(x)dx ~~~~~~~~~~~~~~~~~~~~(1)
\end{equation}
for all $v \in H^1_0(0,1)$.
Now consider discretized problem of…
Thomas345
- 11
1
vote
0 answers
Cylindrical surface with maximum area?
Imagine we have a parametic surface given by:
$$
\Phi(r, \theta) = \begin{pmatrix}
r \cos{\theta} \\
r \sin{\theta} \\
f(r) \\
\end{pmatrix}
$$
$$
r \in \mathbb R^+, \; \; \; \ \theta \in [0, 2\pi]
$$
$f$, will only depend on $r$, for…
Álvaro Rodrigo
- 763
1
vote
1 answer
How do I go about treating this variational problem?
Let's say I have a set of $N$ members (not ordered) $x_i$, and suppose $N$ is even. To each point are associated $k$ characteristics $x_{ij}$, $j=1,...,k$. There is another additional characteristic $x_{i0}$ which is boolean, and there are $N/2$…
1
vote
2 answers
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 simplify the above equation and…
Perry_W
- 85
0
votes
0 answers
What is the point of the Jacobi accessory equation?
In Where does Jacobi's accessory equation come from? Chappers gives two derivations of the Jacobi accessory equation. From the first, it is clear that the Jacobi accessory equation is a means to ensure that a stationary point of the functional is a…
Seb Ellwood
- 35
0
votes
1 answer
Using these definitions how to prove $\gamma =\frac{\mu }{L^{2}}$ and $L=\frac{1}{\gamma }$
A mapping $\psi :H\rightarrow H$ is said to strongly monotone if $\exists $ $%
\mu >0$ such that
$$
\left\langle \psi \left( u\right) -\psi \left( v\right) ,u-v\right\rangle
\geq \mu \left\Vert u-v\right\Vert ^{2}
$$
A mapping $\psi :H\rightarrow H$…
Subhan Siraj
- 3
- 3
0
votes
1 answer
Properties of the $\frac{1}{2}x^2$ function relevant for variational analysis in mathematical physics
There is a theorem in mathematical physics that, by the looks of it, hinges on the nature of the function $f(x)=\frac{1}{2}x^2$
This question is about examining properties of the $f(x)=\frac{1}{2}x^2$ function, that is why I am submitting this…
Cleonis
- 186
0
votes
0 answers
Variational methods
What happens if I use the free parameters in variational methods in a non-linear manner?
I have this question in front of me and I am not sure how to answer it.
hcl734
- 11
0
votes
1 answer
Similarity between the differential of functionals and functions
In the book Calculus of variations by I. M. Gelfand, the differential of a functional is defined in the following way, and its uniqueness is proven:
However, it seems strange to me that the expression $\Delta J[h]=\varphi[h]+\epsilon||h||$…
Wild Feather
- 644