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Results tagged with calculus-of-variations
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user 11260
Questions on the calculus of variations, which deals with the optimization of functionals mostly defined on infinite dimensional spaces.
2
votes
nested integrals in functional derivative
I find it helpful in cases such as this to discretize the integrals, so that $g$ becomes a matrix with elements $g_{ij}=g(y_i,x_j)$ and $f$ becomes a vector with elements $f_i=f(x_i)$. Then $J[f]=\sum …
- 173k
17
votes
Accepted
What variational problem does the parabolic suspension bridge solve?
Quite generally, the shape $y(x)$ of the cable and the shape $w(x)$ of the suspended deck are governed by the Melan equation, a fourth order ODE. A variational formulation is given in Variational form …
- 173k
4
votes
Accepted
Units of time in the gradient flow equation?
If you wish to interpret the gradient flow equation as an equation of physics (rather than mathematics), then you need to introduce a friction coefficient into your problem, which tells you how rapidl …
- 173k
12
votes
Accepted
Who first resolved Hilbert's 20th problem?
A discussion of Tonelli's contributions and their relation to Hilbert's work can be found in this AMS bulletin. The original work was published in Italian, Fondamenti di Calcolo delle Variazioni (Bolo …
- 173k
2
votes
Euler-Lagrange equations and Bellman's principle of optimality
A. multi-dimensional state, one-dimensional time
Multi-dimensional extensions $x\in\mathbb{R}^n$ of the one-dimensional Hamilton-Jacobi-Bellman equations have been considered in Consistency of a Simp …
- 173k
3
votes
Fair surfaces - general mathematical theory
Moreton's thesis work was included in a book that explores fairness measures in modeling from a variety of perspectives: Designing Fair Curves and Surfaces: Shape Quality in Geometric Modeling and Com …
- 173k
2
votes
Unusual problem of calculus-of-variations. Attempt 2
Since the OP states that $\lambda$ is not fixed from the outset, and assuming that $$J_G\equiv\int_{-1}^1 \cos(\tfrac{1}{2}\pi x)G(x)\,dx\neq 0,$$
we can make $I_G$ arbitrarily large by choosing $f(x …
- 173k
6
votes
Accepted
The term for problems "like" Brachistocrone?
"Calculus of variations" seems an accepted umbrella term; at least, looking at the corresponding Wikipedia entry, you'll recognize that most problems in this class are of the type you are looking for: …
- 173k
1
vote
Accepted
Deriving Helfrich's shape equation for closed membranes
The follow-up paper by Helfrich you are searching for is
Bending energy of vesicle membranes: General expressions for the first, second, and third variation of the shape energy and applications to s …
- 173k
7
votes
Accepted
Are all Helmholtz decompositions related?
Q: How are two Helmholtz decompositions related?
A: The scalar fields differ by a harmonic function.
Starting from a first decomposition $\sigma_1,\Gamma_1$, you can construct a second one by adding t …
- 173k
0
votes
Accepted
Definition of Euler-Lagrange equation and properties, where can I find?
These lecture notes by Piotr Hajłasz might have the introductory level you are looking for:
The lectures will be divided into two almost independent streams. One
of them is the theory of Sobolev spac …
- 173k
12
votes
Accepted
Almgren's mimeographed lectures notes on varifolds
Here is the story behind these notes, and a redirect to On the First Variation of a Varifold, W.K. Allard (1972).
a quote from: Selected Works of Frederick J. Almgren
- 173k
4
votes
Gradient descent relaxation dynamics of a Euler-Lagrange equation
The usual way to ensure the convergence of the steepest descent formulation of the Euler-Lagrange equations, is to introduce a friction term, see The Calculus of Variations by Jeff Calder. Instead of
…
- 173k