# Tagged Questions

**-1**

votes

**0**answers

26 views

### Finding the source for minimizer of a functional for all $C^2$-curve $x(t_0)=x_0$ and $x(t_1)=x_1$

I am trying to find where this problem comes from and its corresponding proof for my students, but I cannot find the source anywhere. If anyone can find the source of this, or has any ideas where I ...

**2**

votes

**0**answers

63 views

### Approximating solutions to minima of the discrete Lagrangian

I have been stuck on this problem for a week and I'm not sure whether or not it is hard or I'm just missing something obvious.
General gist of the problem
I have a variational problem on a ...

**4**

votes

**1**answer

220 views

### Minimizing action squared versus action

I have a very basic question in the calculus of variations:
Suppose I want to minimize the functional
$$A[r, r'] = \int_\Omega L(r, r') dx $$
When is it possible to say that extremals of $A$ agree ...

**5**

votes

**1**answer

219 views

### A Lagrangian problem with a countable family of local extrema ?

Dear MO contributors,
let $r > 0, L > 0$. I am interested in maximizing the integral:
$$
\int_0^{2\pi} \frac{f(\alpha)^2 f'(\alpha)^2}{\sqrt{f(\alpha)^2 + f'(\alpha)^2}} \ \mathrm{d} \alpha
$$
...

**2**

votes

**2**answers

205 views

### Equitable division of a contiguous resource

I have come across the following result regarding equitable division of a resource, which is a simple and immediate consequence of linear programming complementarity (in the infinite-dimensional ...

**2**

votes

**1**answer

133 views

### Proving that a constructed curve solves an optimization problem

Caveat up-front: I'm not a mathematician, so please excuse any stupidity/ignorance that follows. First, let me explain what I'm trying to do: I want to choose a function that maximizes the following ...

**1**

vote

**1**answer

185 views

### Proving a variational problem has no solutions

Consider the following integral
$ \int_{0}^{\frac{\pi}{2}} \left( \sqrt{y(x)^2 + y'(x)^2} \left( \ln \left( \frac{\sin(x)}{1 -\cos(x)} \right) + \frac{\pi}{2} \right) + \frac{\pi}{2} y'(x) + 1 ...

**1**

vote

**1**answer

163 views

### Conditions ensuring extrema are twice continuously differentiable?

For a functional $J[y]=\int_{a}^{b}F(t,y,y')dt$, are there any conditions that ensure extrema over the class of piecewise continuously differentiable functions are all in $C^2[a,b]$?

**3**

votes

**3**answers

452 views

### An optimization problem

The following problem optimization problem arose in a project I am working on with a student. I would like to minimize the quantity:
$$ M=\frac{1}{12} + \int_0^\frac{1}{2} \left( \tfrac{1}{2}-x ...

**0**

votes

**1**answer

124 views

### optimize with respect to domain shape

Let $\Gamma$ be the set of all closed $C^2$ curves in the plane which enclose unit area and let $\Omega$ be the set of all subsets of $\mathbb{R}^2$ that are enclosed by some curve in $\Gamma$. Now ...

**13**

votes

**1**answer

709 views

### What braking strategy is most fuel-efficient?

You notice a stop-light ahead of you and it is currently red. You can't run the red light, so you will have to brake, but braking wastes energy and you want to be as fuel efficient as possible. What ...

**4**

votes

**2**answers

1k views

### Minimizing a function containing an integral

I am trying to optimize a function of the following form:
$L = \int_{t=0}^{T}(AR-x)dt$, where A is a system parameter
i.e. I am trying to find the optimum x(t) that minimizes L over all admissible ...