User shrdlu - MathOverflow

A constant potential due to a designed radial force inside a spherical shell

2012-07-20T19:06:27Z

It is well known that the gravitational forces due to a spherical shell of uniform density cancels in the interior of the shell (in three dimensions). Another way to state this is that the gravitational potential is uniform in the interior of the sphere.

Suppose you are given a sphere of fixed radius and you can design a radial force (and associated radial potential) . Can you

(1) Give example of other potentials that are constant in the interior of the sphere?

(2) Characterize all potentials that are uniform in the sphere's interior?

Literature references are particularly appreciated.

I know that the solutions to (2) are a vector space and are a solution to an integral equation, but it is not clear that this is a fruitful line of attack.

Answer by shrdlu for A constant potential due to a designed radial force inside a spherical shell

2012-07-21T14:03:04Z

I'm answering my own question after a night of sleep.

Without loss of generality we consider the unit sphere centered at the origin. Define the potential between two point masses separated by a distance $\ell$ as $Q(\ell)$. The potential at a point at radius $r$ inside the unit sphere can be found by taking a point at $(x,y,z)=(0,0,r)$ and integrating $Q$ over the unit sphere which yields $$\Phi(r) = 2 \pi \int_0^\pi Q(\ell) \sin \phi \, d \phi $$ where $\ell = \sqrt{1+r^2 - 2 r \cos \phi}$. We want $\Phi(r)=\Phi_0$, a constant.

Change variables in the integral from $\phi$ to $\ell$. Note that $\ell^2 = 1+r^2 - 2 r \cos \phi$ so $2 \ell d\ell = 2 r \sin \phi d \phi$. So the integral equation can be rewritten as $$\frac{ \Phi_0}{4 \pi}= \frac{1}{2r} \int_{1-r}^{1+r} \ell Q(\ell) ~ d \ell $$ Now, stare at the righthand side. It is the average value of $\ell Q(\ell)$ over the interval $[1-r,1+r]$, which we want to be constant for $0\le r <1$. The answer is $$Q(\ell) = \frac{ \Phi_0}{4 \pi} \cdot \frac{\left [ 1 +f(\ell-1) \right ]}{\ell} . $$ where $f$ is any odd function. Three examples:

$$\bullet \ F(z)=0 \Rightarrow Q(\ell) = \frac{ \Phi_0}{4 \pi \ell} , \qquad \text{the Newtonian potential},$$ $$\bullet \ F(z)=z \Rightarrow Q(\ell) = \frac{ \Phi_0}{4 \pi},\qquad \text{a constant potential,}$$ $$\bullet \ F(z)=(3z-z^3)/2 \Rightarrow Q(\ell) =\frac{ \Phi_0}{4 \pi} \cdot \frac{3\ell-\ell^2}{2},\qquad \text{a quadratic potential.}$$

Answer by shrdlu for Identifying a system of ODEs

2012-04-01T04:06:11Z

Forgive the brevity - what you are interested in in something called the Osgood condition. Systems of this kind have been studied for a long time, particularly when the $m_i$'s are all equal. If you look at the $n=2$ case and look at $r= (x_1-x_2)$ the problem reduces $r_t =c/r$ and the fact that $c/r$ is not integrable at $r=0$ tells you that you have a finite time collision.

A good starting point is Ruelle's Thermodynamics text.

Answer by shrdlu for Seeking scalar functions in n>=2 variables (preferably as solution to PDE) with limited regularity.

2011-12-17T17:57:50Z

The porous media equation $$u_t=(u^n)_{xx}$$ has solutions with compact support and therefore is a de facto example. Regularity at the edge of the support depends on the value of n (which is >1).

You may wish to look at Hamilton-Jacobi flows also such as $$u_t=|\nabla u |^2$$ Which form cusps in $R^n$ generically.

Answer by shrdlu for How to isolate $f(x)$ in $f(x+a)=f(x)+a\times g(x)$?

2011-11-01T04:41:28Z

The answer is not unique - you can add any function of period $a$ to f. This is what the singularities are trying to tell you. In a distributional sense, the Fourier transform of a function of period a is supported exactly on the zeros of $e^{2 \pi i a \xi}-1$.

Comment by shrdlu

2012-07-20T19:57:53Z

No, not necessarily. It would yield a designer force between particles that might be positive at some distances and negative at others. However inside the sphere the cumulative force would be zero.