Detecting non-linear relationships between distributions? X,Y are r.v's exactly related by some unknown non-linear relationship.
Does there exist a neat analog of Correlation, that gives some information about a this relationship? 
 A: Were you looking for something like $Distance Correlation$? 
http://en.wikipedia.org/wiki/Distance_correlation
This will be non-zero for any sort of relationship between your $x$ and $y$. Therefore this can trap arbitrary non-linear relationships though interpreting the values is harder than interpreting correlation.
If this is what you need try the "energy" library in R. 
set.seed(1234)
x<-rnorm(1000,0,1)
y<-x^2
cor(x,y)
[1] -0.03908369
library(energy)
dcor(x,y, R=500))
[1] 0.5478997

#to get a p-value for the distance correlation:
dcov.test(x,y)

        dCov test of independence

data:  index 1, replicates 500
nV^2 = 140.74, p-value = 0.001996
sample estimates:
     dCov 
0.3751596 

A: Correlation using Spearmans (captures non-linear relationships)
corr = df.corr("spearman")

In the case of monotonic relationship, the variables tend to move in the same relative direction, but not necessarily at a constant rate. In a linear relationship, the variables move in the same direction at a constant rate. In some cases, variables increasing concurrently, but not at the same rate. This relationship is monotonic, but not linear.
The Pearson correlation coefficient for these data is 0.843, but the Spearman correlation is higher 0.948.

Note: It doesn't capture quadratic
