Nearly two centuries before Chebyshev’s systematic approach to moments there lived a mathematical prodigy. A Cambridge graduate, educated on Copernicus and Galileo, he knew about mass and motion but felt bothered by them. You see, they weren’t the mass and motion we know today; they were fickle philosophical concepts — no math, just a general idea of what to look for. That fuzz bothered him so much that he decided to investigate and clarify; by 1687, now a professor of Mathematics, Issac Newton, published Principia. Unknowingly, he had created the core of physics and calculus.
So we are talking about physics today (my favorite application of moments)! But what’s the excitement for? You may be wondering how physics actually has anything to do with moments. We spent the last post covering moments of a PDE to learn the fundamentals; now that we understand the essence of the moment, we begin to play with what we can use. It turns out that we can generalize past a probability distribution to any type of distribution, such as a density function. But what is a density function? Why do moments generalize to it? What does that generalization reveal? Let’s cover that today!
Before we go any further let me introduce a few terms as they will be used in this post:
Density Function:
- An object’s density function, $\rho(x)$ is a function that describes the object’s distribution of mass density at any point in space
Rigid Body:
- An object whose density function does not change (as long as we look from the object’s point of view)
- For example a liquid would not be a rigid body because depending on its position or motion, its distribution of density can change (ie the liquid moves)
Tensor:
- A tensor is a set of numbers. The tensor has discrete numerical indices for each number. The amount of unique indices dictates the order of a tensor
- For example, Bob has a shelf with two rows and four columns. He could represent his shelf as a 2X4 second order tensor (aka a matrix) and make a numerical id for the objects he puts into the shelf.
- Unlike a shelf though, the tensor can’t have any weird ‘shape’. It can only be a parallelogram (like a rectangle or square)
Before we can generalize moments to a density function, we need to expand the concept of moments to multiple variables (after all we live in a 3D world, not a line).
For raw moments we will generally use the tensor product but tensor contractions will be used when calculating skewness and kurtosis (oh yeah those two will show up again…). The tensor product will basically combine our n vectors into an n order tensor.
And finally we will integrate from negative infinity to infinity along each variable in $X$.
As we covered last time, $X$ can be modified by centralization and standardization. We can do the exact same thing here! Centralization is just as straightforward. The center will subtract from $X$ before calculating the moment. Standardization is a bit more complicated. The variance will be a matrix, $\Sigma$. To ‘divide’ by $\sqrt{\Sigma}$ we will multiply $X$ by $\Sigma^{-\frac{1}{2}}$ (This is a thing from linear algebra).
Centralized
$$E(Y); E(Y); Y = X-\mu$$Standardized
$$E(Y); Y = \Sigma^{-\frac{1}{2}}X$$Both
$$E(Y); Y = \Sigma^{-\frac{1}{2}}(X-\mu)$$Now that we are prepared, let’s traverse a density function’s moments!
While the PDE’s zeroth moment would mean nothing, $\rho(X)$’s zeroth moment actually means something. Because n would be zero, the moment would be the full sum of density. That makes the zeroth moment the mass!
You might be thinking that we should logically proceed by dividing out the mass from the first moment. The center of mass is a very important piece of information after all! Sure but doesn’t it seem weird that the first moment essentially gives the product of position and mass. Isn’t that awfully reminiscent of momentum, the products of velocity and mass, or force, acceleration and mass? I personally think this actually makes sense! Think of it like this… each bit of matter has the same ‘influence’. More matter has more ‘influence’. So when we take the zeroth moment, we measure not only the mass but the ‘influence’ the object has. The first moment then becomes the object’s position in a space weighed by ‘influence’. But here is the cool thing. Maybe our world truly exists in this ‘influence’ weighted position. And maybe that is why concepts like momentum, the product of mass(‘influence weight’) and velocity, stay constant within a system. It’s as if, without outside force, the ‘influence’ of an object’s motion stays the same. I think this thought process nicely complements any intuition surrounding conservation of momentum and potentially more complicated topics like relativity (mmm we’ll talk about that one soon…).
But I’m forgetting something. Can’t objects also rotate. You have probably heard of the moment of inertia. $$I=\int_{}^{}\int_{}^{}\int_{}^{}\rho(x,y,z)r^{2}dV$$ $$\text{where r is the distance from axis of rotation}$$ This formula looks awfully reminiscent of elements in the second moment. $$I=\int_{-\infty }^{\infty }\rho(x_{1},x_{2},x_{3})\begin{bmatrix}x_{1}^{2} & x_{1}x_{2} & x_{1}x_{3} \\ x_{1}x_{2} & x_{2}^{2} & x_{2}x_{3} \\ x_{1}x_{3} & x_{2}x_{3} & x_{3}^{2}\end{bmatrix}dX$$ Well, $\rho(X)$’s second centered moment is actually called an inertia tensor (the center is the pivot of rotation)! In 3 dimensions (4d and 2d is a mess of its own), this second order tensor can transform angular velocity into angular momentum. There is so much to talk about rigid body rotation (here’s a really cool video if you’re interested) but here is the key takeaway from what I understand. Unless the point of view is from the inertia plane(just like our weighted plane) where the angular momentum is entirely conserved, in normal space, only the angular momentum’s magnitude is conserved (this occurs because the inertia tensor would be rotating with the angular velocity). The actual direction results from the conservation of both rotational kinetic energy and angular momentum. While linear motion would get a singular direction from conserving both energy and momentum, rotational motion could be a singular direction or an entire set of directions that repeat during rotation (the video has a very pretty way of showing this). This (very oversimplified) understanding explains why certain axes of rotation are unstable.
Past the second moment, we have reached speculation territory. I have never seen anyone cover the uses for the third moment in physics. What I have figured out though is that the skewness (as seen here) would (like the center of mass for linear motion) describe the object’s rotational position, not like an angle but like a kind of vector that would move with rotation. A nice intuition behind this would be that a rotationally symmetrical object, a sphere, would have no skewness meaning that all of its rotational positions would be indistinguishable!