In many areas of math, I've been surprised at how much research, past and present, focuses on second order 'things'. Examples:
- Number theory: quadratic reciprocity, quadratic number fields
- Analysis: Second-order PDE's, Lagrangian equations, Newton's laws
- Geometry: conic sections, quadric surfaces, quadratic forms
- Topology: Surfaces
- Calculus: Second-derivative test
And many more. Introductory PDE classes and number theory classes may spend the entire semester on quadratic things.
Why is this so common? Is it because we can't handle third-order things, or is it because second order things are more important/common than higher-order things?
Best Answer
My numbering below does not correspond to the original numbering in the question.
In many areas of math homogeneous polynomials appear. Diagonal homogeneous polynomials, like diagonal matrices, have a particular simple form. Most matrices are not diagonal, and there are some important theorems in linear algebra that give conditions under which a matrix can be diagonalized by a linear change of variables (spectral theorems). For homogeneous polynomials, those of degree $1$ are diagonal by definition and it's a basic theorem that all homogeneous polynomials of degree $2$ over any field (not of characteristic 2) can be put into diagonal form by a suitable linear change of variables over that field. This is false in every degree greater than $2$. See https://mathoverflow.net/questions/77702/transformation-of-a-cubic-form for an example of a cubic form that is not diagonalizable over the rational numbers. Because a quadratic form in two variables can be diagonalized by completing the square, you could think of diagonalizability of quadratic forms in more than two variables as a generalized form of completing the square, but I think a better way to look at this is geometric: the reason quadratic forms (outside of characteristic $2$) can be diagonalized over their field of definition is because of the existence of reflections in orthogonal groups. That leads in to our next topic.
In group theory, a rich source of groups comes from orthogonal groups of quadratic forms. There is no problem in extending this definition to higher degree: for any homogeneous polynomial $f(\mathbf x)$ in $n$ variables over a field $K$, define the orthogonal group of $f$ over $K$ to be all $A \in {\mathrm GL}_n(K)$ such that $f(A\mathbf x) = f(\mathbf x)$. Jordan proved that for $f$ of degree greater than 2 and "nonsingular" in a suitable sense, and $K$ of characteristic 0, the orthogonal group of $f$ over $K$ is finite. For instance, if $f = x_1^d + \cdots + x_n^d$ and $d > 2$ then the orthogonal group of $f$ over the complex numbers consists of the permutations of the $n$ variables, multiplication of each variable by a $d$th root of unity in $\mathbf C$, and compositions of these operations. For me, this finiteness of orthogonal groups in degree above $2$ is a vague reason why classical geometry is governed by degree $2$ objects. Without an infinite orthogonal group there are not enough interesting transformations, at least algebraically. Moving from algebra to analysis, the Morse lemma provides an important role for quadratic forms in topology.
In physics, I've always thought that most important differential equations are second-order essentially because Newton's laws connect the physical concept of force to acceleration, which is a second derivative. Newton's first law shows force in general couldn't be directly related to velocity, a first derivative, and the next derivative turns out to be what we need. There are some interesting differential equations of physical interest are higher than second-order, such as the KdV equation.
Quadratic fields are popular in number theory because they are computationally tractable and can be described in a more or less uniform way. One nice feature of quadratic fields is that every order in a quadratic field can be described in the computationally concrete form ${\mathbf Z}[\alpha]$ but this is false for any number field of degree greater than $2$: if $[K:\mathbf Q] > 2$ then there are orders in $K$ that are not of the form ${\mathbf Z}[\alpha]$. Also, in any number field $K$ of degree greater than 2 there are infinitely many ${\mathbf Z}$-lattices in $K$ that are not invertible as a fractional ideal for their ring of multipliers. That is, if $L$ is a ${\mathbf Z}$-lattice in $K$ then $L$ is a fractional ideal for the ring $R(L) = \{x \in K : xL \subset L\}$ and infinitely many such $L$ are not invertible $R(L)$-fractional ideals. That never happens for $\mathbf Z$-lattices in quadratic fields and is related to the fact that all orders in quadratic fields have the form ${\mathbf Z}[\alpha]$. Historically, quadratic fields were first studied in number theory as an outgrowth of the study of quadratic forms, and this ties in somewhat with the first item above.
The field $\mathbf R$ has only one proper finite extension, namely $\mathbf C$, and this is responsible for Hamilton's quaternions being the only noncommutative finite-dimensional $\mathbf R$-central division algebra. If you study division rings over other fields, then the simplest nontrivial examples are in degree 4 (necessarily: the dimension over the center is a perfect square if it is finite), and these are the quaternion algebras. Their description is closely related to the quadratic extensions of the field. To describe, concretely, a division ring having dimension greater than 4 over its center means you're dealing with an object of dimension at least 9, and you can appreciate that this is going to be more complicated if you want to study it computationally. (There is a general theory of division rings, or more broadly central simple algebras, with quaternion algebras being the simplest interesting examples in the same way that quadratic fields are the simplest interesting field extensions.)
Here is a minor, but interesting, reason that 2 is special: the only finite extension of $\mathbf R$ has degree $2$, and there is a theorem of Artin and Schreier that if $k$ is any non-algebraically closed field such that its algebraic closure is a finite extension, then the algebraic closure must be a quadratic extension and $k$ must look in some sense like the real numbers ($k$ has to be a real-closed field).