I am always fascinated when a quadratic form (or a quadric) arises naturally. I have some elementary examples, but most of all, I want to learn more examples. I hope this question isn't considered too vague for MO. Most forms I list are really elementary, and all are finite dimensional.

I got most of the following examples from M.Berger, Geometry I & II, and from the truly beautiful book "Eléments de géométrie : actions de groupes" by french author Rached Meinmné.

$(0)$ the discriminant on the affine space of unitary degre 2 polynomials

$(i)$ the determinant on endomorphisms of a 2 dimensional vector space, and $\mathrm{Tr}^2-4\mathrm{det}$

$(ii)$ the radical on the space of quadratic forms on a 2 dimensional vector space, and the isotrope cone (not sure about the name, degenerate cone?).

$(iii)$ the family of hermitian forms (built from the Wronskian) on the solution space of the discrete Schroedinger equation that allow one to show the existence of right and left side $L^2$ solutions, and the Weyl m function.

$(iv)$ If $\Delta$ is any $2$ dimensional complex vector space, then $\mathrm{Herm}(\Delta)$, the real vector space of hermitian forms on $\Delta$, carries a natural quadratic form obtained by constructing an essentially unique morphism $\rho$ from $\mathrm{Herm}(\Delta)$ to $\mathrm{Hom}(\Delta\oplus\overline{\Delta})$ such that for all $h\in\mathrm{Herm}(\Delta),~\rho(h)^2$ is proportional to $\mathrm{Id}$, the proportionality defining the quadratic form. Here, $\rho$ only depends on a choice of a nonzero element $\omega\in\Lambda^2\Delta^*$.

$(v)$ If $V$ is a 4 dimensional vector space, then $\Lambda^2 V$ carries the natural quadric $Q(v)=v\wedge v$ where $\Lambda^4 V$ is identified with the underlying field, which vanishes exactly when $v$ comes from the canonical map $\mathrm{Gr}(2,V)\rightarrow P\Lambda^2V$.

I remember reading about one on the space of circles, but I forgot the details. What other examples of natural quadratic forms are there?

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Second-order approximation to a potential function at a critical point? – Qiaochu Yuan Mar 18 '11 at 20:27
I think the question is a little broad. Can you be more specific about what you are looking to gain by having such a list? – Qiaochu Yuan Mar 18 '11 at 20:34
People are quite fond of :  Symmetric bilinear forms [by] J. Milnor [and] D. Husemoller by John Willard Milnor, 1973,Springer-Verlag edition, in English. – Will Jagy Mar 18 '11 at 21:10
What do you mean by "arise naturally"? In your examples you are often introducing constraints on a dimension to force a quadratic form in what otherwise would really be better considered as a homogeneous polynomial of some degree. For instance, the determinant on endomorphisms of an n-dim. space is a homogeneous polynomial of degree n in n variables. To say the determinant becomes a quadratic form when you set n = 2 seems, to me, to be missing the big picture of what happens in general, since the determinant is usually not a quadratic form. (Continued...) – KConrad Mar 18 '11 at 22:11
I would consider as examples that give incentive to study general quadratic forms those constructions which, in general, produce quadratic forms in potentially any number of variables, not just a small number because you fixed some parameter to force a quadratic form. For instance, the trace form on an associative algebra or the Killing form on a Lie algebra. These are symmetric bilinear forms in possibly many variables (depends on the dimension of the algebra) and correspond to some quadratic forms in many variables. – KConrad Mar 18 '11 at 22:14

Dear Olivier, in line with the more advanced nature of this site, let me give an example of a less elementary nature.

Consider a compact Riemann surface $X$ of genus 2 and on it stable vector bundles $E$ of rank 2 whose determinant bundle $\Lambda ^2E$ is isomorphic to some fixed line bundle $L$ of degree $-1$. Newstead has proved that the moduli space of those vector bundles is the intersection of two quadrics in five-dimesional projective space $\mathbb P^5(\mathbb C)$. And one of those quadrics is the Klein quadric in $\mathbb P^5(\mathbb C)$ parametrizing the lines in some three-dimensional projective space canonically associated to $X$ and $L$. (A Klein quadric is the quadric you mention in number (v) of your list.)

References
P E. Newstead Stable bundles of rank 2 and odd degree over a curve of genus 2, Topology 7 (1968), 205-215.
For a geometric description including the role of the Klein quadric, see:
M. S. Narasimhan and S. Ramanan Moduli of Vector Bundles on a Compact Riemann Surface, Annals of Mathematics, Vol. 89, No. 1, 1969 , pp. 14-51.

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@Olivier: when you ask such a broad, open-ended question, it seems to be good form to wait more than a couple of hours to accept an answer. – Pete L. Clark Mar 19 '11 at 5:55
you're right, what I meant to do is to give his answer a thumbs up, not close the question. – Olivier Bégassat Mar 20 '11 at 0:03

In topology, Poincaré duality implies that given a connected and oriented compact manifold M of dimension 4k, the cup product gives rise to an integral non-degenerate symmetric bilinear form on the "middle" cohomology group $H^{2k}(M,\mathbf Z)$.

This gives rise to the definition of the signature, which will maybe be of interest to you. A variant for 4k+2 dimensional manifolds gives rise to the famous Kervaire invariant.

See https://en.wikipedia.org/wiki/Signature_%28topology%29 for example.

A Good reference is the book by Milnor and Husemoller.

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Quadratic forms occur naturally, as stress energy functions, in Bob Connelly's work on tensegrity structures.

You can find a wealth of details on his website, in particular the book http://www.math.cornell.edu/~web7510/framework.pdf

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Two advanced examples from the theory of abelian varieties (for example, elliptic curves) where it is non-trivial to prove that you even get a quadratic form.

Let $A/k$ be an abelian variety defined over an algebraically closed field. Then the degree map $$\deg : \operatorname{End}(A)\longrightarrow\mathbb Z$$ is a quadratic form. A reference for this is Mumford's Abelian Varieties (or my Arithmetic of Elliptic Curves for the dimension 1 case). To indicate why this is non-trivial, I will mention that if you know it for $k=\overline{\mathbb F}_p$, then you can use basic facts about the Frobenius map (roughly, that $a+b\phi$ is separable if $p\nmid a$) to prove the Weil estimate for $\#A(\mathbb F_{p^n})$.

The second example is when $k$ is the algebraic closure of $\mathbb Q$. Let $D$ be a symmetric divisor on $A$. Then the canonical height function $$\hat h_D : A(k) \longrightarrow \mathbb R$$ of Neron and Tate defined by $$\hat h_D(P) = \lim_{n\to\infty} 4^{-n}h_D(2^nP)$$ is a quadratic form. Further, if $D$ is ample, then $\hat h_D(P)=0$ if and only $P$ is a point of finite order. Canonical heights are of great importance in studying the arithmetic of abelian varieties. For example, they appear prominently in the statement of the Birch-Swinnerton-Dyer conjecture. They are discussed in Lang's Fundamentals of Diophantine Geometry and my book with Hindry Diophantine Geometry: An Introduction. (Again, the elliptic curve case is covered in my AEC.)

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Binary quadratic forms arise in nature as norm forms for a quadratic field. This point of view has various consequences in number theory.

1. For a fixed negative discriminant (the definite case), Gauss discovered that the quadratic forms (or their $\mathrm{SL}_2(\mathbb{Z})$ equivalence classes) can be composed. This led him to the phenomenon of the ideal class group before ideals were invented by Kummer and Dedekind. Besides in the ideal class group for more general number fields, Gauss's composition law has found an extension in Bhargava's higher composition laws. These are based on the representation theory of arithmetic groups ($\mathrm{SL}_2(\mathbb{Z})$ and its generalizations), in which regard they are natural structures in themselves. They have striking applications to old problems regarding mean asymptotics of Selmer ranks of elliptic curves, the $3$-parts of class groups of quadratic fields, etc.

2. The Epstein zeta function takes the shape $\zeta_Q(s) := \sum_{\mathbf{n} \neq \mathbf{0}} Q(\mathbf{n})^{-s}$, for a given signature $(d,0)$ quadratic form $Q$. It has all the right analytical properties (meromorphic with simple pole at $s = 1$ and a functional equation relating $s \leftrightarrow 1-s$), allowing to decompose the zeta function of an imaginary quadratic field over a set of representatives $Q$ for the class group. This has consequences for the arithmetic of these fields, beautifully developed in Siegel's Lectures on Advanced Analytic Number Theory (Tata Institute lecture series, 1961).

3. For $d = 2$, $\zeta_Q(s)$ is in effect an Eisenstein series ($|mz+n|^2$ being a binary quadratic form in $m,n$), which is a natural structure all over mathematics, being a continuum of modular forms in the spectral resolution of the hyperbolic Laplacian. Siegel apparently had much interest in the conceptual role played in number theory by the higher rank quadratic forms and their Epstein zeta function. Much of his work was put on representation theoretic footing in Weil's 1964 paper Sur certains groupes d'operateurs unitaires. Michael Berg's book, The Fourier-Analytic Proof of Quadratic Reciprocity, is a terrific introduction to these ideas.

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Perhaps too trivial an example; the fixed points of a (non-intentity) Mobius transform.

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can you explain please? I looked up "fixed points of Moebius transforms" and only found that (the non identity ones) they either fix two points on the boundary $\partial\mathbb{H}$, one point on the boundary and no other, or two points in $\mathbb{H}$ – Olivier Bégassat Mar 20 '11 at 0:28