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Throughout this post, we will work only at the formal level of analysis, ignoring issues of convergence of integrals, justifying differentiation under the integral sign, and so forth. (Rigorous justification of the conservation laws and other identities arising from the formal manipulations below can usually be established in an a posteriori fashion once the identities are in hand, without the need to rigorously justify the manipulations used to come up with these identities).

It is a remarkable fact in the theory of differential equations that many of the ordinary and partial differential equations that are of interest (particularly in geometric PDE, or PDE arising from mathematical physics) admit a variational formulation; thus, a collection ${\Phi: \Omega \rightarrow M}$ of one or more fields on a domain ${\Omega}$ taking values in a space ${M}$ will solve the differential equation of interest if and only if ${\Phi}$ is a critical point to the functional

$\displaystyle J[\Phi] := \int_\Omega L( x, \Phi(x), D\Phi(x) )\ dx \ \ \ \ \ (1)$

involving the fields ${\Phi}$ and their first derivatives ${D\Phi}$, where the Lagrangian ${L: \Sigma \rightarrow {\bf R}}$ is a function on the vector bundle ${\Sigma}$ over ${\Omega \times M}$ consisting of triples ${(x, q, \dot q)}$ with ${x \in \Omega}$, ${q \in M}$, and ${\dot q: T_x \Omega \rightarrow T_q M}$ a linear transformation; we also usually keep the boundary data of ${\Phi}$ fixed in case ${\Omega}$ has a non-trivial boundary, although we will ignore these issues here. (We also ignore the possibility of having additional constraints imposed on ${\Phi}$ and ${D\Phi}$, which require the machinery of Lagrange multipliers to deal with, but which will only serve as a distraction for the current discussion.) It is common to use local coordinates to parameterise ${\Omega}$ as ${{\bf R}^d}$ and ${M}$ as ${{\bf R}^n}$, in which case ${\Sigma}$ can be viewed locally as a function on ${{\bf R}^d \times {\bf R}^n \times {\bf R}^{dn}}$.

Example 1 (Geodesic flow) Take ${\Omega = [0,1]}$ and ${M = (M,g)}$ to be a Riemannian manifold, which we will write locally in coordinates as ${{\bf R}^n}$ with metric ${g_{ij}(q)}$ for ${i,j=1,\dots,n}$. A geodesic ${\gamma: [0,1] \rightarrow M}$ is then a critical point (keeping ${\gamma(0),\gamma(1)}$ fixed) of the energy functional

$\displaystyle J[\gamma] := \frac{1}{2} \int_0^1 g_{\gamma(t)}( D\gamma(t), D\gamma(t) )\ dt$

or in coordinates (ignoring coordinate patch issues, and using the usual summation conventions)

$\displaystyle J[\gamma] = \frac{1}{2} \int_0^1 g_{ij}(\gamma(t)) \dot \gamma^i(t) \dot \gamma^j(t)\ dt.$

As discussed in this previous post, both the Euler equations for rigid body motion, and the Euler equations for incompressible inviscid flow, can be interpreted as geodesic flow (though in the latter case, one has to work really formally, as the manifold ${M}$ is now infinite dimensional).

More generally, if ${\Omega = (\Omega,h)}$ is itself a Riemannian manifold, which we write locally in coordinates as ${{\bf R}^d}$ with metric ${h_{ab}(x)}$ for ${a,b=1,\dots,d}$, then a harmonic map ${\Phi: \Omega \rightarrow M}$ is a critical point of the energy functional

$\displaystyle J[\Phi] := \frac{1}{2} \int_\Omega h(x) \otimes g_{\gamma(x)}( D\gamma(x), D\gamma(x) )\ dh(x)$

or in coordinates (again ignoring coordinate patch issues)

$\displaystyle J[\Phi] = \frac{1}{2} \int_{{\bf R}^d} h_{ab}(x) g_{ij}(\Phi(x)) (\partial_a \Phi^i(x)) (\partial_b \Phi^j(x))\ \sqrt{\det(h(x))}\ dx.$

If we replace the Riemannian manifold ${\Omega}$ by a Lorentzian manifold, such as Minkowski space ${{\bf R}^{1+3}}$, then the notion of a harmonic map is replaced by that of a wave map, which generalises the scalar wave equation (which corresponds to the case ${M={\bf R}}$).

Example 2 (${N}$-particle interactions) Take ${\Omega = {\bf R}}$ and ${M = {\bf R}^3 \otimes {\bf R}^N}$; then a function ${\Phi: \Omega \rightarrow M}$ can be interpreted as a collection of ${N}$ trajectories ${q_1,\dots,q_N: {\bf R} \rightarrow {\bf R}^3}$ in space, which we give a physical interpretation as the trajectories of ${N}$ particles. If we assign each particle a positive mass ${m_1,\dots,m_N > 0}$, and also introduce a potential energy function ${V: M \rightarrow {\bf R}}$, then it turns out that Newton’s laws of motion ${F=ma}$ in this context (with the force ${F_i}$ on the ${i^{th}}$ particle being given by the conservative force ${-\nabla_{q_i} V}$) are equivalent to the trajectories ${q_1,\dots,q_N}$ being a critical point of the action functional

$\displaystyle J[\Phi] := \int_{\bf R} \sum_{i=1}^N \frac{1}{2} m_i |\dot q_i(t)|^2 - V( q_1(t),\dots,q_N(t) )\ dt.$

Formally, if ${\Phi = \Phi_0}$ is a critical point of a functional ${J[\Phi]}$, this means that

$\displaystyle \frac{d}{ds} J[ \Phi[s] ]|_{s=0} = 0$

whenever ${s \mapsto \Phi[s]}$ is a (smooth) deformation with ${\Phi[0]=\Phi_0}$ (and with ${\Phi[s]}$ respecting whatever boundary conditions are appropriate). Interchanging the derivative and integral, we (formally, at least) arrive at

$\displaystyle \int_\Omega \frac{d}{ds} L( x, \Phi[s](x), D\Phi[s](x) )|_{s=0}\ dx = 0. \ \ \ \ \ (2)$

Write ${\delta \Phi := \frac{d}{ds} \Phi[s]|_{s=0}}$ for the infinitesimal deformation of ${\Phi_0}$. By the chain rule, ${\frac{d}{ds} L( x, \Phi[s](x), D\Phi[s](x) )|_{s=0}}$ can be expressed in terms of ${x, \Phi_0(x), \delta \Phi(x), D\Phi_0(x), D \delta \Phi(x)}$. In coordinates, we have

$\displaystyle \frac{d}{ds} L( x, \Phi[s](x), D\Phi[s](x) )|_{s=0} = \delta \Phi^i(x) L_{q^i}(x,\Phi_0(x), D\Phi_0(x)) \ \ \ \ \ (3)$

$\displaystyle + \partial_{x^a} \delta \Phi^i(x) L_{\partial_{x^a} q^i} (x,\Phi_0(x), D\Phi_0(x)),$

where we parameterise ${\Sigma}$ by ${x, (q^i)_{i=1,\dots,n}, (\partial_{x^a} q^i)_{a=1,\dots,d; i=1,\dots,n}}$, and we use subscripts on ${L}$ to denote partial derivatives in the various coefficients. (One can of course work in a coordinate-free manner here if one really wants to, but the notation becomes a little cumbersome due to the need to carefully split up the tangent space of ${\Sigma}$, and we will not do so here.) Thus we can view (2) as an integral identity that asserts the vanishing of a certain integral, whose integrand involves ${x, \Phi_0(x), \delta \Phi(x), D\Phi_0(x), D \delta \Phi(x)}$, where ${\delta \Phi}$ vanishes at the boundary but is otherwise unconstrained.

A general rule of thumb in PDE and calculus of variations is that whenever one has an integral identity of the form ${\int_\Omega F(x)\ dx = 0}$ for some class of functions ${F}$ that vanishes on the boundary, then there must be an associated differential identity ${F = \hbox{div} X}$ that justifies this integral identity through Stokes’ theorem. This rule of thumb helps explain why integration by parts is used so frequently in PDE to justify integral identities. The rule of thumb can fail when one is dealing with “global” or “cohomologically non-trivial” integral identities of a topological nature, such as the Gauss-Bonnet or Kazhdan-Warner identities, but is quite reliable for “local” or “cohomologically trivial” identities, such as those arising from calculus of variations.

In any case, if we apply this rule to (2), we expect that the integrand ${\frac{d}{ds} L( x, \Phi[s](x), D\Phi[s](x) )|_{s=0}}$ should be expressible as a spatial divergence. This is indeed the case:

Proposition 1 (Formal) Let ${\Phi = \Phi_0}$ be a critical point of the functional ${J[\Phi]}$ defined in (1). Then for any deformation ${s \mapsto \Phi[s]}$ with ${\Phi[0] = \Phi_0}$, we have

$\displaystyle \frac{d}{ds} L( x, \Phi[s](x), D\Phi[s](x) )|_{s=0} = \hbox{div} X \ \ \ \ \ (4)$

where ${X}$ is the vector field that is expressible in coordinates as

$\displaystyle X^a := \delta \Phi^i(x) L_{\partial_{x^a} q^i}(x,\Phi_0(x), D\Phi_0(x)). \ \ \ \ \ (5)$

Proof: Comparing (4) with (3), we see that the claim is equivalent to the Euler-Lagrange equation

$\displaystyle L_{q^i}(x,\Phi_0(x), D\Phi_0(x)) - \partial_{x^a} L_{\partial_{x^a} q^i}(x,\Phi_0(x), D\Phi_0(x)) = 0. \ \ \ \ \ (6)$

The same computation, together with an integration by parts, shows that (2) may be rewritten as

$\displaystyle \int_\Omega ( L_{q^i}(x,\Phi_0(x), D\Phi_0(x)) - \partial_{x^a} L_{\partial_{x^a} q^i}(x,\Phi_0(x), D\Phi_0(x)) ) \delta \Phi^i(x)\ dx = 0.$

Since ${\delta \Phi^i(x)}$ is unconstrained on the interior of ${\Omega}$, the claim (6) follows (at a formal level, at least). $\Box$

Many variational problems also enjoy one-parameter continuous symmetries: given any field ${\Phi_0}$ (not necessarily a critical point), one can place that field in a one-parameter family ${s \mapsto \Phi[s]}$ with ${\Phi[0] = \Phi_0}$, such that

$\displaystyle J[ \Phi[s] ] = J[ \Phi[0] ]$

for all ${s}$; in particular,

$\displaystyle \frac{d}{ds} J[ \Phi[s] ]|_{s=0} = 0,$

which can be written as (2) as before. Applying the previous rule of thumb, we thus expect another divergence identity

$\displaystyle \frac{d}{ds} L( x, \Phi[s](x), D\Phi[s](x) )|_{s=0} = \hbox{div} Y \ \ \ \ \ (7)$

whenever ${s \mapsto \Phi[s]}$ arises from a continuous one-parameter symmetry. This expectation is indeed the case in many examples. For instance, if the spatial domain ${\Omega}$ is the Euclidean space ${{\bf R}^d}$, and the Lagrangian (when expressed in coordinates) has no direct dependence on the spatial variable ${x}$, thus

$\displaystyle L( x, \Phi(x), D\Phi(x) ) = L( \Phi(x), D\Phi(x) ), \ \ \ \ \ (8)$

then we obtain ${d}$ translation symmetries

$\displaystyle \Phi[s](x) := \Phi(x - s e^a )$

for ${a=1,\dots,d}$, where ${e^1,\dots,e^d}$ is the standard basis for ${{\bf R}^d}$. For a fixed ${a}$, the left-hand side of (7) then becomes

$\displaystyle \frac{d}{ds} L( \Phi(x-se^a), D\Phi(x-se^a) )|_{s=0} = -\partial_{x^a} [ L( \Phi(x), D\Phi(x) ) ]$

$\displaystyle = \hbox{div} Y$

where ${Y(x) = - L(\Phi(x), D\Phi(x)) e^a}$. Another common type of symmetry is a pointwise symmetry, in which

$\displaystyle L( x, \Phi[s](x), D\Phi[s](x) ) = L( x, \Phi[0](x), D\Phi[0](x) ) \ \ \ \ \ (9)$

for all ${x}$, in which case (7) clearly holds with ${Y=0}$.

If we subtract (4) from (7), we obtain the celebrated theorem of Noether linking symmetries with conservation laws:

Theorem 2 (Noether’s theorem) Suppose that ${\Phi_0}$ is a critical point of the functional (1), and let ${\Phi[s]}$ be a one-parameter continuous symmetry with ${\Phi[0] = \Phi_0}$. Let ${X}$ be the vector field in (5), and let ${Y}$ be the vector field in (7). Then we have the pointwise conservation law

$\displaystyle \hbox{div}(X-Y) = 0.$

In particular, for one-dimensional variational problems, in which ${\Omega \subset {\bf R}}$, we have the conservation law ${(X-Y)(t) = (X-Y)(0)}$ for all ${t \in \Omega}$ (assuming of course that ${\Omega}$ is connected and contains ${0}$).

Noether’s theorem gives a systematic way to locate conservation laws for solutions to variational problems. For instance, if ${\Omega \subset {\bf R}}$ and the Lagrangian has no explicit time dependence, thus

$\displaystyle L(t, \Phi(t), \dot \Phi(t)) = L(\Phi(t), \dot \Phi(t)),$

then by using the time translation symmetry ${\Phi[s](t) := \Phi(t-s)}$, we have

$\displaystyle Y(t) = - L( \Phi(t), \dot\Phi(t) )$

as discussed previously, whereas we have ${\delta \Phi(t) = - \dot \Phi(t)}$, and hence by (5)

$\displaystyle X(t) := - \dot \Phi^i(x) L_{\dot q^i}(\Phi(t), \dot \Phi(t)),$

and so Noether’s theorem gives conservation of the Hamiltonian

$\displaystyle H(t) := \dot \Phi^i(x) L_{\dot q^i}(\Phi(t), \dot \Phi(t))- L(\Phi(t), \dot \Phi(t)). \ \ \ \ \ (10)$

For instance, for geodesic flow, the Hamiltonian works out to be

$\displaystyle H(t) = \frac{1}{2} g_{ij}(\gamma(t)) \dot \gamma^i(t) \dot \gamma^j(t),$

so we see that the speed of the geodesic is conserved over time.

For pointwise symmetries (9), ${Y}$ vanishes, and so Noether’s theorem simplifies to ${\hbox{div} X = 0}$; in the one-dimensional case ${\Omega \subset {\bf R}}$, we thus see from (5) that the quantity

$\displaystyle \delta \Phi^i(t) L_{\dot q^i}(t,\Phi_0(t), \dot \Phi_0(t)) \ \ \ \ \ (11)$

is conserved in time. For instance, for the ${N}$-particle system in Example 2, if we have the translation invariance

$\displaystyle V( q_1 + h, \dots, q_N + h ) = V( q_1, \dots, q_N )$

for all ${q_1,\dots,q_N,h \in {\bf R}^3}$, then we have the pointwise translation symmetry

$\displaystyle q_i[s](t) := q_i(t) + s e^j$

for all ${i=1,\dots,N}$, ${s \in{\bf R}}$ and some ${j=1,\dots,3}$, in which case ${\dot q_i(t) = e^j}$, and the conserved quantity (11) becomes

$\displaystyle \sum_{i=1}^n m_i \dot q_i^j(t);$

as ${j=1,\dots,3}$ was arbitrary, this establishes conservation of the total momentum

$\displaystyle \sum_{i=1}^n m_i \dot q_i(t).$

Similarly, if we have the rotation invariance

$\displaystyle V( R q_1, \dots, Rq_N ) = V( q_1, \dots, q_N )$

for any ${q_1,\dots,q_N \in {\bf R}^3}$ and ${R \in SO(3)}$, then we have the pointwise rotation symmetry

$\displaystyle q_i[s](t) := \exp( s A ) q_i(t)$

for any skew-symmetric real ${3 \times 3}$ matrix ${A}$, in which case ${\dot q_i(t) = A q_i(t)}$, and the conserved quantity (11) becomes

$\displaystyle \sum_{i=1}^n m_i \langle A q_i(t), \dot q_i(t) \rangle;$

since ${A}$ is an arbitrary skew-symmetric matrix, this establishes conservation of the total angular momentum

$\displaystyle \sum_{i=1}^n m_i q_i(t) \wedge \dot q_i(t).$

Below the fold, I will describe how Noether’s theorem can be used to locate all of the conserved quantities for the Euler equations of inviscid fluid flow, discussed in this previous post, by interpreting that flow as geodesic flow in an infinite dimensional manifold.

I’m continuing my series of articles for the Princeton Companion to Mathematics by uploading my article on the Fourier transform. Here, I chose to describe this transform as a means of decomposing general functions into more symmetric functions (such as sinusoids or plane waves), and to discuss a little bit how this transform is connected to differential operators such as the Laplacian. (This is of course only one of the many different uses of the Fourier transform, but again, with only five pages to work with, it’s hard to do justice to every single application. For instance, the connections with additive combinatorics are not covered at all.)

On the official web site of the Companion (which you can access with the user name “Guest” and password “PCM”), there is a more polished version of the same article, after it had gone through a few rounds of the editing process.

I’ll also point out David Ben-Zvi‘s Companion article on “moduli spaces“. This concept is deceptively simple – a space whose points are themselves spaces, or “representatives” or “equivalence classes” of such spaces – but it leads to the “correct” way of thinking about many geometric and algebraic objects, and more importantly about families of such objects, without drowning in a mess of coordinate charts and formulae which serve to obscure the underlying geometry.

[Update, Oct 21: categories fixed.]

It occurred to me recently that the mathematical blog medium may be a good venue not just for expository “short stories” on mathematical concepts or results, but also for more technical discussions of individual mathematical “tricks”, which would otherwise not be significant enough to warrant a publication-length (and publication-quality) article. So I thought today that I would discuss the amplification trick in harmonic analysis and combinatorics (and in particular, in the study of estimates); this trick takes an established estimate involving an arbitrary object (such as a function f), and obtains a stronger (or amplified) estimate by transforming the object in a well-chosen manner (often involving some new parameters) into a new object, applying the estimate to that new object, and seeing what that estimate says about the original object (after optimising the parameters or taking a limit). The amplification trick works particularly well for estimates which enjoy some sort of symmetry on one side of the estimate that is not represented on the other side; indeed, it can be viewed as a way to “arbitrage” differing amounts of symmetry between the left- and right-hand sides of an estimate. It can also be used in the contrapositive, amplifying a weak counterexample to an estimate into a strong counterexample. This trick also sheds some light as to why dimensional analysis works; an estimate which is not dimensionally consistent can often be amplified into a stronger estimate which is dimensionally consistent; in many cases, this new estimate is so strong that it cannot in fact be true, and thus dimensionally inconsistent inequalities tend to be either false or inefficient, which is why we rarely see them. (More generally, any inequality on which a group acts on either the left or right-hand side can often be “decomposed” into the “isotypic components” of the group action, either by the amplification trick or by other related tools, such as Fourier analysis.)

The amplification trick is a deceptively simple one, but it can become particularly powerful when one is arbitraging an unintuitive symmetry, such as symmetry under tensor powers. Indeed, the “tensor power trick”, which can eliminate constants and even logarithms in an almost magical manner, can lead to some interesting proofs of sharp inequalities, which are difficult to establish by more direct means.

The most familiar example of the amplification trick in action is probably the textbook proof of the Cauchy-Schwarz inequality

$|\langle v, w \rangle| \leq \|v\| \|w\|$ (1)

for vectors v, w in a complex Hilbert space. To prove this inequality, one might start by exploiting the obvious inequality

$\|v-w\|^2 \geq 0$ (2)

but after expanding everything out, one only gets the weaker inequality

$\hbox{Re} \langle v, w \rangle \leq \frac{1}{2} \|v\|^2 + \frac{1}{2} \|w\|^2$. (3)

Now (3) is weaker than (1) for two reasons; the left-hand side is smaller, and the right-hand side is larger (thanks to the arithmetic mean-geometric mean inequality). However, we can amplify (3) by arbitraging some symmetry imbalances. Firstly, observe that the phase rotation symmetry $v \mapsto e^{i\theta} v$ preserves the RHS of (3) but not the LHS. We exploit this by replacing v by $e^{i\theta} v$ in (3) for some phase $\theta$ to be chosen later, to obtain

$\hbox{Re} e^{i\theta} \langle v, w \rangle \leq \frac{1}{2} \|v\|^2 + \frac{1}{2} \|w\|^2$.

Now we are free to choose $\theta$ at will (as long as it is real, of course), so it is natural to choose $\theta$ to optimise the inequality, which in this case means to make the left-hand side as large as possible. This is achieved by choosing $e^{i\theta}$ to cancel the phase of $\langle v, w \rangle$, and we obtain

$|\langle v, w \rangle| \leq \frac{1}{2} \|v\|^2 + \frac{1}{2} \|w\|^2$ (4)

This is closer to (1); we have fixed the left-hand side, but the right-hand side is still too weak. But we can amplify further, by exploiting an imbalance in a different symmetry, namely the homogenisation symmetry $(v,w) \mapsto (\lambda v, \frac{1}{\lambda} w)$ for a scalar $\lambda > 0$, which preserves the left-hand side but not the right. Inserting this transform into (4) we conclude that

$|\langle v, w \rangle| \leq \frac{\lambda^2}{2} \|v\|^2 + \frac{1}{2\lambda^2} \|w\|^2$

where $\lambda > 0$ is at our disposal to choose. We can optimise in $\lambda$ by minimising the right-hand side, and indeed one easily sees that the minimum (or infimum, if one of v and w vanishes) is $\|v\| \|w\|$ (which is achieved when $\lambda = \sqrt{\|w\|/\|v\|}$ when $v,w$ are non-zero, or in an asymptotic limit $\lambda \to 0$ or $\lambda \to \infty$ in the degenerate cases), and so we have amplified our way to the Cauchy-Schwarz inequality (1). [See also this discussion by Tim Gowers on the Cauchy-Schwarz inequality.]