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 of one or more fields on a domain taking values in a space will solve the differential equation of interest if and only if is a critical point to the functional

involving the fields and their first derivatives , where the Lagrangian is a function on the vector bundle over consisting of triples with , , and a linear transformation; we also usually keep the boundary data of fixed in case has a non-trivial boundary, although we will ignore these issues here. (We also ignore the possibility of having additional constraints imposed on and , 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 as and as , in which case can be viewed locally as a function on .

Example 1 (Geodesic flow)Take and to be a Riemannian manifold, which we will write locally in coordinates as with metric for . A geodesic is then a critical point (keeping fixed) of the energy functionalor in coordinates (ignoring coordinate patch issues, and using the usual summation conventions)

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

reallyformally, as the manifold is now infinite dimensional).More generally, if is itself a Riemannian manifold, which we write locally in coordinates as with metric for , then a harmonic map is a critical point of the energy functional

or in coordinates (again ignoring coordinate patch issues)

If we replace the Riemannian manifold by a Lorentzian manifold, such as Minkowski space , 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 ).

Example 2 (-particle interactions)Take and ; then a function can be interpreted as a collection of trajectories in space, which we give a physical interpretation as the trajectories of particles. If we assign each particle a positive mass , and also introduce a potential energy function , then it turns out that Newton’s laws of motion in this context (with the force on the particle being given by the conservative force ) are equivalent to the trajectories being a critical point of the action functional

Formally, if is a critical point of a functional , this means that

whenever is a (smooth) deformation with (and with respecting whatever boundary conditions are appropriate). Interchanging the derivative and integral, we (formally, at least) arrive at

Write for the infinitesimal deformation of . By the chain rule, can be expressed in terms of . In coordinates, we have

where we parameterise by , and we use subscripts on 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 , 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 , where 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 for some class of functions that vanishes on the boundary, then there must be an associated differential identity 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 should be expressible as a spatial divergence. This is indeed the case:

Proposition 1(Formal) Let be a critical point of the functional defined in (1). Then for any deformation with , we havewhere is the vector field that is expressible in coordinates as

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

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

Since is unconstrained on the interior of , the claim (6) follows (at a formal level, at least).

Many variational problems also enjoy one-parameter continuous *symmetries*: given any field (not necessarily a critical point), one can place that field in a one-parameter family with , such that

for all ; in particular,

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

whenever arises from a continuous one-parameter symmetry. This expectation is indeed the case in many examples. For instance, if the spatial domain is the Euclidean space , and the Lagrangian (when expressed in coordinates) has no direct dependence on the spatial variable , thus

then we obtain translation symmetries

for , where is the standard basis for . For a fixed , the left-hand side of (7) then becomes

where . Another common type of symmetry is a *pointwise* symmetry, in which

for all , in which case (7) clearly holds with .

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 is a critical point of the functional (1), and let be a one-parameter continuous symmetry with . Let be the vector field in (5), and let be the vector field in (7). Then we have the pointwise conservation law

In particular, for one-dimensional variational problems, in which , we have the conservation law for all (assuming of course that is connected and contains ).

Noether’s theorem gives a systematic way to locate conservation laws for solutions to variational problems. For instance, if and the Lagrangian has no explicit time dependence, thus

then by using the time translation symmetry , we have

as discussed previously, whereas we have , and hence by (5)

and so Noether’s theorem gives conservation of the *Hamiltonian*

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

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

For pointwise symmetries (9), vanishes, and so Noether’s theorem simplifies to ; in the one-dimensional case , we thus see from (5) that the quantity

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

for all , then we have the pointwise translation symmetry

for all , and some , in which case , and the conserved quantity (11) becomes

as was arbitrary, this establishes conservation of the *total momentum*

Similarly, if we have the rotation invariance

for any and , then we have the pointwise rotation symmetry

for any skew-symmetric real matrix , in which case , and the conserved quantity (11) becomes

since is an arbitrary skew-symmetric matrix, this establishes conservation of the *total angular momentum*

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.

** — 1. Euler’s equations — **

The geometric setup for the geodesic interpretation of Euler’s equations of fluid flow is as follows. We will need two copies and of Euclidean space , with two different structures. Firstly, we will have Lagrangian space

which is (viewed as a smooth manifold), together with the standard volume form . This space should be thought of as the space of “labels” of the particles of the fluid, and its coordinates are known as *Lagrangian coordinates*. The symmetry group of this space is the space of orientation-preserving and volume-preserving diffeomorphisms.

Secondly, we will need the *Eulerian space*

which is the smooth manifold together with the Euclidean metric and the standard volume form . This space is the physical space of “positions” of the particles of the fluid. The symmetry space of this space is the space of orientation-preserving rigid motions of Euclidean space.

Let be the space of diffeomorphisms from Lagrangian space to Eulerian space that preserve volume and orientation; this can be viewed as an infinite-dimensional manifold. A single element of describes the positions of an incompressible fluid at a snapshot in time; an incompressible fluid flow is then described by a curve . The time derivative of such a curve can be viewed in Eulerian coordinates as the *velocity field*

If one then defines the Lagrangian

then one can show that the critical points of this Lagrangian (formally) correspond to solutions to the Euler equations using the correspondence (12): see this previous blog post for details.

Applying a volume-preserving change of coordinates, the Lagrangian can also be expressed as

There are then three types of symmetries that are evident for this Lagrangian: time symmetry; symmetry on the Eulerian space; and symmetry on the Lagrangian space.

We begin with time symmetry, , which comes from the fact that the Lagrangian does not depend explicitly on the time variable . As discussed before, this gives conservation of the Hamiltonian (10), which (formally, at least) becomes

thus giving the familiar energy conservation law for the Euler equations.

Now we use the symmetry group that acts on the Eulerian space , and hence on fluid flows . This is a pointwise symmetry of the Lagrangian, and formally gives conservation of total momentum

and total angular momentum

in exact analogy with the situation with the -body system. (Indeed, one can formally view the Euler equations as an limit of a certain family of -body systems, which is indeed how these equations are physically derived.)

Finally, we consider symmetries on the Lagrangian space . Any divergence-free vector field on gives a one-parameter group of volume-preserving, orientation-preserving diffeomorphisms on , which then act on fluid flows by the formula

This is a pointwise symmetry of the Lagrangian, with infinitesimal derivative

Applying (11), we thus (formally) conclude that the quantity

is conserved. We can write this in coordinates as

We can specialise this conservation law by working with specific choices of divergence-free vector field . For instance, suppose we have a closed loop , which we parameterise by unit speed: . For an infinitesimal , we can then create a divergence-free vector field by setting when lies in the (transverse) -neighbourhood of , and zero otherwise. It is geometrically obvious that this field is divergence-free (up to errors of ). The conserved quantity (13) is then equal to

up to lower order terms, so that the quantity

is conserved. Writing for the curve , we see from the chain rule that this is equal to

giving Kelvin’s circulation theorem.

More generally, we can generate a divergence-free vector field from an alternating -vector by taking a further divergence:

Integrating by parts, we can then write (13) as

since is an arbitrary alternating -tensor, we conclude that for each and , the quantity

is conserved in time. Writing and using the chain rule, this becomes

which after interchange of the indices may be rewritten in terms of the vorticity as

giving the pointwise conservation of the pullback of the vorticity in Lagrangian coordinates. (Of course, this is just the differential form of Kelvin’s circulation theorem; it also implies conservation of the vortex stream lines in Lagrangian coordinates.)

Finally, as the vorticity is divergence-free (when viewed as a polar vector field), the pullback is also. If we then set to be the vector field associated to the conserved quantity , the quantity (13) can then be rewritten as the helicity

which is then also conserved; a similar argument gives conservation of the helicity on any set that is the union of stream lines.

Remark 1The above Lagrangian mechanics calculations can also be recast into a Hamiltonian mechanics formalism; see for instance this paper of Olver for a Hamiltonian perspective on the conservation laws for the Euler equations.

## 19 comments

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3 March, 2014 at 12:27 am

AnonymousJust below (6): “that the left-hand side of (2) is equal to” –> “that (2) is equivalent to” (if I’m not mistaken).

Also, for maps, correct horizontal spacing is achieved by using “\colon” instead of “:”.

3 March, 2014 at 9:09 am

AnonymousThe reason for my comment is that I don’t see how the left-hand side of an equation can be equal to an equation. (I hope I haven’t missed something obvious.)

[Corrected, thanks – T.]3 March, 2014 at 9:50 am

arch1I’m puzzled by Example 2. I just dropped my pen and the actual pen/Earth trajectory combo didn’t seem to be a critical point of the given J over the next second (since a faster drop would have increased the KE and decreased the PE, thus increasing the integrand, for each t; whereas a hover would have *decreased* the integrand for each t). Is V here actually the *negative* of the PE, or (much likelier) am I misinterpreting/misapplying the example?

3 March, 2014 at 9:54 am

Terence TaoOne has to respect the boundary conditions, i.e. the pen has to start and end at fixed locations. If one tries to make the pen hover for a long period of time, then to compensate it must suddenly burst downwards at great speed to reach its designated location at the end of the time interval, resulting in a large surge in kinetic energy which ends up making the functional J larger than if it had just dropped straight down. (It is true, though, that it is variationally advantageous for the pen to be somewhat slower initially than at later times (though not to hover at a standstill) to take some advantage of the higher potential energy there to minimise the action; of course, this behaviour is what one actually sees in real life.)

3 March, 2014 at 10:13 am

arch1Ah. So wherever the pencil ends up at t=1 (or at any t>0), it had to have gotten there via a J-minimizing trajectory. Thanks for clearing up my confusion!

3 March, 2014 at 10:58 am

Terence TaoYes, this is how the principle of least action works, although (a) the minimum may only be a local minimum rather than a global minimum (in much the same fashion that geodesics may minimise length locally, but not necessarily globally), and (b) in fact one may only have a critical point rather than a local minimum (this is particularly the case when performing multidimensional variational problems, in which the domain is not just the time axis).

3 March, 2014 at 1:02 pm

arch1Thanks again. I’m now curious how the critical point is chosen, but it’s time for me to do some reading on my own (the fact that this stuff is already very pretty when only dimly understood will I hope be sufficient motivation).

4 March, 2014 at 9:08 am

DenisVery interesting and useful post. By the occasion, may I ask you about your opinion on the related topic. What do you think about the geometric theory of nonlinear PDE a.k.a. secondary calculus and diffiety theory being developed by these people: http://www.levi-civita.org Can this marginal science be perspective for further development and applications?

4 March, 2014 at 11:33 am

jussilindgrenDear Terry, do you think there is an analogy between the Euler equations and Einstein field equations? In GR I understand that the field equation is an Euler-Lagrange equation for the Hilbert action, whereas the actual path can be obtained by plugging the metric in the geodesic equation (Euler-Lagrange equations as well) ? As the Hilbert action is basically a minimization scheme for “average curvature”, could there be an interesting variational formulation based on e.g. minimizing some integrals of Frobenius norms of the gradient tensors or something along these lines? Best,

Jussi

6 March, 2014 at 9:26 am

Conserved quantities for the surface quasi-geostrophic equation | What's new[…] the previous blog post, the Euler equations for inviscid incompressible fluid flow were interpreted in a Lagrangian […]

7 March, 2014 at 5:42 am

MrCactu5 (@MonsieurCactus)Is equation (12) a sort of logarithmic derivative?

7 March, 2014 at 8:21 am

Terence TaoYes, in the infinite-dimensional Lie group of diffeomorphisms on . For instance, if is the identity map (identifying both and with ), and is a fixed vector field, then the integrated flow map is often denoted , which is consistent with the interpretation of as a logarithmic derivative of .

8 March, 2014 at 6:31 pm

MrCactu5 (@MonsieurCactus)Your Lagrangian J(\phi) is just an L^2 norm whether you integrate in the Lagrangian or Euler space, right? So we expecte a large number of symmetries, like some sort of infinite rotation group?

When I was a freshman, we set the derivative to zero.

And we drew elaborate charts keeping track of the sign of the second derivative.

All discussions I seen using calculus of variations fall into extremes. Either they justifying the critical points or omitting those details entirely.

Also, I highly recommend using Einstein summation convention — if you wish.

9 March, 2014 at 4:17 pm

damtsonThank you for a very interesting posting. A few remarks of historical character, with potential bias toward physics literature: the first paper on the Lagrangian formulation of (relativistic) fluid dynamics seems to be by Taub in 1954 (http://journals.aps.org/pr/abstract/10.1103/PhysRev.94.1468), and the first paper mentioning a Hamiltonian structure of the (compressible) Euler equations was written by the great Lev Landau in 1941 (J. Phys. USSR 5, 71 (1941), reprinted in the book by Khalatnikov “An introduction to the theory of superfluidity”). Kelvin’s circulation theorem was derived from Noether’s theorem in http://arxiv.org/abs/hep-th/0101029.

8 April, 2014 at 3:03 am

Matrix Elements of Lorentzian Hamiltonian Constraint in LQG by Alesci,Liegener and Zipfel | quantumtetrahedron[…] Noether’s theorem, and the conservation laws for the Euler equations (terrytao.wordpress.com) […]

15 April, 2014 at 11:58 am

Sagemath 18: Calculation of the Matrix Elements of Thiemann’s Hamiltonian Constraint in Loop Quantum Gravity | quantumtetrahedron[…] Noether’s theorem, and the conservation laws for the Euler equations (terrytao.wordpress.com) […]

25 June, 2014 at 1:48 am

Navier-Stokes Fluid Computers | Combinatorics and more[…] here is a follow up post on Tao’s blog (and a few more II, III), and a post on Shtetl […]

1 July, 2016 at 9:38 pm

AnonymousI am posting the following papers in case you haven’t seen them before.

It seems like X. He found another conserved integral for Euler (the “alignment”):

http://www.sciencedirect.com/science/article/pii/S0893965999000348

It appears the same author later found a sufficient condition for blowup, using an unrelated technique.

30 June, 2020 at 1:18 am

تقارن،قوانین پایستگی و اِمی نٌودِر - سیتپـــــور[…] Noether’s theorem, and the conservation laws for the Euler equations […]