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 functional
or 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 really formally, 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 have
where 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.
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