A Proper Extension of Noether's Symmetry Theorem for Nonsmooth Extremals of the Calculus of Variations

For nonsmooth Euler-Lagrange extremals, Noether's conservation laws cease to be valid. We show that Emmy Noether's theorem of the calculus of variations is still valid in the wider class of Lipschitz functions, as long as one restrict the Euler-Lagrange extremals to those which satisfy the DuBois-Reymond necessary condition. In the smooth case all Euler-Lagrange extremals are DuBois-Reymond extremals, and the result gives a proper extension of the classical Noether's theorem. This is in contrast with the recent developments of Noether's symmetry theorems to the optimal control setting, which give rise to non-proper extensions when specified for the problems of the calculus of variations.


Introduction
Let L(t, x, v) be a given C 1 ([a, b] × R n × R n ; R) function (the Lagrangian). The fundamental problem of the calculus of variations consists to minimize the integral functional ∂L ∂v (t, x(t),ẋ(t)) = ∂L ∂x (t, x(t),ẋ(t)) , which is a first-order necessary optimality condition. Each solution of (2) is called an Euler-Lagrange extremal. Condition (2) is obtained in most textbooks from the assumption that minimizers are smooth, or assuming they are piecewise smooth functions. In this last situation the Euler-Lagrange equations are interpreted as holding everywhere except possibly at finitely many points.
In 1918 Emmy Noether [8,9] established a general theorem asserting that the invariance of the integral functional (1) under a group of transformations depending smoothly on a parameter s, implies the existence of a conserved quantity along the Euler-Lagrange extremals. As corollaries, all the conservation laws known to classical mechanics are easily obtained. For a survey of Noether's theorem and its generalizations see [10]. Noether's theorem, as is found in the many literature of physics, calculus of variations and optimal control, is formulated with X being smooth. A typical example is x(·) ∈ X = C 2 (cf. e.g. [1,4,6]).
Given that the Euler-Lagrange equation (2) makes sense when x(·) has merely essentially bounded derivative -the biggest class X for which (2) is still valid is the class Lip of Lipschitz functions (cf. e.g. [2, §2.2]) 1 -it is expected that the conclusion of Noether's theorem can still be defended in the wider class of Lipschitz functions. This is indeed the case, as it follows from the Pontryagin maximum principle and the results in [11,12]. As far as for the fundamental problem of the calculus of variations the Pontryagin maximum principle reduces to the Euler-Lagrange necessary condition (2) and to the Weierstrass necessary condition which are distinct necessary conditions even in the C 2 -smooth case, this does not give a proper extension of Noether's theorem to the class of Lipschitz functions (the generalization does not reduce to the classical formulation when X = C 2 , since we are restricting the set of Euler-Lagrange extremals to those which satisfy Weierstrass's necessary optimality condition (3)). In the present article we show that to formulate Noether's theorem for admissible Lipschitz functions, one does not need to restrict the set of Euler-Lagrange extremals to Pontryagin extremals, being enough the restriction to those Euler-Lagrange extremals satisfying the DuBois-Reymond condition: 2 We remark that the DuBois-Reymond first-order necessary optimality condition (4) is valid when X is the class of Lipschitz functions, and that (4) is a consequence of the Euler-Lagrange and Weierstrass conditions. For X = C 2 , (4) follows from the Euler-Lagrange equation (2) alone (every Euler-Lagrange C 2extremal is a DuBois-Reymond C 2 -extremal), 3 and therefore our generalization of Noether's theorem to the class of Lipschitz functions (cf. §4) gives a proper extension of the smooth result.

Review of Noether's Symmetry Theorem
The universal principle described by Noether, asserts that the invariance of a problem with respect to a one-parameter group of transformations implies the existence of a conserved quantity along the smooth Euler-Lagrange extremals.
The following result for nonsmooth extremals, is a trivial corollary from the optimal control results in [11,12]. Theorem 2 restrict the conclusion of Noether's theorem to Pontryagin extremals. The following question comes immediately to mind: Is it really necessary to restrict the set of nonsmooth Euler-Lagrange extremals in order to guarantee that (6) is conserved? In Section 3 we show that a restriction is indeed necessary: we provide an example of a Lipschitz Euler-Lagrange extremal which is not a Weierstrass extremal, and which fails to preserve (6).
While Pontryagin extremals are a natural choice in optimal control, in the context of the calculus of variations such restriction seems to be unnatural: Theorem 2 does not simplify to Theorem 1 in the C 2 smooth case (Euler-Lagrange equation differs from Weierstrass's necessary condition in the C 2 smooth case). This means that Theorem 2 does not give a proper extension of the classical Noether's theorem. In Section 4 we give a proper restriction of the set of nonsmooth Euler-Lagrange extremals for which Noether's theorem can still be asserted (Theorem 3).

Main Result
We formulate our Noether theorem for nonsmooth extremals under a more general notion of invariance than the one in Definition 1. We require the symmetry transformation to leave the problem invariant up to first order terms in the parameter, and up to exact differentials. 4
Remark 1. As in the classical context, we are assuming that the parameter transformations (t, x) −→ (T (t, x,ẋ, s), X(t, x,ẋ, s)) reduce to the identity for s = 0, that is, for any choice of t, x, and v.
Remark 2. It is obvious that the invariance notion used in connection with Theorem 2 implies the quasi-invariance up to a gauge-term in Definition 2.
Remark 3. In the 1918 original paper of Emmy Noether [8,9], Noether explains that the derivatives of the trajectories x may also occur in the parameter group of transformations. This possibility has been widely forgotten in the literature of the calculus of variations, the only exception seeming to be the textbook of I. M. Gelfand and S. V. Fomin [4]. Such possibility is, however, very interesting from the point of view of optimal control (cf. [11,12]) and is included in Definition 2.
An example with relevance in Physics, showing that the dependence of the invariance transformations on the derivatives can be crucial in order to obtain a conservation law, can be found in [7].