THE HAMILTON CARTAN FORMALISM FOR rth-ORDER LAGRANGIANS AND THE INTEGRABILITY OF THE KdV AND MODIFIED KdV EQUATIONS W.F. SHADWICK Department of Mathematics, University of North Carolina, Chapel Hill, N. C 2 7514 ABSTRACT. The Hamilton Cartan formalism for rth order Lagrangians is presented in a form suited to dealing with higher-order conserved currents. Noether's Theorem and its converse are stated and Poisson brackets are defined for conserved charges. An isomorphism between the Lie algebra of conserved currents and a Lie algebra of infinitesimal symmetries of the Cartan form is established. This isomorphism, together with the commutativity of the Biicklund transformations for the KdV and modified KdV equations, allows a simple geometric proof that the infinite collections of conserved charges for these equations are in involution with respect to the Poisson bracket determined by their Lagrangians. Thus, the formal complete integrability of these equations appears as a consequence of the properties of their BScklund transformations. It is noted that the Hamilton Cartan formalism determines a symplectic structure on the space of functionals determined by conserved charges and that, in the case of the KdV equation, the structure is the same as that given by Miura et al. [5]. 1. INTRODUCTION The Hamilton Cartan formalism for first-order Lagrangians was extended in [7] and [8] to deal with higlier-order conserved currents, that is, those which depend on derivatives of arbitrary order. In Section 2 of the present paper, a new invariant characterization of the Hamilton Cartan formalism for rth-order Lagrangians is given. As in the first-order case, one obtains a version of Noether's theorem and its converse for higher-order conserved currents as well as a naturally- defined Poisson bracket on equivalence classes of conserved currents called conserved charges. The conserved charges, with the Poisson bracket, comprise a Lie algebra isomorphic to a Lie algebra of infinitesimal symmetries of the Cartan m-form, and this algebra structure allows one to define a formal symplectic structure on the space of functionals determined by the conserved charges. In Section 3, the formalism is applied to the KdV and modified KdV equations to show that the formal complete integrability of these equations follows from the properties of their Nickhind transformations (c.f. [9] ). This result follows from Steudel's derivation of infinite collections of conserved currents for these equations as Noether currents for infinitesimal Nicklund transformations, together with the isomorphism between the algebras of conserved charges and infinitesimal symmetries. Finally, it is noted that the symplectic structure obtained from the Hamilton Caftan formalism for these equations is precisely that which was developed in [2] and [5]. 137 Letters m Mathematical Physics 5 (1981) 137-141. 0377-9017/81/0052 0137 $00.50. Copyright 9 1981 by D. Reidel Publishing Company.