Some Aspects Concerning the Dynamics Given by Pfaff Forms Paul Popescu, Marcela Popescu Department of Applied Mathematics, University of Craiova paul p popescu@ yahoo.com, marcelacpopescu@ yahoo.com Abstract The aim of the paper is to extend Lagrangian dynamics to Pfaff form dynamics, where a Pfaff form is a differential form on a tangent bundle, non necessary closed. Considering the action of a Pfaff form on curves, given by a second order Lagrangian linear in accelerations, we obtain the equations of the first and second variations, using variational methods. In the non-singular case, considered mainly in the paper, the generalized Euler-Lagrange equation is a third order differential equation. As examples, we find that the solutions of the differential equations of motion of a charge in a field and the Euler equations for the rotational dynamics of a rigid body about its center of mass can be obtained as particular solutions of suitable Pfaff forms, with non-negative second variations. 1 Introduction The Euler-Lagrange equation of a first order Lagrangian is a well-known and widely used variational equation which arises from many problems from mathematics, mechanics, physics and other scientific fields. Its solutions are the critical curves of the action defined by the Lagrangians on curves; in the case when the Lagrangian comes from a Riemannian, a non-Riemannian or a Finslerian metric, these solutions are known as geodesics, since they locally minimise the distance. The second variation can decide if the solution is an extreme one (see [9, Ch.1, Sect.2]). The local expression of the first order Euler-Lagrange equation contains the second derivatives and, in the case of a hyperregular Lagrangian, its solutions are integral curves of a global second order differential equation. In this paper we consider actions on curves of Pfaff forms instead of Lagrangians. Pfaff forms are differential one forms on tangent spaces of manifolds, or on open subsets of the tangent spaces. In fact, the action of a Pfaff form is the same as the action of a second order Lagrangian, linear in accelerations (see, for example, [4, 6]). In the non-singular case, the local expression of the Euler-Lagrange equation, obtained by a variational method on the action, involves the third derivatives; in the regular case the solutions are integral curves of a global third order differential equation (Proposition 3.1). The dynamics of Pfaff forms has different behaviors. In the case when the Pfaff form comes from a non-Lagrangian, the dynamics is given by the classical Euler-Lagrange equation of a suitable first order Lagrangian. For a Pfaff form on a one dimensional manifold, the Euler-Lagrange equation (16), admits a standard Lagrangian description (Proposition 3.2). The case of a non-singular Pfaff form is mainly considered in the paper. If the Pfaff form is regular, its action on curves can be described by the action of a second order Lagrangian, linear in the second order velocities (accelerations). The Euler-Lagrange equation of a higher order Lagrangian is originary due to Ostrogradski, then used in a modern form, as in [10] or [4]. For a second order Lagrangian linear in accelerations, the Euler- Lagrange equations are called in the paper as the generalized Euler-Lagrange equations. A formula for the second variational derivative is also given. As concrete examples, we consider the differential equations of motion of a charge in a field (for- mulas (17) in [3, Section 17]) and the Euler equations for the rotational dynamics of a rigid body about its center of mass [5]. We prove that there are suitable Pfaff forms in each case, such that the solutions of the considered differential equations are also extremal solutions for the generalized Euler-Lagrange equations of the Pfaff forms, i.e. the second variation has a constant sign along these solutions (Propositions 3.4 and 3.6). 195