The consequences of the projectability of Poincar\'e-Cartan forms in a third-order jet bundle $J^3\pi$ to a lower-order jet bundle are analyzed using the constraint algorithm for the Euler-Lagrange equations in $J^3\pi$. The results are applied to the Hilbert Lagrangian for the Einstein equations. Furthermore, the case of higher-order mechanics is also studied as a particular situation.
In the history of mechanics, there have been two points of view for studying mechanical systems: The Newtonian and the Cartesian.
According the Descartes point of view, the motion of mechanical systems is described by the first-order differential equations in the N dimensional configuration space Q.
In this paper we develop the Cartesian approach for mechanical systems with three degrees of freedom and with constraint which are linear with respect to velocity. The obtained results we apply to discuss
the integrability of the geodesic flows on the surface in the three dimensional Euclidian space and to analyze the integrability of a heavy rigid body in the Suslov and the Veselov cases.
In the development of nonholonomic mechanics one can observe recurring confusion over
the very equations of motion as well as the deeper questions associated with the geometry
and analysis of these equations. First of all, as far as the equations of motion themselves are concerned, the confusion mainly centered on whether or not the equations could be derived from a variational principle in the usual sense.
Attempting to dissipate this confusion, in the present paper we deduce a new form of
equations of motion which are suitable for both nonholonomic systems with either linear or nonlinear constraints and holonomic systems (A-model). These equations are deduced from the principle of stationary action (or Hamiltonian principle) with nonzero transpositional relations.
We show that the well-known equations of motion for nonholonomic and holonomic systems
can be deduced from the A-model. For the systems which we call the generalized Vorones-Chaplygin systems we deduce the
equations of motion which coincide with the Vorones and Chaplygin equations for the case in which the constraints are linear with respect to the velocity.
An additional result is that the transpositional relations are different from zero only for those generalized coordinates whose variations (in accordance with the equations of nonholonomic constraints) are dependent. For the remaining coordinates, the transpositional relations may be zero.