Ehresmann Connection Research Articles

Integrability for peaffian constrained systems: A geometrical theory

There exists an Ehresmann connection on the fibred constrained sub-manifold defined by Pfaffian differential constraints. It is proved that curvature of the connection is closely related to the d-σ commutation relation in the classical nonholonomic mechanics. It is also proved that conditions of complete integrability for Pfaffian systems in Frobenius sense are equivalent to the three requirements upon the conditional variations in the classical calculus of variations: (1) the variations belong to the constrained manifold, (2) variational operators commute with differential operators, (3) variations satisfy the Chetaev's conditions. Thus this theory verifies the conjecture or experience of researchers of mechanics on the integrability conditions in terms of variation calculus.

GROWTH OF LEAVES IN FOLIATIONS WITH EHRESMANN CONNECTIONS

In this paper, we obtain a sufficient condition for a foliation ℱ with an Ehresmann connection D=(Tℱ)⊥ on a compact, connected manifold M to have leaves whose universal covers have the same growth type. When such a foliation has a compact leaf L, we show that the growth type of all leaves is bounded from above by the growth types of π1L and HD(L).

Nonholonomic mechanical systems with symmetry

This work develops the geometry and dynamics of mechanical systems with nonholonomic constraints and symmetry from the perspective of Lagrangian mechanics and with a view to control theoretical applications. The basic methodology is that of geometric mechanics applied to the formulation of Lagrange d'Alembert, generalizing the use of connections and momentum maps associated with a given symmetry group to this case. We begin by formulating the mechanics of nonholonomic systems using an Ehresmann connection to model the constraints, and show how the curvature of this connection enters into Lagrange's equations. Unlike the situation with standard configuration space constraints, the presence of symmetries in the nonholonomic case may or may not lead to conservation laws. However, the momentum map determined by the symmetry group still satisfies a useful differential equation that decouples from the group variables. This momentum equation, which plays an important role in control problems, involves parallel transport operators and is computed explicitly in coordinates. An alternative description using a ``body reference frame'' relates part of the momentum equation to the components of the Euler-Poincar\'{e} equations along those symmetry directions consistent with the constraints. One of the purposes of this paper is to derive this evolution equation for the momentum and to distinguish geometrically and mechanically the cases where it is conserved and those where it is not. An example of the former is a ball or vertical disk rolling on a flat plane and an example of the latter is the snakeboard, a modified version of the skateboard which uses momentum coupling for locomotion generation. We construct a synthesis of the mechanical connection and the Ehresmann connection defining the constraints, obtaining an important new object we call the nonholonomic connection. When the nonholonomic connection is a principal connection for the given symmetry group, we show how to perform Lagrangian reduction in the presence of nonholonomic constraints, generalizing previous results which only held in special cases. Several detailed examples are given to illustrate the theory. September 1994 Revised, March 1995 Revised, June 1995

On nonholonomic and vakonomic dynamics of mechanical systems with nonintegrable constraints

The classical nonholomic equations for a mechanical system subject to linear nonintegrable constraints are presented in Section 2. These are displayed in a geodesic form by the use of a suitable covariant derivative (due to Synge) in Section 3. We then express the Frobenius integrability of the constraint distribution by means of a zero torsion condition for the above Synge connection. Section 4 provides a self-contained derivation from a nonholonomic variational problem of the equations of motion for nonholonomic systems. These equations, which are nonequivalent to the previous ones, were first developed by Arnold and Kozlov and called vakonomic (vak) equations. Sections 5 and 6 are concerned with a geometrical interpretation of the terms occurring in the right-hand side of the vak equations. Under quite general assumptions, these latter can be described in terms of the curvature of an Ehresmann (local) connection whose horizontal subspace is precisely the constraint distribution. Furthermore, by introducing a suitable Lie group action on the configuration manifold, the local Ehresmann connection can be made into a global one which coincides with the mechanical connection of Smale-Marsden. Section 7 gives a motivation, in terms of Hopf-Rinow and Ambrose-Singer theorems, for the nonclassical requirement of the assignment of the reaction forces' values in the initial kinematical state in vakonomic mechanics. Section 8 develops a fundamental approach to the description of holonomic, i.e. geometrical constraints. We describe the reaction forces by using the Poincaré dual (a class of closed 1-forms) of the orientable constraint submanifold. As an instance of the construction developed in Sections 5 and 6, we consider in Section 9 the disk rolling without sliding on the plane.

Geometric characterization of linearisable second-order differential equations

AbstractGiven an Ehresmann connection on the tangent bundle τ: TM → M we define a linear connection on the pull-back bundle τ*(TM). With the aid of this tool, necessary and sufficient conditions are derived for the existence of local coordinates in which a system of second-order differential equations is linear.

Ehresmann connection for the canonical foliation on a manifold over a local algebra

For a canonical foliation on a manifoldMA over a local algebra, theA-affine horizontal distribution complementary to the leaves, similar to the horizontal distribution of a higher order connection on the fiber bundle ofA-jets, is defined. In the case of a complete manifoldMA, theA-affine horizontal distribution is proved to be an Ehresmann connection in the sense of Blumental-Hebda. It is shown that theA-affine horizontal distribution onMA exists if and only if the Atiyah class of a certain foliated principal bundle vanishes.

Ehresmann connections for lagrangian foliations

The notion of an Ehresmann connection was introduced by Ehresmann (1950). In recent years it has been extensively studied by some authors. The main aim of this paper is to demonstrate that under some relatively natural assumptions lagrangian foliation admit Ehresmann connections.

The connection whose holonomy is the classical adiabatic angles of Hannay and Berry and its generalization to the non-integrable case

We show how averaging defines an Ehresmann connection whose holonomy is the classical adiabatic angles which Hannay defined for families of completely integrable systems. The averaging formula we obtain for the connection only requires that the family of Hamiltonians has a continuous symmetry group. This allows us to extend the notion of the Hannay angles to families of non-integrable systems with symmetry. We state three geometric axioms satisfied by the connection. These axioms uniquely determine the connection, thus enabling us to find new formulas for the connection and its curvature. Two examples are given.

Universal connections, gauge anomalies and Lie group cohomology

The geometric interpretation, due to Thierry-Mieg (1980) of the ghost field as certain components of an Ehresmann connection on the total space of a principal bundle, and of the BRS transformation as part of the exterior derivative on the total space, is discussed and amended to incorporate the spacetime dependence of the ghost field. Using this interpretation it is shown that non-trivial solutions to the Wess-Zumino consistency condition for gauge anomalies are related to non-trivial de Rham cohomology of the Lie group.

General prolongations and (x, t)-depending pseudopotentials for the KdV equation

Given an exterior differential system on a manifold M, we study general prolongations of the system on a locally trivial fiber bundle (M̃, π̃, M) by a Cartan–Ehresmann connection. We characterize such prolongations for the system associated with the KdV equation without any assumption of ‘‘(x, t) independence.’’ The partial Lie algebra discovered by Wahlquist–Estabrook [J. Math. Phys. 16, 1 (1975)] appears by this way as an intrinsic tool. Simple analytic pseudopotentials are classified up to diffeomorphism.

Moving frames and prolongation algebras

We consider differential ideals generated by sets of 2-forms which can be written with constant coefficients in a canonical basis of 1-forms. By setting up a Cartan–Ehresmann connection, in a fiber bundle over a base space in which the 2-forms live, one finds an incomplete Lie algebra of vector fields in the fibers. Conversely, given this algebra (a prolongation algebra), one can derive the differential ideal. The two constructs are thus dual, and analysis of either derives properties of both. Such systems arise in the classical differential geometry of moving frames. Examples of this are discussed, together with examples arising more recently: the Korteweg–de Vries and Harrison–Ernst systems.

A prolongation structure for the AKNS system and its generalization

A prolongation structure is determined for the AKNS system of equations. Using an interpretation in terms of Cartan–Ehresmann connections, a generalization is then constructed which leads to an inverse scattering problem for the multicomponent nonlinear Schrodinger equation iqαt= (1/2) qαxx +qα(Σnβ=1‖qβ‖2) (α=1,...,n).