The mechanical ideas

  The key underlying idea is to find a finite-dimensional embedding for the global equilibrium path. Commonly the global path is embedded into an infinite-dimensional function space, which is theoretically perfect but practically useless. We appeal to the Uniqueness Theorem of Peano [2] stating that trajectories of explicit ODEs satisfying the Lifschitz condition are uniquely determined by the Cauchy-type initial conditions. Although our ODEs do satisfy these requirements, Peano's result can not be applied immediately since we are interested in the solution of BVPs instead of IVPs. However, let us consider that the boundary conditions prescribe Cauchy-type conditions on both ends. Let us regard only those BC which apply at the origin (s=0). These conditions will fix some of the initial conditions. The space spanned by the remaining initial conditions serves us as a finite-dimensional embedding space for the global equilibrium path: The map from the solution set of the BVP onto this space is unique. Our method is based on the piece-wise linearization of this finite-dimensional space. The points of the n-dimensional space corresponding to BVP solutions (this is the point set we are looking for) is specified by n-1 scalar conditions of the form , where the -s denote the variable initial conditions at s=0 and the -s the conditions imposed at the far end (s=1). We illustrate these ideas on a simple example. Let us regard the ODE describing the static equilibrium of a slender, linearly elastic beam. This equation was first described by Euler :

where EI denotes the (constant) bending stiffness of the beam, P denotes the load parameter (compressive force applied at the end), denotes the angle between the bar axis and the line of application of P and ' denotes differentiation with respect to the arc-length. The IVP of this ODE is completely determined by the initial conditions . If we choose the boundary conditions to be (cantilever beam) then the embedding space for the global equilibrium path is spanned by . There are more complicated cases where this space is more than 2 dimensional, this depends on the ODE and the boundary conditions. Since n=2, we have a single condition to meet: .