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Finite reduction and Morse index estimates for mechanical systems
A simple version of exact finite dimensional reduction for the variational
setting of mechanical systems is presented. It is worked out by means of a
thorough global version of the implicit function theorem for monotone
operators. Moreover, the Hessian of the reduced function preserves all the
relevant information of the original one, by Schur's complement, which
spontaneously appears in this context. Finally, the results are
straightforwardly extended to the case of a Dirichlet problem on a bounded
domain.Comment: 13 pages; v2: minor changes, to appear in Nonlinear Differential
Equations and Application
Between Laws and Models: Some Philosophical Morals of Lagrangian Mechanics
I extract some philosophical morals from some aspects of Lagrangian
mechanics. (A companion paper will present similar morals from Hamiltonian
mechanics and Hamilton-Jacobi theory.) One main moral concerns methodology:
Lagrangian mechanics provides a level of description of phenomena which has
been largely ignored by philosophers, since it falls between their accustomed
levels--``laws of nature'' and ``models''. Another main moral concerns
ontology: the ontology of Lagrangian mechanics is both more subtle and more
problematic than philosophers often realize.
The treatment of Lagrangian mechanics provides an introduction to the subject
for philosophers, and is technically elementary. In particular, it is confined
to systems with a finite number of degrees of freedom, and for the most part
eschews modern geometry. But it includes a presentation of Routhian reduction
and of Noether's ``first theorem''.Comment: 106 pages, no figure
Optimality conditions applied to free-time multi-burn optimal orbital transfers
While the Pontryagin Maximum Principle can be used to calculate candidate
extremals for optimal orbital transfer problems, these candidates cannot be
guaranteed to be at least locally optimal unless sufficient optimality
conditions are satisfied. In this paper, through constructing a parameterized
family of extremals around a reference extremal, some second-order necessary
and sufficient conditions for the strong-local optimality of the free-time
multi-burn fuel-optimal transfer are established under certain regularity
assumptions. Moreover, the numerical procedure for computing these optimality
conditions is presented. Finally, two medium-thrust fuel-optimal trajectories
with different number of burn arcs for a typical orbital transfer problem are
computed and the local optimality of the two computed trajectories are tested
thanks to the second-order optimality conditions established in this paper
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