Abstract
From the global analytical point of view a onedimensionalvariational problem consists
in extremizing a differentiable action/cost function f: X →R, where X is an
infinite dimensional manifold of paths in a manifold M, overa subset Ω C X of
admissible paths, for example those satisfying someregularity conditions, boundary
conditions or other constraints. Thus a solution to thevariational problem is a
critical point of the restriction fΩ.
A standard criterion for existence of critical points is thePalaisSmale condition.
If this condition is satisfied then the gradient flowassociated with f is well behaved,
and we are guaranteed not only existence of critical pointsbut also existence of a
minimum. Moreover it is then possible to relate the totalnumber of critical points
to topological properties of Ω
Original language  English 

Qualification  Doctor of Philosophy 
Awarding Institution 

Award date  15 Jul 2016 
Publication status  Unpublished  2016 
Fingerprint
Cite this
}
Global analysis of onedimensional variational problems. / Schrader, Philip John.
2016.Research output: Thesis › Doctoral Thesis
TY  THES
T1  Global analysis of onedimensional variational problems
AU  Schrader, Philip John
PY  2016
Y1  2016
N2  [Truncated] From the global analytical point of view a onedimensionalvariational problem consistsin extremizing a differentiable action/cost function f: X →R, where X is aninfinite dimensional manifold of paths in a manifold M, overa subset Ω C X ofadmissible paths, for example those satisfying someregularity conditions, boundaryconditions or other constraints. Thus a solution to thevariational problem is acritical point of the restriction fΩ.A standard criterion for existence of critical points is thePalaisSmale condition.If this condition is satisfied then the gradient flowassociated with f is well behaved,and we are guaranteed not only existence of critical pointsbut also existence of aminimum. Moreover it is then possible to relate the totalnumber of critical pointsto topological properties of Ω
AB  [Truncated] From the global analytical point of view a onedimensionalvariational problem consistsin extremizing a differentiable action/cost function f: X →R, where X is aninfinite dimensional manifold of paths in a manifold M, overa subset Ω C X ofadmissible paths, for example those satisfying someregularity conditions, boundaryconditions or other constraints. Thus a solution to thevariational problem is acritical point of the restriction fΩ.A standard criterion for existence of critical points is thePalaisSmale condition.If this condition is satisfied then the gradient flowassociated with f is well behaved,and we are guaranteed not only existence of critical pointsbut also existence of aminimum. Moreover it is then possible to relate the totalnumber of critical pointsto topological properties of Ω
KW  Calculus of variations
KW  Global analysis
KW  Elastica
KW  Morse theory
KW  Geometric control
KW  Riemannian cubics
KW  SubRiemannian
KW  PalaisSmale condition
M3  Doctoral Thesis
ER 