Invariant Higher-Order Variational Problems II
Copyright policy )Invariant Higher-Order Variational Problems II
Motivated by applications in computational anatomy, we consider a second-order problem in the calculus of variations on object manifolds that are acted upon by Lie groups of smooth invertible transformations. This problem leads to solution curves known as Riemannian cubics on object manifolds that are endowed with normal metrics. The prime examples of such object manifolds are the symmetric spaces. We characterize the class of cubics on object manifolds that can be lifted horizontally to cubics on the group of transformations. Conversely, we show that certain types of non-horizontal geodesics on the group of transformations project to cubics. Finally, we apply second-order Lagrange--Poincar�� reduction to the problem of Riemannian cubics on the group of transformations. This leads to a reduced form of the equations that reveals the obstruction for the projection of a cubic on a transformation group to again be a cubic on its object manifold.
40 pages, 1 figure. First version -- comments welcome!
- Panthéon-Assas University France
- University of Paris France
- PSL Research University France
- French National Centre for Scientific Research France
- École Polytechnique France
Mathematics - Differential Geometry, Constrained dynamics, Dirac's theory of constraints, Higher-order theories, Optimal control problems involving partial differential equations, higher-order theories, 510, 515, FOS: Mathematics, Group actions and symmetry properties, Existence theories for problems in abstract spaces, optimal control problems involving partial differential equations, Higher-order theories for problems in Hamiltonian and Lagrangian mechanics, Mathematics - Optimization and Control, constrained dynamics, other variational principles, Other variational principles, Hamilton’s principle, [PHYS.MECA]Physics [physics]/Mechanics [physics], Abstract critical point theory (Morse theory, Lyusternik-Shnirel'man theory, etc.) in infinite-dimensional spaces, [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA], Analyse, Hamilton's principle, Differential Geometry (math.DG), Optimization and Control (math.OC), Other variational principles in mechanics, Constrained dynamics, [PHYS.MECA] Physics [physics]/Mechanics [physics], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Mathematics - Differential Geometry, Constrained dynamics, Dirac's theory of constraints, Higher-order theories, Optimal control problems involving partial differential equations, higher-order theories, 510, 515, FOS: Mathematics, Group actions and symmetry properties, Existence theories for problems in abstract spaces, optimal control problems involving partial differential equations, Higher-order theories for problems in Hamiltonian and Lagrangian mechanics, Mathematics - Optimization and Control, constrained dynamics, other variational principles, Other variational principles, Hamilton’s principle, [PHYS.MECA]Physics [physics]/Mechanics [physics], Abstract critical point theory (Morse theory, Lyusternik-Shnirel'man theory, etc.) in infinite-dimensional spaces, [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA], Analyse, Hamilton's principle, Differential Geometry (math.DG), Optimization and Control (math.OC), Other variational principles in mechanics, Constrained dynamics, [PHYS.MECA] Physics [physics]/Mechanics [physics], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
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