Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Invariant higher-order variational problems   We investigate higher-order geometric $k$-splines for template matching on
Lie groups. This is motivated by the need to apply diffeomorphic template
matching to a series of images, e.g., in longitudinal studies of Computational
Anatomy. Our approach formulates Euler-Poincar\'e theory in higher-order
tangent spaces on Lie groups. In particular, we develop the Euler-Poincar\'e
formalism for higher-order variational problems that are invariant under Lie
group transformations. The theory is then applied to higher-order template
matching and the corresponding curves on the Lie group of transformations are
shown to satisfy higher-order Euler-Poincar\'{e} equations. The example of
SO(3) for template matching on the sphere is presented explicitly. Various
cotangent bundle momentum maps emerge naturally that help organize the
formulas. We also present Hamiltonian and Hamilton-Ostrogradsky Lie-Poisson
formulations of the higher-order Euler-Poincar\'e theory for applications on
the Hamiltonian side.

31
The categories of this paper are: ao-sci
20
The categories of this paper are: acc-phys
32
The categories of this paper are: atom-ph
37
The categories of this paper are: supr-con
21
The categories of this paper are: adap-org
11
The categories of this paper are: hep-lat
23
The categories of this paper are: solv-int
13
The categories of this paper are: quant-ph
28
The categories of this paper are: alg-geom
25
The categories of this paper are: patt-sol
5
The categories of this paper are: gr-qc
16
The categories of this paper are: stat
24
The categories of this paper are: dg-ga
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The categories of this paper are: funct-an
4
The categories of this paper are: cond-mat
36
The categories of this paper are: cmp-lg