The class of integral variational functionals for paths in smooth manifolds, whose extremals are (nonparameterized) sets, is considered in this study. Recently, it was shown that for the functionals depending on tangent vectors, this property follows from any of the following two equivalent conditions: (a) the Lagrange function, defined on the tangent bundle, is positively homogeneous in the components of tangent vectors and (b) the Lepage differential form of the Lagrange function is projectable onto the Grassmann fibrations of rank 1 and order 1 (projective bundle); the classical Hilbert form was rediscovered this way as the projection of the Lepage form. In this paper, we extend these results to variational functionals of any order. Projectability conditions onto Grassmann fibrations of any order are found. The case of projective bundles is then studied in full generality. The proofs are based on the Euler–Zermelo conditions and the properties of higher-order Grassmann fibrations of rank 1. As an application, equations for set solutions in Riemannian geometry are derived.

1.
S. S.
Chern
, “
Finsler Geometry is just Riemannian geometry without the quadratic restriction
,”
Not. AMS
1966
,
959
963
.
2.
S. S.
Chern
,
W. H.
Chen
, and
K. S.
Lam
,
Lectures on Differential Geometry
(
World Scientific
,
2000
).
3.
M.
Hohmann
,
Ch.
Pfeifer
, and
N.
Voicu
, “
Finsler gravity action from variational completion
,”
Phys. Rev. D
100
,
064035
(
2019
).
4.
D.
Krupka
, “
Lepage forms in Kawaguchi spaces and the Hilbert form, paper in Honor of Professor Lajos Tamassy
,”
Publ. Math. Debrecen
84
,
147
164
(
2014
).
5.
Z.
Urban
and
D.
Krupka
, “
Foundations of higher-order variational theory on Grassmann fibrations
,”
Int. J. Geom. Methods Mod. Phys.
11
,
1460023
(
2014
).
6.
Z.
Urban
and
D.
Krupka
, “
The Zermelo conditions and higher-order homogeneous functions
,”
Publ. Math. Debrecen
82
,
59
76
(
2013
).
7.
D.
Krupka
, “
Higher-order homogeneous functions: Classification
,”
Publ. Math. Debrecen
(unpublished).
8.
C.
Ehresmann
, “
Les prolongements d’une variete différentiable I
,”
C. R. Acad. Sc. Paris
223
,
598
600
(
1951
);
C.
Ehresmann
, “
Les prolongements d’une variete différentiable II
,”
C. R. Acad. Sc. Paris
223
,
777
779
(
1951
);
C.
Ehresmann
, “
Les prolongements d’une variete différentiable III
,”
C. R. Acad. Sc. Paris
223
,
1081
1083
(
1951
);
C.
Ehresmann
, “
Les prolongements d’une variete différentiable IV
,”
C. R. Acad. Sc. Paris
234
,
1028
1030
(
1952
);
C.
Ehresmann
, “
Les prolongements d’une variete différentiable V
,”
C. R. Acad. Sc. Paris
234
,
1424
1425
(
1952
).
9.
D. R.
Grigore
and
D.
Krupka
, “
Invariants of velocities and higher order Grassmann bundles
,”
J. Geom. Phys.
24
,
244
264
(
1998
).
10.
D.
Krupka
and
M.
Krupka
, “
Jets and contact elements
,” in
Proceedings of the Seminar on Differential Geometry
, Mathematical Publications Vol. 2, edited by
D.
Krupka
(
Silesian University in Opava
,
Czech Republic
,
2000
), pp.
39
85
.
11.
D.
Krupka
,
Introduction to Global Variational Geometry
, Atlantis Studies in Variational Geometry (
Atlantis Press
,
2015
), p.
371
.
12.
D.
Krupka
, “
Some geometric aspects of variational problems in fibered manifolds
,”
Folia Fac. Sci. Nat. UJEP Brunensis, Physica
14
,
65
(
1973
); arXiv:math-ph/0110005 (
2001
).
13.
D.
Krupka
,
O.
Krupkova
, and
D.
Saunders
, “
Cartan-Lepage forms in geometric mechanics
,”
Int. J. Nonlinear Mech.
47
,
1154
1160
(
2012
).
14.
D.
Krupka
,
Z.
Urban
, and
J.
Volná
, “
Variational submanifolds of Euclidean spaces
,”
J. Math. Phys.
59
,
032903
(
2018
).
15.
P.
Dedecker
, “
On the generalization of symplectic geometry to multiple integrals in the calculus of variations
,” in
Differential Geometrical Methods in Mathematical Physics
, Lecture Notes in Mathematics Vol. 570 (
Springer
,
Berlin
,
1977
), pp.
395
456
.
16.
D.
Krupka
and
M.
Krupka
, “
Grassmann fibrations and the calculus of variations
,”
Balkan J. Geom. Appl.
15
,
68
79
(
2010
).
17.
D.
Saunders
, “
Homogeneous variational problems: A minicourse
,”
Commun. Math.
19
,
91
128
(
2010
).
18.
Z.
Urban
, “
Variational principles for immersed submanifolds
,” in
The Inverse Problem of the Calculus of Variations
, Atlantis Studies in Variational Geometry, edited by
D. V.
Zenkov
(
Atlantis
,
2015
), pp.
103
170
.
You do not currently have access to this content.