In this paper we examine the existence of Lie groups, whose canonical geodesic flows are variational with respect to a left-invariant regular—but not necessarily quadratic (i.e., metric)—Lagrange function. We give effective necessary and sufficient conditions for the existence of an invariant variational principle generating the canonical flow. With these results, taken in conjunction with the classification of Lie algebras, we solve the inverse problem of invariant Lagrangian dynamics in dimensions up to four.

1.
Cartan
,
E.
and
Schouten
,
J. A.
, “
On the geometry of the group-manifold of simple and semi-simple groups
,”
Proc. Akad. Wekensch, Amsterdam
29
,
803
815
(
1926
).
2.
Granam
,
R.
,
Miller
,
E. J.
, and
Thompson
,
G.
, “
Variationality of four-dimensional Lie group connections
,”
J. Lie Theory
14
,
395
425
(
2004
).
3.
Godbillon
,
C.
,
Géométrie Différentielle et Mécanique Analytique
(
Hermann
, Paris,
1969
).
4.
Grifone
,
J.
, “
Structure presque-tangente et connexions I, II
,”
Ann. Inst. Fourier
22
,
287
334
(
1972
);
Grifone
,
J.
,
Ann. Inst. Fourier
22
,
291
338
(
1972
).
5.
Grifone
,
J.
and
Muzsnay
,
Z.
,
Variational Principles for Second-order Differential Equations
(
World Scientific
, Singapore,
2000
).
6.
Jacobson
,
N.
,
Lie Algebras
(
Interscience
, New York,
1962
).
7.
Muzsnay
,
Z.
and
Thompson
,
G.
, “
Inverse problem of the calculus of variations on Lie groups
,”
Diff. Geom. Applic.
23
,
257
281
(
2005
).
8.
Patera
,
J.
,
Sharp
,
R. T.
,
Winternitz
,
P.
, and
Zassenhaus
,
H.
, “
Invariants of real low dimension Lie algebras
,”
J. Math. Phys.
17
,
986
994
(
1976
).
9.
Thompson
,
G.
, “
Variational connections on Lie groups
,”
Diff. Geom. Applic.
18
,
255
270
(
2003
).
You do not currently have access to this content.