Contact geometry has been applied to various mathematical sciences, and it has been proposed that a contact manifold and a strictly convex function induce a dually flat space that is used in information geometry. Here, such a dually flat space is related to a Legendre submanifold in a contact manifold. In this paper, contact geometric descriptions of vector fields on dually flat spaces are proposed on the basis of the theory of contact Hamiltonian vector fields. Based on these descriptions, two ways of lifting vector fields on Legendre submanifolds to contact manifolds are given. For some classes of these lifted vector fields, invariant measures in contact manifolds and stability analysis around Legendre submanifolds are explicitly given. Throughout this paper, Legendre duality is explicitly stated. In addition, to show how to apply these general methodologies to applied mathematical disciplines, electric circuit models and some examples taken from nonequilibrium statistical mechanics are analyzed.

1.
V. I.
Arnold
,
Mathematical Methods of Classical Mechanics
(
Springer
,
Berlin
,
1976
).
2.
R.
Hermann
,
Geometry, Systems and Physics
(
Dekker
,
New York
,
1973
).
3.
R.
Mrugala
,
Suken kokyuroku
1142
,
167
181
(
2000
).
4.
D.
Eberard
,
B. M.
Maschke
, and
A. J.
Van Der Schaft
,
Rep. Math. Phys.
60
,
175
198
(
2007
).
5.
A.
Bravetti
,
C. S.
Lopez-Monsalvo
, and
F.
Nettel
,
Ann. Phys.
361
,
377
400
(
2015
).
6.
S.
Goto
,
J. Math. Phys.
56
,
073301
(
2015
).
7.
R.
Mrugala
,
J. D.
Nulton
,
J. C.
Schon
, and
P.
Salamon
,
Phys. Rev. A
41
,
3156
3160
(
1990
).
8.
J.
Jurkowski
,
Phys. Rev. E
62
,
1790
1798
(
2000
).
9.
A.
Bravetti
and
C. S.
Lopez-Monsalvo
,
J. Phys. A
48
,
125206
(
2015
); e-print arXiv:1408.5443v3.
10.
R.
Ghrist
, in
Handbook of Mathematical Fluid Dynamics
(
Elsevier
,
2007
), Vol.
4
, Chap. 1, pp.
1
37
.
11.
M.
Dahl
,
Prog. Electromagn. Res.
46
,
77
104
(
2004
).
13.
A.
Bravetti
and
D.
Tapias
,
J. Phys. A
48
,
245001
(
2014
); e-print arXiv:1412.0026v2.
14.
S. I.
Amari
and
H.
Nagaoka
,
Methods of Information Geometry
,
Translations of Mathematical Monographs
Vol.
191
(
American Mathematical Society
,
Providence
,
2000
).
15.
A.
Facache
,
V. S. D. S.
Martins
,
D.
Dochain
, and
B.
Maschke
,
IEEE Trans. Autom. Control
54
,
2341
2351
(
2009
).
16.
H. R.
Estay
,
B.
Mashke
, and
D.
Sbarbaro
, in Preprints of the 18th IFAC World Congress, Mirano, Italy, 28 August–2 September, 2011.
17.
G. S.
Ezra
,
J. Math. Chem.
32
,
339
359
(
2002
).
18.
19.
R.
Mrugala
,
J. D.
Nulton
,
J. C.
Schon
, and
P.
Salamon
,
Rep. Math. Phys.
29
,
109
121
(
1991
).
20.
A.
Ohara
,
N.
Sunada
, and
S.
Amari
,
Linear Algebra Appl.
247
,
31
53
(
1996
).
21.
A.
Fujiwara
and
S. I.
Amari
,
Physica D
80
,
317
327
(
1995
).
22.
Y.
Nakamura
,
Jpn. J. Ind. Appl. Math.
11
,
21
30
(
1994
).
23.
D.
Eberard
,
B. M.
Maschke
, and
A. J.
Van Der Schaft
, in Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems, Kyoto, Japan, 24–28 July, 2006.
24.
M. W.
Hirsch
and
S.
Smale
,
Differential Equations, Dynamical Systems, and Linear Algebra
(
Academic Press
,
1974
).
25.
L.
Onsager
and
S.
Machlup
,
Phys. Rev.
91
,
1505
1512
(
1953
).
26.
D.
Eberard
,
B. M.
Maschke
, and
A. J.
Van Der Schaft
, in Proceedings of the 44th Conference on Decision and Control, and the European Control Conference (IEEE, 2005), pp. 5977–5982.
27.
J. H.
Doty
and
K. L.
Yerkes
, Proceedings of the 44th AIAA (American Institute of Aeronautics & Astronautics,
2011
).
28.
T.
Noda
,
J. Aust. Math. Soc.
90
,
371
384
(
2011
).
29.
A.
Ohara
and
T.
Wada
,
J. Phys. A
43
,
035002
(
2010
).
You do not currently have access to this content.