In this article, we provide a Hamilton–Jacobi formalism on locally conformally symplectic (lcs) manifolds. We are interested in the Hamilton–Jacobi as an alternative method for formulating the dynamics, while our interest in the locally conformal character will account for physical theories described by Hamiltonians defined on well-behaved line bundles, whose dynamic takes place in open subsets of the general manifold. We present a lcs Hamilton–Jacobi equation in subsets of the general manifold and then provide a global view by using the Lichnerowicz–deRham differential. We show a comparison between the global and local description of a lcs Hamilton–Jacobi theory, and how actually the local behavior can be glued to retrieve the global behavior of the Hamilton–Jacobi theory. A particular example is the case of Gaussian isokinetic dynamics in which we apply our structure in certain submanifolds of the phase space.
REFERENCES
1.
R.
Abraham
and J. E.
Marsden
, Foundations of Mechanics
, 2nd ed., revised and enlarged (Benjamin/Cummings Publishing Co., Inc., Advanced Book Program
, Reading, MA
, 1978
).2.
V. I.
Arnold
, Mathematical Methods of Classical Mechanics
, Graduate Texts in Mathematics Vol. 60, 2nd ed. (Springer-Verlag
, New York
, 1989
).3.
A.
Awane
, “k-symplectic structures
,” J. Math. Phys.
33
(12
), 4046
–4052
(1992
).4.
A.
Banyaga
, “Some properties of locally conformal symplectic structures
,” Comment. Math. Helvetici
77
(2
), 383
–398
(2002
).5.
G.
Bazzoni
, “Locally conformally symplectic and Kahler geometry
,” Math. Sci.
5
, 129
–154
(2018
).6.
F.
Belgun
, O.
Goertsches
, and D.
Petrecca
, “Locally conformally symplectic convexity
,” J. Geom. Phys.
135
, 235
–252
(2019
).7.
R. C.
Calleja
, A.
Celletti
, and R.
de la Llave
, “A KAM theory for conformally symplectic systems: Efficient algorithms and their validation
,” J. Differ. Equations
255
, 978
–1049
(2013
).8.
J. F.
Cariñena
, X.
Gràcia
, G.
Marmo
, E.
Martínez
, M. C.
Muñoz-lecanda
, and N.
Román-Roy
, “Geometric Hamilton-Jacobi theory
,” Int. J. Geom. Methods Mod. Phys.
03
(07
), 1417
–1458
(2006
).9.
B.
Chantraine
and E.
Murphy
, “Conformal symplectic geometry of cotangent bundles
,” J. Symplectic Geom.
17
(3
), 639
–661
(2019
).10.
D.
Chinea
, M.
de León
, and J. C.
Marrero
, “Locally conformal cosymplectic manifolds and time-dependent Hamiltonian systems
,” Comment. Math. Univ. Carolin.
32
(2
), 383
–387
(1991
).11.
C. P.
Dettmann
and G. P.
Morriss
, “Proof of Lyapunov exponent pairing for systems at constant kinetic energy
,” Phys. Rev. E
53
(6
), R5545
(1996
).12.
13.
F.
Guedira
and A.
Lichnerowicz
, “Géométrie des algébres de Lie locales de Kirillov
,” J. Math. Pures Appl.
63
(4
), 407
–484
(1984
).14.
S.
Grillo
and E.
Padrón
, “A Hamilton–Jacobi theory for general dynamical systems and integrability by quadratures in symplectic and Poisson manifolds
,” J. Geom. Phys.
110
, 101
–129
(2016
).15.
S.
Grillo
and E.
Padrón
, “Extended Hamilton-Jacobi theory, contact manifolds, and integrability by quadratures
,” J. Math. Phys.
61
, 012901
(2020
).16.
S.
Haller
and T.
Rybicki
, “Reduction for locally conformal symplectic manifolds
,” J. Geom. Phys.
37
(3
), 262
–271
(2001
).17.
S.
Haller
and T.
Rybicki
, “On the group of diffeomorphisms preserving a locally conformal symplectic structure
,” Ann. Global Anal. Geom.
17
, 475
–502
(1999
).18.
A.
Ibort
, M.
de Léon
, and G.
Marmo
, “Reduction of Jacobi manifolds
,” J. Phys. A: Math. Gen.
30
(8
), 2783
(1997
).19.
N.
Istrati
, “A characterisation of toric LCK manifolds
,” J. Symplectic Geom.
17
(5
), 1297
–1316
(2019
).20.
H.-C.
Lee
, “A kind of even dimensional differential geometry and its applications to exterior calculus
,” Am. J. Math.
65
, 433
–438
(1943
).21.
M.
de León
, D. M.
de Diego
, J. C.
Marrero
, M.
Salgado
, and S.
Vilariño
, “Hamilton-Jacobi theory in k-symplectic field theories
,” Int. J. Geom. Methods Mod. Phys.
07
(8
), 1491
–1507
(2010
).22.
M.
de Léon
, D. M.
de Diego
, and M.
Vaquero
, “Hamilton–Jacobi theory, symmetries and coisotropic reduction
,” J. Math. Pures Appl.
107
(5
), 591
–614
(2017
).23.
M.
de León
, J. C.
Marrero
, and E.
Padrón
, “Lichnerowicz Jacobi cohomology
,” J. Phys. A: Math. Gen.
30
(17
), 6029
–6055
(1997
).24.
M.
de León
, D.
Martín de Diego
, and M.
Vaquero
, “A Hamilton–Jacobi theory on Poisson manifolds
,” J. Geom. Mech.
6
(1
), 121
–140
(2014
).25.
M.
de León
and P. R.
Rodrigues
, Methods of Differential Geometry in Analytical Mechanics
. North-Holland Mathematics Studies Vol. 158 (North-Holland Publishing Co.
, Amsterdam
, 1989
).26.
M.
de León
, M.
Salgado
, and S.
Vilariño
, Methods of Differential Geometry in Classical Field Theories k-Symplectic and k-Cosymplectic Approaches
(World Scientific
, Singapore
, 2015
).27.
M.
de León
and C.
Sardón
, “Cosymplectic and contact structures for time-dependent and dissipative Hamiltonian systems
,” J. Phys. A: Math. Theor.
50
(25
), 255205
(2017
).28.
P.
Libermann
and C. M.
Marle
, Symplectic Geometry and Analytical Mechanics in Mathematics and Its Applications
(D. Reidel Publishing Co.
, Dordrecht
, 1987
), Vol. 35.29.
C.
Liverani
and M. P.
Wojtkowski
, “Conformally symplectic dynamics and symmetry of the Lyapunov spectrum
,” Commun. Math. Phys.
194
, 47
–60
(1998
).30.
C. M.
Marle
, “On Jacobi manifolds and Jacobi bundles
,” in Symplectic Geometry, Groupoids, and Integrable Systems
(Springer
, New York, NY
, 1991
), pp. 227
–246
.31.
D.
McDuff
and D.
Salamon
, Introduction to Symplectic Topology
, Oxford Mathematical Monographs, 2nd ed. (The Clarendon Press; Oxford University Press
, New York
, 1998
).32.
A.
Otiman
and M.
Stanciu
, “Darboux-Weinstein theorem for locally conformally symplectic manifolds
,” J. Geom. Phys.
111
, 1
–5
(2017
).33.
M.
Stanciu
, “Locally conformally symplectic reduction
,” Ann. Global Anal. Geom.
56
(2
), 245
–275
(2019
).34.
I.
Vaisman
, “Locally conformal symplectic manifolds
,” Int. J. Math. Math. Sci.
8
, 521
–536
(1985
).35.
I.
Vaisman
, Lectures on the Geometry of Poisson Manifolds
, Progress in Mathematics Vol. 118 (Birkhäuser Verlag
, Basel
, 1994
).36.
I.
Vaisman
, “Hamiltonian vector fields on almost symplectic manifolds
,” J. Math. Phys.
54
(9
), 092902
(2013
).© 2021 Author(s).
2021
Author(s)
You do not currently have access to this content.
