TY - JOUR
T1 - Hamiltonization of higher-order nonlinear ordinary differential equations and the Jacobi last multiplier
AU - Guha, Partha
AU - Ghose Choudhury, A.
N1 - Funding Information:
Acknowledgements We wish to thank Pepin Cariñena, Manuel Rañada, Gerrado Torres del Castillo and Basil Grammaticos for their valuable remarks. In particular, we are extremely grateful to Joe Poveromo for his correspondences on conformal Weyl gravity equation. We wish to thank especially Gerrado Torres del Castillo for careful reading of the manuscript. In addition AGC wishes to acknowledge the support provided by the S.N. Bose National Centre for Basic Sciences, Kolkata in the form of an Associateship.
PY - 2011/11
Y1 - 2011/11
N2 - It is known that Jacobi's last multiplier is directly connected to the deduction of a Lagrangian via Rao's formula (Madhava Rao in Proc. Benaras Math. Soc. (N.S.) 2:53-59, 1940). In this paper we explicitly demonstrate that it also plays an important role in Hamiltonian theory. In particular, we apply the recent results obtained by Torres del Castillo (J. Phys. A Math. Theor. 43:265202, 2009) and deduce the Hamiltonian of a second-order ODE of the Lienard type, namely, ẍ+f(x)x·2 + g(x)=0. In addition, we consider cases where the coefficient functions may also depend on the independent variable t. We illustrate our construction with various examples taken from astrophysics, cosmology and the Painlevé-Gambier class of differential equations. Finally we discuss the Hamiltonization of third-order equations using Nambu-Hamiltonian mechanics.
AB - It is known that Jacobi's last multiplier is directly connected to the deduction of a Lagrangian via Rao's formula (Madhava Rao in Proc. Benaras Math. Soc. (N.S.) 2:53-59, 1940). In this paper we explicitly demonstrate that it also plays an important role in Hamiltonian theory. In particular, we apply the recent results obtained by Torres del Castillo (J. Phys. A Math. Theor. 43:265202, 2009) and deduce the Hamiltonian of a second-order ODE of the Lienard type, namely, ẍ+f(x)x·2 + g(x)=0. In addition, we consider cases where the coefficient functions may also depend on the independent variable t. We illustrate our construction with various examples taken from astrophysics, cosmology and the Painlevé-Gambier class of differential equations. Finally we discuss the Hamiltonization of third-order equations using Nambu-Hamiltonian mechanics.
KW - Conformal Weyl gravity
KW - Jacobi's last multiplier
KW - Nambu-Hamiltonians
KW - Time-dependent Hamiltonian system
KW - White dwarf equation
UR - http://www.scopus.com/inward/record.url?scp=80855139755&partnerID=8YFLogxK
U2 - 10.1007/s10440-011-9637-3
DO - 10.1007/s10440-011-9637-3
M3 - Article
AN - SCOPUS:80855139755
SN - 0167-8019
VL - 116
SP - 179
EP - 197
JO - Acta Applicandae Mathematicae
JF - Acta Applicandae Mathematicae
IS - 2
ER -