scholarly journals Revisiting Lie integrability by quadratures from a geometric perspective

2016 ◽  
Vol 110 ◽  
pp. 25-40
Author(s):  
José F. Cariñena ◽  
Manuel F. Rañada ◽  
Fernando Falceto ◽  
Janusz Grabowski
Keyword(s):  
Integrability By Quadratures

scholarly journals Geometry of Lie integrability by quadratures

2015 ◽  
Vol 48 (21) ◽  
pp. 215206 ◽  
Author(s):  
J F Cariñena ◽  
F Falceto ◽  
J Grabowski ◽  
M F Rañada
Keyword(s):  
Integrability By Quadratures

scholarly journals $\lambda$ -Symmetries and integrability by quadratures

2017 ◽  
Vol 82 (5) ◽  
pp. 1061-1087 ◽  
Author(s):  
C. Muriel ◽  
J. L. Romero ◽  
A. Ruiz
Keyword(s):  
Integrability By Quadratures

scholarly journals Integrability by quadratures of Hamiltonian systems and Picard-Fuchs type equations: The modern differential-geometric aspects

2005 ◽  
Vol 6 (1) ◽  
pp. 65 ◽  
Author(s):  
Y. A. Prykarpatsky ◽  
A. M. Samoilenko ◽  
D. L. Blackmore ◽  
A. K. Prykarpatsky
Keyword(s):  
Hamiltonian Systems ◽  
Integrability By Quadratures

scholarly journals Extended Hamilton–Jacobi Theory, Symmetries and Integrability by Quadratures

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1357
Author(s):  
Sergio Grillo ◽  
Juan Carlos Marrero ◽  
Edith Padrón
Keyword(s):  
Lie Groups ◽  
Canonical Projection ◽  
Open Dense Subset ◽  
Semisimple Lie Groups ◽  
Invariant Vector ◽  
Integral Curves ◽  
Invariant Vector Field ◽  
Integrability By Quadratures ◽  
Hamilton Jacobi Equation ◽  
Complete Solutions

In this paper, we study the extended Hamilton–Jacobi Theory in the context of dynamical systems with symmetries. Given an action of a Lie group G on a manifold M and a G-invariant vector field X on M, we construct complete solutions of the Hamilton–Jacobi equation (HJE) related to X (and a given fibration on M). We do that along each open subset U⊆M, such that πU has a manifold structure and πU:U→πU, the restriction to U of the canonical projection π:M→M/G, is a surjective submersion. If XU is not vertical with respect to πU, we show that such complete solutions solve the reconstruction equations related to XU and G, i.e., the equations that enable us to write the integral curves of XU in terms of those of its projection on πU. On the other hand, if XU is vertical, we show that such complete solutions can be used to construct (around some points of U) the integral curves of XU up to quadratures. To do that, we give, for some elements ξ of the Lie algebra g of G, an explicit expression up to quadratures of the exponential curve expξt, different to that appearing in the literature for matrix Lie groups. In the case of compact and of semisimple Lie groups, we show that such expression of expξt is valid for all ξ inside an open dense subset of g.

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