# Solvability of a Lie algebra of vector fields implies their integrability by quadratures

@article{Cariena2016SolvabilityOA,
title={Solvability of a Lie algebra of vector fields implies their integrability by quadratures},
author={Jos{\'e} F. Cari{\~n}ena and Fernando Falceto and Janusz Grabowski},
journal={Journal of Physics A: Mathematical and Theoretical},
year={2016},
volume={49}
}
• Published 8 June 2016
• Mathematics
• Journal of Physics A: Mathematical and Theoretical
We present a substantial generalisation of a classical result by Lie on integrability by quadratures. Namely, we prove that all vector fields in a finite-dimensional transitive and solvable Lie algebra of vector fields on a manifold can be integrated by quadratures.
10 Citations
• Mathematics
• 2020
We present a local and constructive differential geometric description of finite-dimensional solvable and transitive Lie algebras of vector fields. We show that it implies a Lie's conjecture for such
• Mathematics
Journal of Physics A: Mathematical and Theoretical
• 2023
Motivated by the time-dependent Hamiltonian dynamics, we extend the notion of Arnold–Liouville and noncommutative integrability of Hamiltonian systems on symplectic manifolds to that on cosymplectic
• Mathematics
Symmetry
• 2021
The general theory of the Jacobi last multipliers in geometric terms is reviewed and the theory is applied to different problems in integrability and the inverse problem for one-dimensional mechanical systems.
A geometric approach to integrability and reduction of dynamical system is developed from a modern perspective. The main ingredients in such analysis are the inﬁnitesimal symmetries and the tensor
The Gauss-Manin equations are solved for a class of flat-metrics defined by Novikov algebras, this generalizing a result of Balinskii and Novikov who solved this problem in the case of commutative
• S. Grillo
• Mathematics
Analysis and Mathematical Physics
• 2021
The non-commutative integrability (NCI) is a property fulfilled by some Hamiltonian systems that ensures, among other things, the exact solvability of their corresponding equations of motion. The
• Mathematics
Journal of Physics A: Mathematical and Theoretical
• 2022
A stratified Lie system is a nonautonomous system of first-order ordinary differential equations on a manifold M described by a t-dependent vector field X=∑α=1rgαXα , where X 1, …, X r are vector
• Mathematics
• 2019
A $\mathcal{F}$- foliated Lie system is a first-order system of ordinary differential equations whose particular solutions are contained in the leaves of the foliation $\mathcal{F}$ and all
• Mathematics
International Journal of Geometric Methods in Modern Physics
• 2019
The theory of quasi-Lie systems, i.e. systems of first-order ordinary differential equations that can be related via a generalized flow to Lie systems, is extended to systems of partial differential
• Mathematics
• 2020
Transitive local Lie algebras of vector fields can be easily constructed from dilations of $\mathbb{R}^n$ associating with coordinates positive weights (give me a sequence of $n$ positive integers

## References

SHOWING 1-10 OF 14 REFERENCES

• Mathematics
• 2014
In this paper, we extend the Lie theory of integration by quadratures of systems of ordinary differential equations in two different ways. First, we consider a finite-dimensional Lie algebra of
This paper addresses a class of problems associated with the conditions for exact integrability of systems of ordinary differential equations expressed in terms of the properties of tensor
Preface.- Basic Concepts.- Semisimple Lie Algebras.- Root Systems.- Isomorphism and Conjugacy Theorems.- Existence Theorem.- Representation Theory.- Chevalley Algebras and Groups.- References.-
• Mathematics
• 1978
In this paper we shall show that the equations of motion of a solid, and also Liouville's method of integration of Hamiltonian systems, appear in a natural manner when we study the geometry of level
is integrable by quadratures [1] (for details, see also [2, 3]). More precisely, all of its solutions can be found by “algebraic operations” (including inversion of functions) and “quadratures,” that
In [1] a local description of analytic vector fields finitely generating a transitive nilpotent Lie algebra L on a manifold is given. Our aim is to generalize this result by (i) omitting the
Let M be a real analytic manifold, and let L be a transitive Lie algebra of real analytic vector fields on M. A concept of completeness is introduced for such Lie algebras. Roughly speaking, L is
We study finite-dimensional Lie algebras of polynomial vector fields in $n$ variables that contain the vector fields ${\partial}/{\partial x_i} \; (i=1,\ldots, n)$ and \$x_1{\partial}/{\partial x_1}+
Part 1 Newtonian mechanics: experimental facts investigation of the equations of motion. Part 2 Lagrangian mechanics: variational principles Lagrangian mechanics on manifolds oscillations rigid