Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0807.2796

An old Method of Jacobi to find Lagrangians

M. C. Nucci·P. G. L. Leach
View original source ↗Read source document ↗Export metadata ↓

Abstract

In a recent paper by Ibragimov [N. H. Ibragimov, Invariant Lagrangians and a new method of integration of nonlinear equations, J. Math. Anal. Appl. 304 (2005) 212--235] a method was presented in order to find Lagrangians of certain second-order ordinary differential equations admitting a two-dimensional Lie symmetry algebra. We present a method devised by Jacobi which enables to derive (many) Lagrangians of any second-order differential equation. The method is based on the search of the Jacobi Last Multipliers of the equations. We exemplify the simplicity and elegance of Jacobi's method by applying it to the same two equations as did Ibragimov. We show that the Lagrangians obtained by Ibragimov are particular cases of some of the many Lagrangians that can be obtained by Jacobi's method.

Explore related subjects

nlin.SI

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

M. C. Nucci, P. G. L. Leach. 2008-07-17. An old Method of Jacobi to find Lagrangians. https://doi.org/10.1142/s1402925109000467

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the geometry of WDVV equations and their Hamiltonian formalism in arbitrary dimension

It is known that in low dimensions WDVV equations can be rewritten as commuting quasilinear bi-Hamiltonian systems. We extend some of these results to arbitrary dimension $N$ and arbitrary scalar product $η$. In particular, we show that suitable subsets of WDVV equations can be interpreted as a set of linear line congruences in suitable Plücker embeddings. This form leads to their representation as $N-2$ Hamiltonian systems of conservation laws. Moreover, we show that WDVV equations can be reduced to an orthonomic form, which is also passive in low dimensions $N\leq 5$. This also leads to the commutativity of the Hamiltonian systems of conservation laws ($N\leq 5$), after which we can find a solution of the WDVV equations from a joint solution of the Hamiltonian systems. Finally, we conjecture that passivity holds in all dimensions.

nlin.SI↗
arXiv2026

Multivariable Painleve'-II equation: connection formulas for arbitrary system size

Connection formulas for the asymptotic solutions of a system of n> 1 coupled Painleve'-II equations with symmetry-breaking parameters are written explicitly. An asymptotically exact WKB approach to these formulas relies on the quantum-mechanical independent crossing approximation for an explicitly time-dependent Schroedinger equation.

nlin.SI↗
arXiv2026

A Classification of Hirota-Integrable Supersymmetric Bilinear KdV-Type Equations

We present a classification of supersymmetric bilinear KdV-type equations admitting unconstrained three-super-soliton solutions. Extending Hirota's classical three-soliton criterion to the supersymmetric setting, we derive the complete bosonic and fermionic compatibility conditions governing the existence of three-super-soliton solutions. We prove that every supersymmetric bilinear KdV-type equation possesses unconstrained one- and two-super-soliton solutions, whereas three-super-soliton solutions exist only when eight integrability conditions are satisfied. These conditions provide a supersymmetric analogue of Hirota's classical integrability criterion and naturally recover the fermionic relations previously introduced by Carstea, revealing their structural origin. As a consequence, we classify the supersymmetric extensions of Hirota bilinear KdV-type equations and show that only a subset of Hietarinta's classical classification remains valid in the unrestricted supersymmetric framework.

nlin.SI↗