Invariants of two-dimensional systems via complex Lagrangians with applications

Differential invariants of a class of Lagrangian systems with two degrees of freedom

“…System (17) may be reduced to a decoupled form if it has invariants of the form (15). It is readily seen that system (17) has vanishing invariants I 4 , I 5 when b/a = −1, ω 2 = ω 1 , in which case system (17) breaks up in the variables q 1 ± q 2 , as it is noticed in [26].…”

“…In [12] for a system of n second-order ODEs some fundamental relative invariants are introduced and the criteria of its equivalence to the simplest formẍ = 0 are obtained. Note that for systems (1) with two degrees of freedom some symmetry properties (which may be used for their integration) were previously studied in [13,14,15]. More precisely, in [13] one deals with Lie point symmetries of autonomous systems (6), (7).…”

“…More precisely, in [13] one deals with Lie point symmetries of autonomous systems (6), (7). The recent papers [14,15] are devoted to constructing the Noether type symmetries and first integrals for those particular systems (6) which are equivalent in a complex domain to a single equation…”

Hypercomplex analysis and integration of systems of ordinary differential equations

“…These two processes are facilitated by analyticity. In the case where , these procedures have been used to solve various problems associated to differential equations . In the latter works, the scalar equation from which the systems stem is called the base equation .…”

Two-dimensional systems that arise from the Noether classification of Lagrangians on the line

“…Now by differentiating (26) with respect of f and g and using Eqs. (28) in it gives us the following solution C 1 = χ 1 = 0 = χ 2 = C 2 , and ς 1 = C 3 , ς 2 = C 4 , while the gauge functions are determined to be A 1 = C 5 , A 2 = C 6 . Therefore, we obtain a single Noether-like operator which is translation in x of system (17).…”