A separability theorem for dynamical systems admitting alternative Lagrangian descriptions

A class of recursion operators on a tangent bundle

“…Indeed, the type (1,1) tensors we consider here, are getting close to the kind of tensors discussed in [19,20]. An extra assumption we have to make is that the tensors must be algebraically diagonisable.…”

“…Equally evident then is the property 20) which is a coordinate form of (4.9). remark that these relations express that K is the Hessian of a certain function.…”

“…The theory about the existence of alternative Lagrangians of the same second-order differential equations (not just by addition of a total time derivative or multiplication by a constant) is already extensively examined (see for example [11], [19] and [20]). It is known that one can associate to a pair of alternative Lagrangians ℓ and ℓ ′ of a dynamical system Γ, a type (1,1) tensor field T , uniquely defined by…”

Application of the theory of separability of second-order differential equations

“…The extra termQ 2 2 , on the other hand, would seem to contradict equation (13). However, under the present circumstances, the theory merely guarantees the existence of coordinates which will bring the "forces" in the form (13) (we have β 2 + µ = 0 here) and of course does not preclude that separation may be achieved in other coordinates as well.…”

“…For the history of the problem of full separability, we should also mention contributions by Ferrario et al (see e.g. [1,2]) for the special case of Lagrangian systems.…”