Ermakov–Ray–Reid systems with additive noise

Relativistic Ermakov-Milne-Pinney Systems and First Integrals

“…Finally, an intriguing question remains about the RRR system defined by Eqs. (24) and (25). If one starts from the Lagrangian (13) adding an Ermakov term U(y/x)/(x 2 + y 2 ) to the potential V, we find precisely the system (24)- (25), where f , g are suitable related to U, but with a factor 1/γ instead of 1/γ 2 on all terms of the right-hand side.…”

“…It is apparent that Eqs. (24), (25) and (26) define a RRR system and its invariant, showing a complete symmetry between the x and y variables and recovering the RR system and invariant in the formal NR limit c → ∞, as shown by comparison with Eqs. (10)- (12).…”

“…in Eqs. (24) and 25, maintaining the constancy of the RRR invariant. Similar remarks apply to the NR case [46][47][48].…”

“…then it can be checked after some algebra that the RRR system (24) and 25is recovered, with κ 2 = ω 2 /γ, and that the invariant (28) transforms into (26). The procedure gives a more transparent derivation of the RRR system from a dynamical rescaling of time starting from the RR system.…”

“…For instance, sometimes it can be referred just as Pinney equation. Diverse generalizations of the EMP equation have been proposed, as for instance allowing C(t) to have a dependence on time [8,19], the inclusion of dissipation [20,21], unbalanced systems of EMP equations with different frequency functions [22], modified nonlinearities [23], stochastic differential equations with additive noise [24]. Following the generalization trend, it would be relevant to extend the EMP equation into the special relativity domain, with potential applications for the dynamics of charged particles in high energy density fields [25].…”