Inspired by the equations of motion of the gyroscope, where a Lagrangian and
conservative system may appear to mimic a dissipative one when focusing on a single
degree of freedom for finite time intervals, we introduce a gyroscopic-type coupling
between harmonic oscillators. The aim is to propose a tentative scheme of solution for
the problem of finding a higher-dimensional Lagrangian system approximating a
lower-dimensional dissipative one. Specifically, we consider a certain family of
Lagrangian systems, for which the time evolution of the first Lagrangian parameter is
conjectured to be a good approximation for the evolution of a one-dimensional linear
dissipative system in finite time intervals, up to a fixed precision. The behavior of the
selected family of gyroscopic couplings is compared with a given dissipative system,
properly optimizing a family of parameters according to a described scheme.
Numerical calculations are reported, suggesting the validity of the proposed
conjecture.
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