Group properties of the finite-dimensional linearization operator of a dynamical system in the model of orbital tether system motion
DOI:
https://doi.org/10.20397/2177-6652/2023.v23i0.2600Palavras-chave:
group approach, symplectic structure, quadratic invariant, linear Hamiltonian system, Lie algebraResumo
The article considers an autonomous linear dynamical system represented by ordinary differential equations that define the motion of an orbital tether system. The object of the study are group properties of the spectrum of a finite-dimensional linearization operator, which smoothly depends on the control parameter k, in the case of general position – without degeneracy. Conditions are defined under which there emerges a group of symplectomorphisms generating in a linear system the first integral in the form of a nondegenerate quadratic form – Hamilton functions. The existence of a symplectic structure and a quadratic invariant in a dynamical system allows to reduce it on the basis of the variational principle to a divergent Hamiltonian form of the equations of motion with the linearization operator represented in a certain "canonical" form. The distinguished category of systems with a single invariant allows both to construct a Lie algebra of the corresponding Lie group and to move on to the study of stability and gyroscopic stabilization in a mechanical systemReferências
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