Abstract
In the present paper, two new generating sets, of homology invariant functions will be established. Moreover, by the aid of two independent homology invariant functions of each set we established the transformed first order Lane-Emden equation. The first equation for polytropic index n ≠–1, ±∞ depends on five free parameters, while the other equation is for, n ＝ ±∞ and depends on three free parameters.
Highlights
The reduction of the differential equations is probably the most challenging problem in dynamics and physics
Homology is a powerful tool used by mathematicians to study the properties of spaces and maps that are insensitive to small perturbations
Kaczynski et al [4] presented the conceptual background for computational homology and indicated how homology can be used to study nonlinear dynamics
Summary
The reduction of the differential equations is probably the most challenging problem in dynamics and physics. A general interpretation of reducibility includes various transformations and changes the original problem along mathematical lines and in a physical sense. Such transformations will be achieved using homology theorem. Kaczynski et al [4] presented the conceptual background for computational homology and indicated how homology can be used to study nonlinear dynamics. The important consequence of the use of homology theorem, is that, if we can find two independent homology invariant functions, say u and v, the Lane-Emden equation transformed to u and v variables is of order one. By the aid of two independent homology invariant functions of each set we established the transformed first order Lane-Emden equation. The first equation for polytropic index n 1, depends on five free parameters, while, the other equation is for, n and depends on three free parameters
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