The inverse problem of a mixed Liénard-type nonlinear oscillator equation from symmetry perspective. (English) Zbl 1358.70022
Summary: In this paper, we discuss the inverse problem for a mixed Liénard-type nonlinear oscillator equation
\[
\ddot{x}+f(x)\dot{x}^2+g(x)\dot{x}+h(x)=0,
\]
where \(f(x)\), \(g(x)\) and \(h(x)\) are arbitrary functions of \(x\). Very recently, we have reported the Lie point symmetries of this equation. By exploiting the interconnection between Jacobi last multiplier, Lie point symmetries and Prelle-Singer procedure, we construct a time-independent integral for the case exhibiting maximal symmetry from which we identify the associated conservative nonstandard Lagrangian and Hamiltonian functions. The classical dynamics of the nonlinear oscillator is also discussed, and certain special properties including isochronous oscillations are brought out.
MSC:
| 70H03 | Lagrange’s equations |
| 70H05 | Hamilton’s equations |
| 34A55 | Inverse problems involving ordinary differential equations |
| 34C14 | Symmetries, invariants of ordinary differential equations |
| 37J15 | Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010) |
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