Generalization of approximate partial Noether approach in phase space. (English) Zbl 1373.70016
Summary: The approximate partial Noether theorem proposed earlier for the ordinary differential equations (ODEs) [I. Naeem and F. M. Mahomed, Nonlinear Dyn. 57, No. 1–2, 303–311 (2009; Zbl 1176.70031)] is generalized in phase space for the perturbed Hamiltonian-type systems. The notion of approximate partial Hamiltonian is developed. An approximate partial Hamiltonian gives rise to an approximate Hamiltonian-type perturbed dynamical system of first-order ODEs. An approximate Legendre-type transformation connects the approximate partial Lagrangian and what we term as approximate partial Hamiltonian. The formulas for approximate partial Hamiltonian operators determining equations and first integrals are provided explicitly. We have explained our approach with the help of simple illustrative example. Then, it is applied to establish the approximate first integrals, reductions and exact solutions of two perturbed cubically coupled Duffing-Van der Pol oscillators. Both resonant and nonresonant cases are considered.
MSC:
| 70H33 | Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics |
| 70H03 | Lagrange’s equations |
| 70K60 | General perturbation schemes for nonlinear problems in mechanics |
| 34D10 | Perturbations of ordinary differential equations |
| 34C14 | Symmetries, invariants of ordinary differential equations |
Keywords:
approximate partial Lagrangian; approximate partial Hamiltonian; approximate first integrals; approximate partial Noether theorem; perturbed cubically coupled Duffing-van der Pol oscillatorsCitations:
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