Abstract:

There are a lot of nonlinear problems in nature and engineering technology, which need to be described by nonlinear differential equations. Conservation laws play an important role in solving, reducing and qualitative analysis of differential equations. Therefore, it is of great significance to study the approximate conservation laws of nonlinear dynamical equations. In this paper, we apply the Noether symmetry method to the study of approximate conservation laws of weakly nonlinear dynamical equations. Firstly, the weakly nonlinear dynamical equations are transformed into the Lagrange equations of general holonomic system. Under the Lagrangian framework, the definition of Noether quasi-symmetry and the generalized Noether identities are established, and the approximate Noether conservation laws are obtained. Secondly, the weakly nonlinear dynamical equations are transformed into the Hamilton equations of general holonomic system in phase space. Under the Hamiltonian framework, the definition of Noether quasi-symmetry and the generalized Noether identities are established, and the approximate Noether conservation laws are obtained. Thirdly, the weakly nonlinear dynamical equations are transformed into the generalized Birkhoff's equations. Under the Birkhoffian framework, the definition of Noether quasi-symmetry and the generalized Noether identities are established, and the approximate Noether conservation laws are obtained. Finally, taking the famous Van der Pol equation, the Duffing equation and the weakly nonlinear coupled oscillators as examples, the computation of Noether quasi-symmetries and approximate conservation laws for weakly nonlinear systems under three different frameworks is analyzed. The results show that the same weakly nonlinear dynamical equation can be reduced to different general holonomic systems or different generalized Birkhoff systems. The result under the Hamiltonian framework is a special case of the Birkhoffian framework, while the result under the Lagrangian framework is equivalent to that under the Hamiltonian framework. Using Noether symmetry method to find approximate conservation laws of weakly nonlinear dynamical equations is not only convenient and effective, but also has great flexibility.

Key words: weakly nonlinear dynamics, approximate Noether conservation laws, Noether quasi-symmetry, generalized Noether identity