The nonlinear dynamics based on the nonstandard Hamiltonians

The new action of a nonstandard \(S=\int _a^b {e^{[p\dot{q}-H(p,q,t)]}dt}\) with a nonstandard Hamiltonian is introduced. The nonstandard Hamiltonian equations are obtained by using the standard variational method. Some new dynamical properties that the nonlinear systems hold are obtained. It is demonstrated that several constrained Hamiltonian systems have been identified to possess some interesting properties, and some additional features are discussed in details.