SUMMARYThe notions of partial Lagrangians, partial Noether operators and partial Euler-Lagrange equations are used in the construction of first integrals for ordinary differential equations that need not be derivable from variational principles. We obtain a Noether-like theorem that provides the first integral by means of a formula which has the same structure as the Noether integral. However, the invariance condition for the determination of the partial Noether operators is different as we have a partial Lagrangian and as a result partial Euler-Lagrange equations. Applications given include those that admit a standard Lagrangian such as the harmonic oscillator, modified Emden and Ermakov-Pinney equations and systems of two second-order equations that do not have standard Lagrangians.

The non-standard Hamiltonian system, also referred to as a partial Hamiltonian system in the literature, of the form ${\dot q^i} = \frac{{\partial H}}{{\partial {p_i}}},{\text{ }}{\dot p^i} = - \frac{{\partial H}}{{\partial {q_i}}} + {\Gamma ^i}(t,{\text{ }}{q^i},{\text{ }}{p_i})$ appears widely in economics, physics, mechanics, and other fields. The non-standard (partial) Hamiltonian systems arise from physical Hamiltonian structures as well as from artificial Hamiltonian structures. We introduce the term ‘artificial Hamiltonian’ for the Hamiltonian of a model having no physical structure. We provide here explicitly the notion of an artificial Hamiltonian for dynamical systems of ordinary differential equations (ODEs). Also, we show that every system of second-order ODEs can be expressed as a non-standard (partial) Hamiltonian system of first-order ODEs by introducing an artificial Hamiltonian. This notion of an artificial Hamiltonian gives a new way to solve dynamical systems of first-order ODEs and systems of second-order ODEs that can be expressed as a non-standard (partial) Hamiltonian system by using the known techniques applicable to the non-standard Hamiltonian systems. We employ the proposed notion to solve dynamical systems of first-order ODEs arising in epidemics.

Using geometric methods for linearizing systems of second order cubically non-linear in the first derivatives ordinary differential equations, we extend to the third order by differentiating the second order equation. This yields criteria for conditional linearizability via point transformation with respect to a second order equation of classes of third order ordinary differential equations, which are distinct from the classes available in the literature. Some examples are given and discussed.

We study here the Lie symmetries, conservation laws, reductions, and new exact solutions of (2+1) dimensional Zakharov-Kuznetsov (ZK), Gardner Kadomtsev-Petviashvili (GKP), and Modified Kadomtsev-Petviashvili (MKP) equations. The multiplier approach yields three conservation laws for ZK equation. We find the Lie symmetries associated with the conserved vectors, and three different cases arise. The generalized double reduction theorem is then applied to reduce the third-order ZK equation to a second-order ordinary differential equation (ODE) and implicit solutions are established. We use the Sine-Cosine method for the reduced second-order ODE to obtain new explicit solutions of ZK equation. The Lie symmetries, conservation laws, reductions, and exact solutions via generalized double reduction theorem are computed for the GKP and MKP equations. Moreover, for the GKP equation, some new explicit solutions are constructed by applying the first integral method to the reduced equations.

Different approaches to construct first integrals for ordinary differential equations and systems of ordinary differential equations are studied here. These approaches can be grouped into three categories: direct methods, Lagrangian or partial Lagrangian formulations, and characteristic (multipliers) approaches. The direct method and symmetry conditions on the first integrals correspond to first category. The Lagrangian and partial Lagrangian include three approaches: Noether’s theorem, the partial Noether approach, and the Noether approach for the equation and its adjoint as a system. The characteristic method, the multiplier approaches, and the direct construction formula approach require the integrating factors or characteristics or multipliers. The Hamiltonian version of Noether’s theorem is presented to derive first integrals. We apply these different approaches to derive the first integrals of the harmonic oscillator equation. We also study first integrals for some physical models. The first integrals for nonlinear jerk equation and the free oscillations of a two-degree-of-freedom gyroscopic system with quadratic nonlinearities are derived. Moreover, solutions via first integrals are also constructed.