Noether's Theorem and the Lie Symmetries of Goodwin-model

The dynamic behavior of a physical system can concisely be described by the least action principle. In the centrum of its mathematical presentations is a specific function of the coordinates and velocities, i.e., the Lagrangian. If the integral of a Lagrangian is stationary, then the system is moving along an extremal path through the phase space. All Lie symmetries of a Lagrangian correspond to a conserved quantity, and the conservation principle is explained by variation symmetry. Briefly, that is the meaning of Noether's theorem. After showing that R. H. Goodwin's cyclical growth model has a Lagrangian we introduce the generalized Noether-theorem and apply to Goodwin's 2D model in order to get its Hamiltonian. We prove that the cyclical motion in his model derives from its dynamic Lie symmetries. These cyclical trajectories are extremal trajectories in the phase space and along these trajectories the first integral of the model's Lagrangian is stationary which by the principle of least action also means that they satisfy the first-order necessary conditions. The optimality still needs satisfying the sufficient conditions. Since the Legendre's second order sufficient conditions are not applicable here we show satisfying the other sufficient condition, the local convex surface of Lagrangian with the minimum non-trivial fixed-point and contour lines of the extremal trajectories. Our conclusion is that all systems' solutions described by first order nonlinear ordinary differential equations system are optimal if they have a Lagrangian which satisfies the sufficient and necessary conditions.