3 - Flows in phase space

Summary

… the historical value of a science depends not upon the number of particular phenomena it can present but rather upon the power it has of coordinating diverse facts and subjecting them to one simple code.

E. L. Ince, in Ordinary Differential Equations

Solvable vs integrable

In this chapter we will consider n coupled and generally nonlinear differential equations written in the form dx_i/dt = V_i(x₁,…,x_n). Since Newton's formulation of mechanics via differential equations, the idea of what is meant by a solution has a short but very interesting history (see Ince's appendix (1956), and also Wintner (1941)). In the last century, the idea of solving a system of differential equations was generally the ‘reduction to quadratures’, meaning the solution of n differential equations by means of a complete set of n independent integrations (generally in the form of n – 1 conservation laws combined with a single final integration after n – 1 eliminating variables). Systems of differential equations that are discussed analytically in mechanics textbooks are almost exclusively restricted to those that can be solved by this method. Jacobi (German, 1804–1851)systematized the method, and it has become irreversibly mislabeled as ‘integrability’. Following Jacobi and also contemporary ideas of geometry, Lie (Norwegian, 1842–1899) studied first order systems of differential equations from the standpoint of invariance and showed that there is a universal geometric interpretation of all solutions that fall into Jacobi's ‘integrable’ category.