Stanford University Department of Aeronautics and Astronautics Introduction to Symmetry Analysis Brian Cantwell Department of Aeronautics and Astronautics Stanford University Chapter 3 - Systems of ODEs and First-Order PDEs - State-Space Analysis

Stanford University Department of Aeronautics and Astronautics 3.1 Autonomous Systems of ODEs in the Plane Consider the system of coupled first-order ODEs Integrate the coupled nonlinear right hand sides (3.1) (3.2)

Stanford University Department of Aeronautics and Astronautics The result is two parametric functions for x and y (3.3) Initial conditions (3.4) In general s 0 can be arranged to be 0.

Stanford University Department of Aeronautics and Astronautics 3.2 Characteristics Eliminate s between the functions F and g to produce the family of curves The value of a characteristic is determined by the initial values of x and y. (3.5) (3.6)

Stanford University Department of Aeronautics and Astronautics Form the total differential Replace the differentials dx and dy The characteristics satisfy the first order PDE (3.7) (3.8) (3.9)

Stanford University Department of Aeronautics and Astronautics 3.3 First Order ODEs The family is also the set of solution curves of the ODE Equation (3.10) can be written in the form (3.10) (3.11)

Stanford University Department of Aeronautics and Astronautics We can rearrange (3.11) to read (3.12) There is a temptation to regard (3.12) as a perfect differential but this is generally incorrect since the integrability condition is usually not satisfied. That is (3.13) (3.14)

Stanford University Department of Aeronautics and Astronautics The vector field defined by the slopes and that defined by are identical up to a scalar multiplying factor in the magnitude of the displacement vector along the characteristics This implies that the partial derivatives and the vector components must have a common multiplying factor (3.23) (3.24) (3.25) (3.26)

Stanford University Department of Aeronautics and Astronautics Pfaff’s theorem

Stanford University Department of Aeronautics and Astronautics (3.30) (3.31) or (3.32) (3.33) Let

Stanford University Department of Aeronautics and Astronautics The family of solution characteristics is with total differential and corresponding system of characteristics To solve the first equality we have to find a particular solution of (3.34) (3.35) (3.36) The solution of (3.33) is determined by solving the characteristic equations (3.37)

Stanford University Department of Aeronautics and Astronautics

3.10 Exercises

Stanford University Department of Aeronautics and Astronautics