Introduction to Symmetry Analysis

1. Introduction to Symmetry Analysis Chapter 8 - Ordinary Differential Equations Brian Cantwell Department of Aeronautics and Astronautics Stanford University

2. 8.1 Extension of Lie Groups in the Plane

3. The Extended Transformation is a Group Two transformations of the extended group Compose the two transformations

4. The last relation is rearranged to read Differentiating F and G gives Comparing the expressions in parentheses we have The composed transformation is in exactly the same form as the original transformation!

5. Finite transformation of the second derivative The twice extended finite transformation is

6. Finite transformation of higher derivatives

7. The p-th order extended group is

8. Infinitesimal transformation of the second derivative Recall the infinitesimal transformation of coordinates where Substitute Expand and retain only the lowest order terms

9. The once-extended infinitesimal transformation in the plane is where the infinitesimal function fully written out is

10. The twice-extended infinitesimal transformation in the plane is Expand and retain only the lowest order terms where

11. The infinitesimal function transforming third derivatives is

12. The infinitesimal transformation of higher order derivatives Expand and retain only the lowest order terms. The p times extended infinitesimal transformation is where

13. 8.2 Expansion of an ODE in a Lie Series - the Invariance Condition for ODEs

15. The characteristic equations associated with extended groups are

16. The general second-order ordinary differential equation is invariant under the twice-extended group if and only if

17. Consider the case of the simplest second-order ODE The invariance condition is Fully written out the invariance condition is For invariance this equation must be satisfied subject to the condition that y is a solution of

18. The determining equations of the group are These equations can be used to work out the unknown infinitesimals.

19. Assume that the infinitesimals can be written as a multivariate power series Insert these series into the determining equations

20. The coefficients must satisfy the following algebraic system Finally the infinitesimals are

21. The two parameter group of the Blasius equation The invariance condition

22. Written out the invariance condition is

23. Now gather coefficients of like products of derivatives of y

24. The function y[x] is a solution of the Blasius equation. This is a constraint on the invariance condition that can be used to eliminate the third derivative.

25. Further simplify and

26. Finally the determining equations are From which the two parameter group of the Blasius equation is determined to be