1. Symmetries and Conservation Laws Thank you, Emmy 1 Symmetries and Conservation Laws

2. An example of conservation Symmetries and Conservation Laws 2

3. What just happened? Symmetries and Conservation Laws 3

4. Somewhat more generally Symmetries and Conservation Laws 4

5. Emmy Noether (1882-1935) German mathematician and colleague of Hilbert Symmetries and Conservation Laws 5

6. Types of Coordinate Transformations Symmetries and Conservation Laws 6

7. Symmetry Symmetries and Conservation Laws 7

8. Translation Symmetry Symmetries and Conservation Laws 8

9. It All Depends Note that the presence (or absence) of a symmetry depends upon both 1.the form of the Lagrangian, and 2.the nature of the transformation translation symmetry does not alter velocities We refer to the Lagrangian as invariant under the transformation. Symmetries and Conservation Laws 9

10. More General Invariance Symmetries and Conservation Laws 10

11. Rotational Symmetry Symmetries and Conservation Laws 11

12. Variations on a Theme Symmetries and Conservation Laws 12

13. Shrinking the Change Symmetries and Conservation Laws 13

14. Continuous Transformations Symmetries and Conservation Laws 14

15. Continuous Transformations Symmetries and Conservation Laws 15

16. Example: Infinitesimal Rotation Symmetries and Conservation Laws 16

17. Infinitesimal Rotation (cont.) Symmetries and Conservation Laws 17

18. Infinitesimal Rotation (cont.) Symmetries and Conservation Laws 18

19. More General Symmetries Symmetries and Conservation Laws 19

20. Continuous Symmetries (II) An infinitesimal transformation for which the change in the Lagrangian, including terms of first order in infinitesimals, is zero. Symmetries and Conservation Laws 20

21. Consequences of Symmetry Symmetries and Conservation Laws 21

22. Consequences of Symmetry (cont.) Symmetries and Conservation Laws 22

23. Example: Translation Symmetries and Conservation Laws 23

24. Example: Rotation Symmetries and Conservation Laws 24