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

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Presentation transcript:

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

An example of conservation Symmetries and Conservation Laws 2

What just happened? Symmetries and Conservation Laws 3

Somewhat more generally Symmetries and Conservation Laws 4

Emmy Noether ( ) German mathematician and colleague of Hilbert Symmetries and Conservation Laws 5

Types of Coordinate Transformations Symmetries and Conservation Laws 6

Symmetry Symmetries and Conservation Laws 7

Translation Symmetry Symmetries and Conservation Laws 8

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

More General Invariance Symmetries and Conservation Laws 10

Rotational Symmetry Symmetries and Conservation Laws 11

Variations on a Theme Symmetries and Conservation Laws 12

Shrinking the Change Symmetries and Conservation Laws 13

Continuous Transformations Symmetries and Conservation Laws 14

Continuous Transformations Symmetries and Conservation Laws 15

Example: Infinitesimal Rotation Symmetries and Conservation Laws 16

Infinitesimal Rotation (cont.) Symmetries and Conservation Laws 17

Infinitesimal Rotation (cont.) Symmetries and Conservation Laws 18

More General Symmetries Symmetries and Conservation Laws 19

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

Consequences of Symmetry Symmetries and Conservation Laws 21

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

Example: Translation Symmetries and Conservation Laws 23

Example: Rotation Symmetries and Conservation Laws 24