Symmetry and Conservation Laws in Mathematical Objects Transformation

1. Conservation

2. A mathematical object that remains invariant under a transformation exhibits symmetry. Geometric objects Algebraic objects Functions A D B A D C C B Symmetry n an integer

3. The generalized momentum derives from the Lagrangian. Independent variable Conjugate momentum If the coordinate is ignorable the conjugate momentum is conserved. Momentum Conservation since if then

4. Lagrangian Invariance • A coordinate transformation changes the Lagrangian. • An invariant Lagrangian exhibits symmetry. • Infinitessimal coordinate transformations • Conserved quantities emerge • Energy conservation when time-independent

5. Translated Coordinates • Kinetic energy is unchanged by a coordinate translation. • Motion independent of coordinate choice • Look at the Lagrangian for an infinitessimal translation. • Shift amount dx, dy • 2 dimensional case y (x, y) = (x’,y’) y’ x x’

6. Expand the transformed Lagrangian. Assume invariance L = L’ Apply EL equation to each coordinate. Coordinates are independent Vanishing infinitessmals Linear momentum is conserved. Translational Invariance

7. Rotated Coordinates • Central forces have rotational symmetry. • Potential independent of coordinate rotation. • Kinetic energy also independent - magnitude of the velocity • Look at the Lagrangian for an infinitessimal rotation. • Pick the z-axis for rotation y (x, y) = (x’,y’) y’ x’ x

8. Make a Taylor’s series expansion. Invariant Lagrangian L = L’ The infinitessimal vanishes. EL equation substitution The angular momentum is conserved. Rotational Invariance

9. Noether’s Theorem • Each symmetry of a physical system has a corresponding conservation law. • Generalizes to any number of variables • Lagrangian invariance leads to conservation next