1. Conservation

2. Symmetry  A mathematical object that remains invariant under a transformation exhibits symmetry. Geometric objects Algebraic objects Functions AB DC DA CB n an integer

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

4. Lagrangian Invariance  A coordinate transformation changes the Lagrangian.  An invariant Lagrangian exhibits symmetry. Infinitessimal coordinate transformationsInfinitessimal coordinate transformations Conserved quantities emergeConserved quantities emerge Energy conservation when time-independentEnergy 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  x,  y 2 dimensional case y (x, y) = (x’,y’) x x’ y’

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

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’) x x’ y’

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

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