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