NoetherGeneralized MomentumRotated CoordinatesRotational InvarianceTranslated CoordinatesTranslational InvarianceConservationGeneralized TransformationsConservation in GeneralNoether’s TheoremNoetherNoetherGeneralized MomentumGeneralized MomentumVariables Variables qq, , q’q’ are not are not functionally independent.functionally independent.The Lagrangian provides The Lagrangian provides canonically conjugate canonically conjugate variable.variable.•generalized momentumgeneralized momentum•need not be a momentumneed not be a momentumIgnorable coordinates imply Ignorable coordinates imply a conserved quantity.a conserved quantity.dtdqqjjjjjjqLqqp),(apj0jqLifthen0jjqLdtdqLsinceRotated CoordinatesRotated CoordinatesFor a central force the kinetic For a central force the kinetic energy depends on the energy depends on the magnitude of the velocity.magnitude of the velocity.•Independent of coordinate Independent of coordinate rotationrotation•Find ignorable coordinatesFind ignorable coordinatesLook at the Lagrangian for Look at the Lagrangian for an infinitessimal rotation.an infinitessimal rotation.•Pick the Pick the zz-axis for rotation-axis for rotation)(22221zyxvvvT yxyxx sincosxyxyy sincosy(x, y)=(x’,y’)xx’y’Rotational InvarianceRotational InvarianceRotate the Lagrangian, and expandRotate the Lagrangian, and expandMake a Taylor’s series expansionMake a Taylor’s series expansionThe Lagrangian must be invariant, so The Lagrangian must be invariant, so LL = = LL’.’.With the Euler equation this simplifies.With the Euler equation this simplifies.),,,,,,( tzxyyxzxyyxLLzxyJypxp )(2OxLyyLxxLyyLxLL  xyypxpdtdxLyyLxxLdtdyyLdtdx 0is constantTranslated CoordinatesTranslated CoordinatesKinetic energy is unchanged Kinetic energy is unchanged by a coordinate translation.by a coordinate translation.Look at the Lagrangian for Look at the Lagrangian for an infinitessimal translation.an infinitessimal translation.•Shift amount Shift amount x, x, yy•Test in 2 dimensionsTest in 2 dimensionsxxxyyyy(x, y)=(x’,y’)xx’y’xxyy),,,,,,( tzyxzyyxxLLTranslational InvarianceTranslational InvarianceAs with rotation, make a Taylor’s series As with rotation, make a Taylor’s series expansionexpansionAgain Again LL = = LL’, and each displacement acts ’, and each displacement acts separately.separately.•Euler equation is appliedEuler equation is appliedMomentum is conserved in each coordinate.Momentum is conserved in each coordinate.)()(22yOxOyLyxLxLLxLdtdxL0yLdtdyL0constantxpxLconstantypyLRotational invariance around any axis implies Rotational invariance around any axis implies constant angular momentum.constant angular momentum.Translational invariance implies constant linear Translational invariance implies constant linear momentum.momentum.These are symmetries of the transformation, and These are symmetries of the transformation, and there are corresponding constants of motion.there are corresponding constants of motion.•These are conservation lawsThese are conservation lawsConservationConservationGeneralized Generalized TransformationsTransformationsConsider a continuous Consider a continuous transformation.transformation.•Parameterized by Parameterized by ss•Solution to E-L equation Solution to E-L equation QQ((ss,,tt))Look at the Lagrangian for Look at the Lagrangian for as a function of the change.as a function of the change.Assume it is invariant under Assume it is invariant under the transformation.the transformation.ssQ)),,(),,(( ttsQtsQLL)(),0( tqtQ ),( tsQ0)),,(),,(( ttsQtsQLdsdConservation in GeneralConservation in GeneralThe invariant Lagrangian The invariant Lagrangian can be expanded.can be expanded.•Drop Drop tt for this example for this exampleApply the E-L equations.Apply the E-L equations.Since it is invariant it implies Since it is invariant it implies a constant.a constant.•Evaluate at Evaluate at ss = 0 = 0 •pp is a conserved quantity is a conserved quantitysQQLsQQLdsdLsQdtdQLsQQLdtddsdL0sQQLdtddsdLconstantsQpsQQLThe one variable argument can be extended for an The one variable argument can be extended for an arbitrary number of generalized variables.arbitrary number of generalized variables.Any differentiable symmetry of the action of a Any differentiable symmetry of the action of a physical system has a corresponding conservation physical system has a corresponding conservation law.law.Noether’s TheoremNoether’s