Noether

Generalized Momentum  Variables q, q’ are not functionally independent.  The Lagrangian provides canonically conjugate variable. generalized momentumgeneralized momentum need not be a momentumneed not be a momentum  Ignorable coordinates imply a conserved quantity. if then since

Rotated Coordinates  For a central force the kinetic energy depends on the magnitude of the velocity. Independent of coordinate rotationIndependent of coordinate rotation Find ignorable coordinatesFind ignorable coordinates  Look at the Lagrangian for an infinitessimal rotation. Pick the z-axis for rotationPick the z-axis for rotation y (x, y)=(x’,y’) x x’ y’

Rotational Invariance  Rotate the Lagrangian, and expand  Make a Taylor’s series expansion  The Lagrangian must be invariant, so L = L ’.  With the Euler equation this simplifies. is constant

Translated Coordinates  Kinetic energy is unchanged by a coordinate translation.  Look at the Lagrangian for an infinitessimal translation. Shift amount  x,  y Test in 2 dimensions y (x, y)=(x’,y’) x x’ y’

Translational Invariance  As with rotation, make a Taylor’s series expansion  Again L = L ’, and each displacement acts separately. Euler equation is appliedEuler equation is applied  Momentum is conserved in each coordinate.

 Rotational invariance around any axis implies constant angular momentum.  Translational invariance implies constant linear momentum.  These are symmetries of the transformation, and there are corresponding constants of motion. These are conservation lawsThese are conservation laws Conservation

Generalized Transformations  Consider a continuous transformation. Parameterized by s Solution to E-L equation Q(s,t)  Look at the Lagrangian for as a function of the change.  Assume it is invariant under the transformation.

Conservation in General  The invariant Lagrangian can be expanded. Drop t for this exampleDrop t for this example  Apply the E-L equations.  Since it is invariant it implies a constant. Evaluate at s = 0Evaluate at s = 0 p is a conserved quantityp is a conserved quantity

 The one variable argument can be extended for an arbitrary number of generalized variables.  Any differentiable symmetry of the action of a physical system has a corresponding conservation law. Noether’s Theorem next