Noether

1
Noether
2
Generalized Momentum

• Variables q, q are not functionally independent.
• The Lagrangian provides canonically conjugate
  variable.
• generalized momentum
• need not be a momentum
• Ignorable coordinates imply a conserved quantity.

since
if
then
3
Rotated Coordinates

• For a central force the kinetic energy depends on
  the magnitude of the velocity.
• Independent of coordinate rotation
• Find ignorable coordinates
• Look at the Lagrangian for an infinitessimal
  rotation.
• Pick the z-axis for rotation

y
y
(x, y)(x,y)
x
x
4
Rotational Invariance

• Rotate the Lagrangian, and expand
• Make a Taylors series expansion
• The Lagrangian must be invariant, so L L.
• With the Euler equation this simplifies.

is constant
5
Translated Coordinates

• Kinetic energy is unchanged by a coordinate
  translation.
• Look at the Lagrangian for an infinitessimal
  translation.
• Shift amount dx, dy
• Test in 2 dimensions

y
(x, y)(x,y)
y
x
x
6
Translational Invariance

• As with rotation, make a Taylors series
  expansion
• Again L L, and each displacement acts
  separately.
• Euler equation is applied
• Momentum is conserved in each coordinate.

7
Conservation

• 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 laws

8
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.

9
Conservation in General

• The invariant Lagrangian can be expanded.
• Drop t for this example
• Apply the E-L equations.
• Since it is invariant it implies a constant.
• Evaluate at s 0
• p is a conserved quantity

10
Noethers Theorem

• 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.

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