Physics+ Lagrangian, Noether and Momentum: UNIZOR.COM - Physics+ 4 All - Lagrangian

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Noether's Theorem
Symmetry and Momentum Conservation

Let's introduce a concept of momentum in generalized coordinates.

We are familiar with a vector of momentum in Euclidean three-dimensional space with Cartesian coordinates (x,y,z) for a point-mass m, as a vector with three components
px(t) = m·x'(t)
py(t) = m·y'(t)
pz(t) = m·z'(t)

Another approach, that uses the kinetic energy of this object
T=½m·(x'²+y'²+z'²)
would be to define
px = ∂T/∂x'
py = ∂T/∂y'
pz = ∂T/∂z'

Both definitions are equivalent, but the latter leads us to the third definition using the Lagrangian
L=T−U
instead of just kinetic energy T, because potential energy U does not depend on velocity:
px = ∂L/∂x'
py = ∂L/∂y'
pz = ∂L/∂z'

Since the Lagrangian of a mechanical system is usable in both Cartesian and non-Cartesian (generalized) coordinates, we can define a generalized momentum
(p1,...,pn)
as a set of partial derivatives of the Lagrangian L by corresponding component of generalized velocity (∂L(...)/q1',...,∂L(...)/qn').

The time-dependent function
pk(t) = ∂L(...)/∂qk'
is called the kth component of the generalized momentum.

Consider a mechanical system with n degrees of freedom and its trajectory in generalized coordinates
q(t) = (q1(t),...,qn(t)).

Theorem
If the Lagrangian of this system
L(q1,...,qn,q1',...,qn',t)
is invariant under translation of qk by infinitesimal value ε
qk → qk + ε
then the kth coordinate of the generalized momentum
pk = ∂L/∂qk'
is conserved.