Noether Theorem and Momentum Conservation: UNIZOR.COM -> Physics+ 4 All -> Lagrangian - Noether p = m·v const

Notes to a video lecture on UNIZOR.COM

Noether's Theorem
Symmetry and Momentum Conservation

History

In 1918, German mathematician Emmy Noether published the proof of a very important theorem named after her.
Albert Einstein and other physicists considered her one of the most significant mathematicians of her time.
Noether's theorem has been called one of the most important mathematical theorems guiding the development of modern physics.
This lecture is about a particular case of Noether's theorem as it applies to classical mechanics using the Lagrangian approach.

Background

Recall these important items from the prior lectures of this course Physics+ 4 All on UNIZOR.COM.

1. Configuration space with generalized coordinates q(t)={q_i(t)}, where i∈[1,n] and n is the number of degrees of freedom.
Generalized velocities (time derivatives of generalized coordinates) q(t)={q_i'(t)}.

2. Lagrangian
L(q,q') = T(q,q') − U(q)
- the difference between the total kinetic T and total potential U energies of a system.
Optionally, the Lagrangian might be explicitly dependent on time, but in this and subsequent lectures we assume that energies and, therefore, the Lagrangian do not explicitly depend on time, but only on positions and velocities. This makes the Lagrangian depend on time implicitly through the motion q(t).
Systems with the Lagrangian explicitly depending on time (like in case of variable gravitational field, driven oscillations or time-dependent electromagnetic field) are not considered here.

3. Action functional
Φ[L(q(t),q'(t))_t∈[t1,t2]] =
= ∫_[t1,t2] L(q(t),q'(t))dt

4. Euler-Lagrange equations
d/dt ∂L/∂q_i' = ∂L/∂q_i (i∈[1,n])
These are differential equations with unknown position functions q(t).
Their solutions are the extremals of the action functional above and, at the same time, the only candidates for real physical trajectories of a mechanical system in its configuration space with generalized coordinates.

Symmetry and Momentum Conservation

Our purpose is to prove a particular case of Noether's theorem about the law of conservation of momentum.

More precisely, if the Lagrangian of a mechanical system is invariant under the translation of generalized coordinates, the generalized momentum of this system is invariant under this translation as well, which constitutes the Momentum Conservation law.

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
p_x(t) = m·x'(t)
p_y(t) = m·y'(t)
p_z(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
p_x = ∂T/∂x'
p_y = ∂T/∂y'
p_z = ∂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:
p_x = ∂L/∂x'
p_y = ∂L/∂y'
p_z = ∂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,...,p_n)
as a set of partial derivatives of the Lagrangian L by corresponding component of generalized velocity (∂L(...)/q₁',...,∂L(...)/q_n').

The time-dependent function
p_k(t) = ∂L(...)/∂q_k'
is called the k^th component of the generalized momentum.

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

Theorem
If the Lagrangian of this system
L(q₁,...,q_n,q₁',...,q_n',t)
is invariant under translation of q_k by infinitesimal value ε
q_k → q_k + ε
then the k^th coordinate of the generalized momentum
p_k = ∂L/∂q_k'
is conserved.

Proof
The invariance of our Lagrangian under translation of q_k means
L(...q_k...,q') = L(...q_k+ε...,q')
Let
Δ_kL=L(...q_k+ε...)−L(...q_k...)=0
Then the partial derivative of the Lagrangian by q_k is
∂L/∂q_k = lim_ε→0 Δ_kL/ε = 0
When a partial derivative of the Lagrangian by a coordinate is zero, this coordinate is called cyclic.
The corresponding Euler-Lagrange equation for this coordinate is
d/dt ∂L/∂q_k' = ∂L/∂q_k

The right side is zero, as we stated above, which results in
d/dt ∂L/∂q_k' = 0
from which, in turn, follows
∂L/∂q_k' is constant.

According to a definition of a generalized momentum, we have proved the conservation of the k^th component of a generalized momentum when the Lagrangian is invariant under the translation along the k^th generalized coordinate.

∂L/∂q_k=0 ⇒ ∂L/∂q_k'=p_k=const
translation symmetry ⇒
⇒ conserved momentum

End of Proof.

Note 1:
In a more general case, if the Lagrangian is invariant under the translation of several generalized coordinates, all the corresponding components of the generalized momentum are conserved.

Note 2:
A deeper interpretation of this result is that momentum is the generator of spatial translations: an infinitesimal shift of a coordinate is produced by the corresponding component of momentum.
Symmetry produces conservation laws, and the conserved quantities generate the corresponding symmetry transformations.

Example
A stone is thrown in the air horizontally along X-axis with speed v from the height H.
Assume, the origin of Cartesian coordinates is on a ground levelimmediately under the initial position of a stone with X and Y axes are on the ground level with Z axis going vertically upward.

The coordinates of the stone trajectory are
x(t) = v·t
y(t) = 0
z(t) = H−½g·t²

The velocity components of the stone trajectory are
x'(t) = v
y'(t) = 0
z'(t) = −g·t

Kinetic and potential energies of the stone are
T = ½m·(x'²+y'²+z'²)
U = m·z

Lagrangian and its partial derivatives by coordinates areis
L=T−U
∂L/∂x = 0
∂L/∂y = 0
∂L/∂z = m ≠ 0

According to the theorem above, the linear momentum components are
p_x=const=∂L/∂x'=m·x'=m·v
p_y=const=∂L/∂y'=m·y'=0
p_z≠const; ∂L/∂z'=m·z'=−g·t