Derivation of Noether Theorem
Background
The previous lectures of the Noether Theorem part of the course Physics+ 4 All have introduced the concepts of symmetry, parameterized group of continuous transformations and a concept of an extended configuration space that combines time and generalized coordinates into one set of coordinates.
We suggested that the symmetries relevant to the laws of motion are transformations of extended time-space coordinates that leave the action functional invariant.
This lecture is about mathematical derivation of certain conservation laws as logical consequences from the symmetries of transformations.
Summary of Assumptions
(1) Let us consider an extended configuration space of a mechanical system with coordinates {t,q}, where t is time and q is a set of generalized coordinates qi(t) (i∈[1,n]).
(2) This system is described by its Lagrangian L(t,q(t),q'(t)) where q'(t) is a set of generalized velocities {qi'(t)} (time derivatives of generalized coordinates).
(3) The trajectory of the movement of this system, a curve in the extended configuration space, is described by parameterized functions
t(x) and
q(x)={qi(x)}, i∈[1,n]
where x is an abstract parameter changing from real value x=a to x=b with {t(a),q(a)} being the start and {t(b),q(b)} being the finish point of the movement.
(4) Given a group of continuous transformations of the extended configuration space parameterized by ε
t⟶t(ε)=T(ε,t,q)
q⟶q(ε)=Q(ε,t,q)
with ε=0 causing a transformation to be the identity transformation, that is t(0)=t and q(0)=q.
We assume, the transformation functions T(ε,t,q) and Q(ε,t,q) are sufficiently differentiable.
(5) This transformation of points (t,q)⟶(t(ε),q(ε))
induces the transformation of every trajectory
{t(x),q(x)}⟶{t(ε)(x),q(ε)(x)}
where x∈[a,b].
We further assume that the transformed trajectory
{t(ε)(x),q(ε)(x)}
belongs to the same class of physically admissible trajectories of a mechanical system defined by its properties and the laws of physics.
(6) Let's assume that the action functional of the movement of this mechanical system
| Φ[t,q] = |
|
L(t,q,q')dt |
is invariant under this induced transformation of trajectories as parameter of transformation ε is infinitesimally changing from zero.
This assumption can be expressed as
(d/dε)Φ[t(ε),q(ε)]|ε=0 = 0
and we assume sufficient differentiability of the action functional by parameter ε.
Derivation of Noether Theorem
The problem with the above representation of the action functional is that not only an expression under an integral is transformed, but the limits of integration t(a) and t(b) change as well, which significantly complicates the analysis of the behavior of the action functional under
The road to simplification is the parameterized representation of a trajectory {t(x),q(x)}.
Using this, we can replace
(a) dt = (dt/dx)·dx
(b) q' = dq/dt = (dq/dx)/(dt/dx)
(c) integration by t on [t(a),t(b)] can be now replaced with an integration by x - the parameter changing on a fixed segment [a,b].
Let's rewrite the action functional as the integral by x using abbreviations
dt/dx=tx (so, dt=tx·dx) and
dq/dx={dqi/dx}={qix}=qx
for brevity
| Φ[t,q] = |
|
L(t,q,qx/tx)·tx·dx |
At this point we would like to bring some time-space uniformity.
Since both time t and generalized space coordinates q={qi} (i∈[1,n]) are all functions of one parameter x (x∈[a,b]) and all have equal standing as coordinates in an extended time-space configuration space, it makes sense to use a single letter
y={yi} (i∈[0,n]) with
y0=t and
yi=qi for all i∈[1,n]).
Also, we replace derivatives
dt/dx=tx with dy0/dx=y0x
and
dq/dx={dqi/dx}={qix}=qx
for i≠0 with
dy/dx={dyi/dx}={yix}=yx.
So, a set of all derivatives {tx,qix} can be written as
dy/dx={dyi/dx}={yix}=yx
where i∈[0,n].
Now we can simplify the formula for action functional by replacing the Lagrangian under the integration with a function that treats all time-space coordinates equally:
| Φ[y] = |
|
𝓛(y,yx)·dx |
where
y(x)={yi(x)} (i∈[0,n]) signifies a set of all time-space coordinates parameterized by x∈[a,b], that is a trajectory in extended configuration space, with y0(x)=t(x), and yi(x)=qi(x) for i≠0 and
yx(x)={yix(x)} (i∈[0,n]) signifies a set of all derivatives of time-space coordinates by parameter x with y0x(x)=tx(x), and yix(x)=qix(x) for i≠0
and a new function 𝓛() is defined for i∈[0,n] as
𝓛(y,yx) = 𝓛({yi},{yix}) =
= L(t,q,qx/tx)·tx =
= L(t,{qi},{qix/tx})·tx
This representation of the same action functional is simpler because a new function 𝓛() under the integral symmetrically depends on n+1 functions {yi(x)} (i∈[0,n]) that encompass t(x) and all {qi(x)} (i∈[1,n]) functions of parameter x and n+1 derivatives of these functions by x, and the parameter x is not a subject of
In addition, the limits of integration by x are from a to b which are constant and not affected by the
The latter form of function 𝓛() under an integral allows to express the assumption about the invariance of the action functional under
y⟶y(ε)
which means t⟶t(ε), q⟶q(ε)),
as
(d/dε)Φ[y(ε)]|ε=0 = 0
in a symmetrical way relative to all time-space coordinates in extended configuration space and use the known apparatus of Calculus to perform all the required operations.
Since our transformations are continuous, ε-transformations with infinitesimal ε are infinitesimal, that is the increments in coordinates
Δy(ε)=y(ε)−y
are infinitesimal as well.
At the same time, the
Expressions
ζ={ζi}={dyi(ε)/dε|ε=0}
are called generators of the transformation.
We will use them below.
Also,
d/dε[dy(ε)/dx]|ε=0 =
= d/dx[dy(ε)/dε]|ε=0 = dζ/dx
Let's apply our assumption about invariance of the action functional under
We'll abbreviate derivatives with subscriptors for brevity (like ζx for dζ/dx) and omit
0 = (d/dε)Φ[y(ε)] =
| = (d/dε) |
|
𝓛(y(ε),yx(ε))·dx = |
We can change the order of differentiation by ε and integration by x because we assumed that the function under the integral is sufficiently smooth, so the convergence theorem holds.
| = |
|
(d/dε) | 𝓛(y(ε),yx(ε))·dx = |
use the rules for differentiation of multi-variable functions, subscriptions to indicate the corresponding derivative, definition ζ=dy(ε)/dε|ε=0 and the rule for interchanging the differentiation by two different variables
dyx(ε)/dε = d/dε[dy(ε)/dx] =
= d/dx[dy(ε)/dε] = dζ/dx = ζx
| = |
|
(𝓛y·dy(ε)/dε+𝓛yx·dyx(ε)/dε)·dx |
| = |
|
(𝓛y·ζ+𝓛yx·ζx)·dx |
where group item 𝓛y·ζ represents
a sum Σi𝓛yi·ζi with i∈[0,n]
which in expanded form is
Σi∂𝓛/∂yi·[dyi(ε)/dε|ε=0]
and group item 𝓛yx·ζx represents
a sum Σi(𝓛yix·ζix) with i∈[0,n]
which expands analogously with a subscript x indicating a derivative by x
We have derived with a fundamental identity
|
[Σ(i𝓛yi·ζi)+Σi(𝓛yix·ζix)]·dx = 0 |
(d/dx)[𝓛yx·ζ] =
= (d𝓛yx/dx)·ζ + 𝓛yx·ζx
⇒
|
(d/dx)[𝓛yx·ζ]·dx = |
| = |
|
(d𝓛yx/dx)·ζ·dx + |
|
𝓛yx·ζx·dx |
⇒
[𝓛yx·ζ]|[a,b] =
| = |
|
(d𝓛yx/dx)·ζ·dx + |
|
𝓛yx·ζx·dx |
⇒
| [𝓛yx·ζ]|[a,b] − |
|
(d𝓛yx/dx)·ζ·dx = |
| = |
|
𝓛yx·ζx·dx |
The above expression for
|
𝓛yx·ζx·dx |
can be substituted into the fundamental identity presented above
|
(𝓛y·ζ+𝓛yx·ζx)·dx = 0 |
where group item
𝓛y·ζ represents Σi𝓛yi·ζi
getting
0 = [𝓛yx·ζ]|[a,b] +
| + |
|
[𝓛y − d/dx(𝓛yx)]·ζ·dx |
Since the ε-transformation leaves the action functional invariant, it preserves the set of stationary trajectories of the action mapping one stationary trajectory onto itself (shift along a trajectory) or to another one (jump to another trajectory).
Since all physical trajectories are stationary for the action functional and, therefore, are the solutions of the Euler–Lagrange equations, the transformed trajectory, being stationary as well, also satisfies the Euler–Lagrange equation:
𝓛y = ∂𝓛/∂y = d/dx(∂𝓛/∂yx) =
= d𝓛yx/dx
which nullifies an integral in the last identity.
Therefore,
[𝓛yx·ζ]|[a,b] = 0 or
𝓛yx(b)·ζ(b) − 𝓛yx(a)·ζ(a) = 0
where group parameter
𝓛yx·ζ represents Σi𝓛yix·ζi with the sum by i∈[0,n].
This means that the value of 𝓛yx·ζ is the same at x=a and x=b.
This holds for any subinterval [a,b] of any physically admissable trajectory.
Since the endpoints can be chosen arbitrarily along the trajectory, the above expression is constant along the trajectory and
d/dx[𝓛yx·ζ] = 0
Noether Theorem
Every continuous symmetry of the action functional (every ε-transformation of the extended configuration space that leaves the action functional invariant) is associated with a conserved quantity along physical trajectories.
In extended configuration space, the conserved quantity is
J = 𝓛yx·ζ
or, in expanded by coordinates format,
J = Σi𝓛yix·ζi with i∈[0,n]
Recall that
y={yi} is a set of time-space coordinates with i∈[0,n];
y0 is time t;
{yi} are a set of generalized coordinates {qi} with i∈[1,n];
ζ={ζi}={dyi(ε)/dε|ε=0} is a set of generators for each time-space coordinate;
𝓛(y,yx) = L(t,tx,q,qx/tx)·tx
where x in a subscript indicates a derivative of a corresponding function by parameter x:
tx=dt/dx and
{qix}={dqi/dx}
We have just proven that
J is a constant of motion and, therefore,
dJ/dx = 0
