Linearly-independent Couplings

One thus has ρ linearly-independent couplings of the Lagrangian expressions with divergences; the linear independence follows from the fact that, from (9), it would follow that δ u = 0, ∆u = 0, ∆x = 0, so there would be a dependency between the infinitesimal transformations.

However, by assumption, such a thing is not fulfilled for any parameter values, since otherwise the Gρ that further arises from the infinitesimal transformations by integration would depend upon less than ρ essential parameters.

The further possibility that δ u = 0, Div(f ∆x) = 0 was, however, excluded.

These conclusions are also still true in the limiting case of infinitely many parameters.

Now, let G be an infinite, continuous group; δ u and its derivatives, and therefore also B, will be linear in the arbitrary functions p(x) and their derivatives 1).

By substituting the values of δ u , still independently of (12), let:

One may now, analogously to the formula for partial integration, replace the derivatives of p with p itself and divergences that are linear in p and its derivatives using the identity: φ(x, u, …)

One thus gets:

The fact that it is no restriction to assume that the p are free of the u, ∂u / ∂x, … shows the converse.Noether – Invariant variational problems

I now construct the n-fold integral of (15), taken over any domain, and choose the p(x) such that they vanish on the boundary of (B – Γ), along with all of the derivatives that appear.

Since the integral of a divergence reduces to a boundary integral, the integral of the left-hand side of (15) thus also vanishes for arbitrary p(x) that only vanish on the boundary, along with sufficiently many derivatives, and from this, it follows, by a well-known argument, that the integrands vanish for any p(x), so one has the ρ relations:

These are the desired dependencies between the Lagrangian expressions and their derivatives for the invariance of I under G∞ρ ; the linear independence is clear, as above, since the inverse leads back to (12), and since one can again go from the infinitesimal transformations back to the finites ones, as will be done more thoroughly in § 4. Thus, ρ arbitrary transformations already appear in the infinitesimal transformations for a G∞ρ .

From (15) and (16), it then follows that Div(B – Γ) = 0.

If one correspondingly assumes a “mixed group” of ∆x and ∆u that are linear in the ε and the p(x) then one sees, when one sets the p(x) equal to zero and then the ε, that the divergence relations (13) exist, as well as the dependencies (16).