The evolutionary derivative or “Fréchet derivative of a tuple of differential functions” (Olver 93, def. 5.24)) is the derivative of a section of some vector bundle $V$ depending on jets of a “field bundle” $E$ (def. below) along the prolongation of an evolutionary vector field on $E$. Equivalently this is a jet-dependent differential operator on the vertical tangent bundle of $E$ and as such is usefully related to the Euler-Lagrange derivative on $E$ (example and prop. below).
In the following fiber bundles are considered in differential geometry and in particular vector bundle means smooth vector bundle.
For
a fiber bundle, regarded as a field bundle, and for
any other fiber bundle over the same base space (spacetime), we write
for the space of sections of the pullback of bundles of $E'$ to the jet bundle $J^\infty_\Sigma(E) \overset{jb}{\longrightarrow} \Sigma$ along $jb$.
(Equivalently this is the space of differential operators from sections of $E$ to sections of $E'$. )
In (Olver 93, section 5.1, p. 288) the field dependent sections of def. , considered in local coordinates, are referred to as tuples of differential functions.
(source forms and evolutionary vector fields are field-dependent sections)
For $E \overset{fb}{\to} \Sigma$ a field bundle, write $T_\Sigma E$ for its vertical tangent bundle and $T_\Sigma^\ast E$ for its dual vector bundle, the vertical cotangent bundle.
Then the field-dependent sections of these bundles according to def. are identified as follows:
the space $\Gamma_{J^\infty_\Sigma(E)}(T_\Sigma E)$ contains the space of evolutionary vector fields $v$ as those bundle morphism which respect not just the projection to $\Sigma$ but also its factorization through $E$:
$\Gamma_{J^\infty_\Sigma(E)}( T^\ast_\Sigma E) \otimes \wedge^{p+1}_\Sigma(T^\ast \Sigma)$ contains the space of source forms $E$ as those bundle morphisms which respect not just the projection to $\Sigma$ but also its factorization through $E$:
This makes manifest the duality pairing between source forms and evolutionary vector fields
which in local coordinates is given by
for $v^a, \omega_a \in C^\infty(J^\infty_\Sigma(E))$ smooth functions on the jet bundle.
(evolutionary derivative of field-dependent section)
Let
be a fiber bundle regarded as a field bundle and let
be a vector bundle. Then for
a field-dependent section of $E$ accoring to def. , its evolutionary derivative is the morphism
which, under the identification of example , sense an evolutionary vector field $v$ to the derivative of $P$ along the prolongation tangent vector field $\hat v$ of $v$.
In the case that $E$ and $V$ are trivial vector bundles over Minkowski spacetime with coordinates $((x^\mu), (\phi^a))$ and $((x^\mu), (\rho^b))$, respectively, then this is given by
This makes manifest that $\mathrm{D}P$ may equivalently be regarded as a $J^\infty_\Sigma(E)$-dependent differential operator from the vertical tangent bundle $T_\Sigma E$ to $V$, namely a morphism of the form
in that
(evolutionary derivative of Lagrangian function)
Over a (pseudo-)Riemannian manifold $\Sigma$, let $\mathbf{L} = L dvol \in \Omega^{p,0}_\Sigma(E)$ be a Lagrangian density, with coefficient function regarded as a field-dependent section (def. ) of the trivial real line bundle:
Then the formally adjoint differential operator
of its evolutionary derivative, def. , regarded as a $J^\infty_\Sigma(E)$-dependent differential operator $\mathrm{D}_P$ from $T_\Sigma$ to $V$ and applied to the constant section
is the Euler-Lagrange derivative
(Euler-Lagrange derivative is derivation via evolutionary derivatives)
Let $V \overset{vb}{\to} \Sigma$ be a vector bundle and write $V^\ast \overset{}{\to} \Sigma$ for its dual vector bundle.
For field-dependent sections (def. )
and
we have that the Euler-Lagrange derivative of their canonical pairing to a smooth function on the jet bundle is the sum of the derivative of either one via the formally adjoint differential operator of the evolutionary derivative (def. ) of the other:
It is sufficient to check this in local coordinates. By the product law for differentiation we have
(evolutionary derivative of Euler-Lagrange forms is formally self-adjoint)
Let $(E,\mathbf{L})$ be a Lagrangian field theory over Minkowski spacetime and regard the Euler-Lagrange derivative
as a field-dependent section of the vertical cotangent bundle
as in example . Then the corresponding evolutionary derivative field-dependent differential operator $D_{\delta_{EL}L}$ (def. ) is formally self-adjoint:
(evolutionary derivative of Euler-Lagrange forms is formally self-adjoint)
Let $(E,\mathbf{L})$ be a Lagrangian field theory over Minkowski spacetime and regard the Euler-Lagrange derivative
as a field-dependent section of the vertical cotangent bundle
as in example . Then the corresponding evolutionary derivative field-dependent differential operator $D_{\delta_{EL}L}$ (def. ) is formally self-adjoint:
(Olver 93, theorem 5.92) The following proof is due to Igor Khavkine.
By definition of the Euler-Lagrange form we have
Applying the variational derivative $\delta$ to both sides of this equation yields
It follows that for $v,w$ any two evolutionary vector fields the contraction of their prolongations $\hat v$ and $\hat w$ into the differential 2-form on the left is
by inspection of the definition of the evolutionary derivative (def. ) and their contraction into the form on the right is
by the fact (prop. ) that contraction with prolongations of evolutionary vector fields coommutes with the total spacetime derivative.
Hence the last two equations combined give
This is the defining condition for $\mathrm{D}_{\delta_{EL}}$ to be formally self-adjoint differential operator.
Peter Olver, Applications of Lie groups to Differential equations, Graduate Texts in Mathematics, Springer 1993
Glenn Barnich, equation(3) of A note on gauge systems from the point of view of Lie algebroids, in P. Kielanowski, V. Buchstaber, A. Odzijewicz,
M.. Schlichenmaier, T Voronov, (eds.) XXIX Workshop on Geometric Methods in Physics, vol. 1307 of AIP Conference Proceedings, 1307, 7 (2010) (arXiv:1010.0899, doi:/10.1063/1.3527427)
Igor Khavkine, starting with p. 45 of Characteristics, Conal Geometry and Causality in Locally Covariant Field Theory (arXiv:1211.1914)
Last revised on December 5, 2017 at 23:07:40. See the history of this page for a list of all contributions to it.