nLab evolutionary derivative

Idea

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).

Definition

In the following fiber bundles are considered in differential geometry and in particular vector bundle means smooth vector bundle.

Definition

(field-dependent sections)

For

$$E \overset{fb}{\longrightarrow} \Sigma$$

a fiber bundle, regarded as a field bundle, and for

$$E' \overset{fb'}{\longrightarrow} \Sigma$$

any other fiber bundle over the same base space (spacetime), we write

$$\Gamma_{J^\infty_\Sigma(E)}(E') \;\coloneqq\; \Gamma_{J^\infty_\Sigma(E)}( jb^\ast E' ) \;=\; Hom_\Sigma(J^\infty_\Sigma(E), E') \;\simeq\; DiffOp(E,E')$$

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 $jb$.

$$\Gamma_{J^\infty_\Sigma(E)}(E') \;=\; \left\{ \array{ && E' \\ & {}^{\mathllap{}}\nearrow & \downarrow \mathrlap{fb'} \\ J^\infty_\Sigma(E) &\underset{jb}{\longrightarrow}& \Sigma } \phantom{A}\,\, \right\} \,.$$

(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.

Example

(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 $E$:

  $$\left( \array{ && T_\Sigma E \\ & {}^{\mathllap{v}}\nearrow & \downarrow^{\mathrlap{tb_\Sigma}} \\ J^\infty_\Sigma(E) &\underset{jb_{\infty,0}}{\longrightarrow}& E & \overset{fb}{\longrightarrow}& \Sigma } \right) \;\in\; \Gamma_{J^\infty_\Sigma(E)}(T_\Sigma E)$$

• $\Gamma_{J^\infty_\Sigma(E)}( T^\ast_\Sigma E) \otimes \wedge^{p+1}_\Sigma(T^\ast \Sigma)$ contains the space of source forms $E$ as those bundle morphisms which respect not just the projection to $\Sigma$ but also its factorization through $E$:

  $$\left( \array{ && T^\ast_\Sigma E \\ & {}^{E}\nearrow & \downarrow^{\mathrlap{ctb_\Sigma}} \\ J^\infty_\Sigma(E) &\underset{jb_{\infty,0}}{\longrightarrow}& E & \overset{fb}{\longrightarrow}& \Sigma } \right) \;\in\; \Gamma_{J^\infty_\Sigma(E)}(T^\ast_\Sigma E)$$

This makes manifest the duality pairing between source forms and evolutionary vector fields

$$\array{ \Gamma_{J^\infty_\Sigma(E)}(T_\Sigma E) \otimes \Gamma_{J^\infty_\Sigma(E)}(T^\ast_\Sigma E) &\longrightarrow& C^\infty(J^\infty_\Sigma(E)) }$$

which in local coordinates is given by

$$(v^a \partial_{\phi^a} \,,\, \omega_a \delta \phi^a) \mapsto v^a \omega_a$$

for $v^a, \omega_a \in C^\infty(J^\infty_\Sigma(E))$ smooth functions on the jet bundle.

Definition

(evolutionary derivative of field-dependent section)

Let

$$E \overset{fb}{\to} \Sigma$$

be a fiber bundle regarded as a field bundle and let

$$V \overset{vb}{\to} \Sigma$$

be a vector bundle. Then for

$$P \in \Gamma_{J^\infty_\Sigma(E)}(V)$$

a field-dependent section of $E$ accoring to def. , its evolutionary derivative is the morphism

$$\array{ \Gamma_{J^\infty_\Sigma(E)}(T_\Sigma E) & \overset{ \mathrm{D}P }{\longrightarrow} & \Gamma_{J^\infty_\Sigma(E)}(V) \\ v &\mapsto& \hat v(P) }$$

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

$$((\mathrm{D}P)(v))^b \;=\; \left( v^a \frac{\partial P^b}{\partial \phi^a} + \frac{d v^a}{d x^\mu} \frac{\partial P^b}{\partial \phi^a_{,\mu}} + \frac{d^2 v^a}{d x^\mu d x^\nu} \frac{\partial P^b}{\partial \phi^a_{,\mu \nu}} + \cdots \right)$$

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

$$\mathrm{D}_P \;\colon\; J^\infty_\Sigma(E) \times_\Sigma J^\infty_\Sigma T_\Sigma E \longrightarrow V$$

in that

(1)$$\mathrm{D}_P(-,v) = \mathrm{D}P(v) = \hat v (P) \,.$$

(Olver 93, def. 5.24)

Examples

Example

(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:

$$L \;\in \; \Gamma_{J^\infty_\Sigma}(E)(\Sigma \times \mathbb{R}) \,,$$

Then the formally adjoint differential operator

$$(\mathrm{D}_L)^\ast \;\colon\; J^\infty_\Sigma(E)\times_\Sigma (\Sigma \times \mathbb{R})^\ast \longrightarrow T_\Sigma^\ast E$$

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

$$1 \in \Gamma_\Sigma(\Sigma \times \mathbb{R}^\ast)$$

is the Euler-Lagrange derivative

$$\delta_{EL}\mathbf{L} \;=\; \left(\mathrm{D}_{L}\right)^\ast(1) \;\in\; \Gamma_{J^\infty_\Sigma(E)}(T_\Sigma^\ast) \simeq \Omega^{p+1,1}_\Sigma(E)_{source}$$

via the identification from example .

(Olver 93, above (5.80))

Properties

Proposition

(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. )

$$\alpha \in \Gamma_{J^\infty_\Sigma(E)}(V)$$

and

$$\beta^\ast \in \Gamma_{J^\infty_\Sigma(E)}(V^\ast)$$

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:

$$\delta_{EL}( \alpha \cdot \beta^\ast ) \;=\; (\mathrm{D}_\alpha)^\ast(\beta^\ast) + (\mathrm{D}_{\beta^\ast})^\ast(\alpha)$$

(Olver 93 (5.80))

Proof

It is sufficient to check this in local coordinates. By the product law for differentiation we have

$$\begin{aligned} \frac{ \delta_{EL} \left(\alpha \cdot \beta^\ast \right) } { \delta \phi^a } & = \frac{\partial \left(\alpha \cdot \beta^\ast \right)}{\partial \phi^a} - \frac{d}{d x^\mu} \left( \frac{\partial \left( \alpha \cdot \beta^\ast \right)}{\partial \phi^a_{,\mu}} \right) + \frac{d}{d x^\mu d x^\nu} \left( \frac{\partial \left( \alpha \cdot \beta^\ast \right) }{\partial \phi^a_{,\mu \nu}} \right) - \cdots \\ & = \phantom{+} \frac{\partial \alpha }{\partial \phi^a} \cdot \beta^\ast - \frac{d}{d x^\mu} \left( \frac{\partial \alpha }{\partial \phi^a_{,\mu}} \cdot \beta^\ast \right) + \frac{d}{d x^\mu d x^\nu} \left( \frac{\partial \alpha }{\partial \phi^a_{,\mu \nu}} \cdot \beta^\ast \right) - \cdots \\ & \phantom{=} + \frac{\partial \beta^\ast }{\partial \phi^a} \cdot \alpha - \frac{d}{d x^\mu} \left( \frac{\partial \beta^\ast }{\partial \phi^a_{,\mu}} \cdot \alpha \right) + \frac{d}{d x^\mu d x^\nu} \left( \frac{\partial \beta^\ast }{\partial \phi^a_{,\mu \nu}} \cdot \alpha \right) - \cdots \\ & = (\mathrm{D}_\alpha)^\ast(\beta^\ast) + (\mathrm{D}_{\beta^\ast})^\ast(\alpha) \end{aligned}$$

Proposition

(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

$$\delta_{EL}\mathbf{L} \;=\; \delta_{EL}L \wedge dvol_\Sigma$$

as a field-dependent section of the vertical cotangent bundle

$$\delta_{EL}L \;\in\; \Gamma_{J^\infty_\Sigma(E)}(T^\ast_\Sigma E)$$

as in example . Then the corresponding evolutionary derivative field-dependent differential operator $D_{\delta_{EL}L}$ (def. ) is formally self-adjoint:

$$(D_{\delta_{EL}L})^\ast \;=\; D_{\delta_{EL}L}$$

(Olver 93, theorem 5.92)

Proposition

(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

$$\delta_{EL}\mathbf{L} \;=\; \delta_{EL}L \wedge dvol_\Sigma$$

as a field-dependent section of the vertical cotangent bundle

$$\delta_{EL}L \;\in\; \Gamma_{J^\infty_\Sigma(E)}(T^\ast_\Sigma E)$$

as in example . Then the corresponding evolutionary derivative field-dependent differential operator $D_{\delta_{EL}L}$ (def. ) is formally self-adjoint:

$$(D_{\delta_{EL}L})^\ast \;=\; D_{\delta_{EL}L}$$

(Olver 93, theorem 5.92) The following proof is due to Igor Khavkine.

Proof

By definition of the Euler-Lagrange form we have

$$\frac{\delta_{EL} L }{\delta \phi^a} \delta \phi^a \, \wedge dvol_\Sigma \;=\; \delta L \,\wedge dvol_\Sigma \;+\; d(...) \,.$$

Applying the variational derivative $\delta$ to both sides of this equation yields

$$\left(\delta \frac{\delta_{EL} L }{\delta \phi^a}\right) \wedge \delta \phi^a \, \wedge dvol_\Sigma \;=\; \underset{= 0}{\underbrace{\delta \delta L}} \wedge dvol_\Sigma \;+\; d(...) \,.$$

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

$$\left( \delta \frac{\delta_{EL} L }{\delta \phi^a} \wedge \delta \phi^a \right)(v,w) = w^a (\mathrm{D}_{\delta_{EL}})_a(v) - v^b(\mathrm{D}_{\delta_{EL}})_b(w) \,,$$

by inspection of the definition of the evolutionary derivative (def. ) and their contraction into the form on the right is

$$\iota_{\hat v} \iota_{\hat w} d(...) \;=\; d(...)$$

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

$$w^a (\mathrm{D}_{\delta_{EL}})_a(v) - v^b(\mathrm{D}_{\delta_{EL}})_b(w) \;=\; d(...) \,.$$

This is the defining condition for $\mathrm{D}_{\delta_{EL}}$ to be formally self-adjoint differential operator.

References

• 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)