nLab
evolutionary derivative

Contents

Contents

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 VV depending on jets of a “field bundleEE (def. below) along the prolongation of an evolutionary vector field on EE. Equivalently this is a jet-dependent differential operator on the vertical tangent bundle of EE and as such is usefully related to the Euler-Lagrange derivative on EE (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

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

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

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

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

Γ J Σ (E)(E)Γ J Σ (E)(jb *E)=Hom Σ(J Σ (E),E)DiffOp(E,E) \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 EE' to the jet bundle J Σ (E)jbΣJ^\infty_\Sigma(E) \overset{jb}{\longrightarrow} \Sigma along jbjb.

Γ J Σ (E)(E)={ E fb J Σ (E) jb ΣA}. \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 EE to sections of EE'. )

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 EfbΣE \overset{fb}{\to} \Sigma a field bundle, write T ΣET_\Sigma E for its vertical tangent bundle and T Σ *ET_\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 Γ J Σ (E)(T ΣE)\Gamma_{J^\infty_\Sigma(E)}(T_\Sigma E) contains the space of evolutionary vector fields vv as those bundle morphism which respect not just the projection to Σ\Sigma but also its factorization through EE:

    ( T ΣE v tb Σ J Σ (E) jb ,0 E fb Σ)Γ J Σ (E)(T ΣE) \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)
  • Γ J Σ (E)(T Σ *E) Σ p+1(T *Σ)\Gamma_{J^\infty_\Sigma(E)}( T^\ast_\Sigma E) \otimes \wedge^{p+1}_\Sigma(T^\ast \Sigma) contains the space of source forms EE as those bundle morphisms which respect not just the projection to Σ\Sigma but also its factorization through EE:

    ( T Σ *E E ctb Σ J Σ (E) jb ,0 E fb Σ)Γ J Σ (E)(T Σ *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

Γ J Σ (E)(T ΣE)Γ J Σ (E)(T Σ *E) C (J Σ (E)) \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 ϕ a,ω aδϕ a)v aω a (v^a \partial_{\phi^a} \,,\, \omega_a \delta \phi^a) \mapsto v^a \omega_a

for v a,ω aC (J Σ (E))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

EfbΣ E \overset{fb}{\to} \Sigma

be a fiber bundle regarded as a field bundle and let

VvbΣ V \overset{vb}{\to} \Sigma

be a vector bundle. Then for

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

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

Γ J Σ (E)(T ΣE) DP Γ J Σ (E)(V) v v^(P) \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 vv to the derivative of PP along the prolongation tangent vector field v^\hat v of vv.

In the case that EE and VV are trivial vector bundles over Minkowski spacetime with coordinates ((x μ),(ϕ a))((x^\mu), (\phi^a)) and ((x μ),(ρ b))((x^\mu), (\rho^b)), respectively, then this is given by

((DP)(v)) b=(v aP bϕ a+dv adx μP bϕ ,μ a+d 2v adx μdx νP bϕ ,μν a+) ((\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 DP\mathrm{D}P may equivalently be regarded as a J Σ (E)J^\infty_\Sigma(E)-dependent differential operator from the vertical tangent bundle T ΣET_\Sigma E to VV, namely a morphism of the form

D P:J Σ (E)× ΣJ Σ T ΣEV \mathrm{D}_P \;\colon\; J^\infty_\Sigma(E) \times_\Sigma J^\infty_\Sigma T_\Sigma E \longrightarrow V

in that

(1)D P(,v)=DP(v)=v^(P). \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 Σ\Sigma, let L=LdvolΩ Σ p,0(E)\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:

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

Then the formally adjoint differential operator

(D L) *:J Σ (E)× Σ(Σ×) *T Σ *E (\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 Σ (E)J^\infty_\Sigma(E)-dependent differential operator D P\mathrm{D}_P from T ΣT_\Sigma to VV and applied to the constant section

1Γ Σ(Σ× *) 1 \in \Gamma_\Sigma(\Sigma \times \mathbb{R}^\ast)

is the Euler-Lagrange derivative

δ ELL=(D L) *(1)Γ J Σ (E)(T Σ *)Ω Σ p+1,1(E) source \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 VvbΣV \overset{vb}{\to} \Sigma be a vector bundle and write V *ΣV^\ast \overset{}{\to} \Sigma for its dual vector bundle.

For field-dependent sections (def. )

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

and

β *Γ J Σ (E)(V *) \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:

δ EL(αβ *)=(D α) *(β *)+(D β *) *(α) \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

δ EL(αβ *)δϕ a =(αβ *)ϕ addx μ((αβ *)ϕ ,μ a)+ddx μdx ν((αβ *)ϕ ,μν a) =+αϕ aβ *ddx μ(αϕ ,μ aβ *)+ddx μdx ν(αϕ ,μν aβ *) =+β *ϕ aαddx μ(β *ϕ ,μ aα)+ddx μdx ν(β *ϕ ,μν aα) =(D α) *(β *)+(D β *) *(α) \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,L)(E,\mathbf{L}) be a Lagrangian field theory over Minkowski spacetime and regard the Euler-Lagrange derivative

δ ELL=δ ELLdvol Σ \delta_{EL}\mathbf{L} \;=\; \delta_{EL}L \wedge dvol_\Sigma

as a field-dependent section of the vertical cotangent bundle

δ ELLΓ J Σ (E)(T Σ *E) \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 δ ELLD_{\delta_{EL}L} (def. ) is formally self-adjoint:

(D δ ELL) *=D δ ELL (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,L)(E,\mathbf{L}) be a Lagrangian field theory over Minkowski spacetime and regard the Euler-Lagrange derivative

δ ELL=δ ELLdvol Σ \delta_{EL}\mathbf{L} \;=\; \delta_{EL}L \wedge dvol_\Sigma

as a field-dependent section of the vertical cotangent bundle

δ ELLΓ J Σ (E)(T Σ *E) \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 δ ELLD_{\delta_{EL}L} (def. ) is formally self-adjoint:

(D δ ELL) *=D δ ELL (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

δ ELLδϕ aδϕ advol Σ=δLdvol Σ+d(...). \frac{\delta_{EL} L }{\delta \phi^a} \delta \phi^a \, \wedge dvol_\Sigma \;=\; \delta L \,\wedge dvol_\Sigma \;+\; d(...) \,.

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

(δδ ELLδϕ a)δϕ advol Σ=δδL=0dvol Σ+d(...). \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,wv,w any two evolutionary vector fields the contraction of their prolongations v^\hat v and w^\hat w into the differential 2-form on the left is

(δδ ELLδϕ aδϕ a)(v,w)=w a(D δ EL) a(v)v b(D δ EL) b(w), \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

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

This is the defining condition for D δ EL\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)

Last revised on December 5, 2017 at 18:07:40. See the history of this page for a list of all contributions to it.