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Throughout, let

  1. Σ\Sigma be a smooth manifold of dimension p+1p+1, thought of as spacetime;

  2. EfbΣE \overset{fb}{\longrightarrow} \Sigma a fiber bundle thought of as a field bundle

  3. LΩ Σ p+1,0(E)\mathbf{L} \in \Omega^{p+1,0}_\Sigma(E) a Lagrangian density.


(evolutionary vector fields)

Let EfbΣE \overset{fb}{\to} \Sigma be a field bundle. Then an evolutionary vector field vv on EE is “variational vertical vector field” on EE, hence a smooth bundle homomorphism out of the jet bundle (def. )

J Σ E v T ΣE jb ,0 E \array{ J^\infty_\Sigma E && \overset{v}{\longrightarrow} && T_\Sigma E \\ & {}_{\mathllap{jb_{\infty,0}}}\searrow && \swarrow_{\mathllap{}} \\ && E }

to the vertical tangent bundle T ΣEΣT_\Sigma E \overset{}{\to} \Sigma of EfbΣE \overset{fb}{\to} \Sigma.

In the special case that the field bundle is a trivial vector bundle over Minkowski spacetime as in example , this means that an evolutionary vector field is a tangent vector field (example ) on J Σ (E)J^\infty_\Sigma(E) of the special form

v =v a ϕ a =v a((x μ),(ϕ a),(ϕ ,μ a),) ϕ a, \begin{aligned} v & = v^a \partial_{\phi^a} \\ & = v^a\left( (x^\mu), (\phi^a), (\phi^a_{,\mu}), \cdots \right) \partial_{\phi^a} \end{aligned} \,,

where the coefficients v aC (J Σ (E))v^a \in C^\infty(J^\infty_\Sigma(E)) are general smooth functions on the jet bundle (while the cmponents are tangent vectors along the field coordinates (ϕ a)(\phi^a), but not along the spacetime coordinates (x μ)(x^\mu) and not along the jet coordinates ϕ ,μ 1μ k a\phi^a_{,\mu_1 \cdots \mu_k}).

We write

Γ E ev(T ΣE)Ω Σ 0,0(E)Mod \Gamma_E^{ev}\left( T_\Sigma E \right) \;\in\; \Omega^{0,0}_\Sigma(E) Mod

for the space of evolutionary vector fields, regarded as a module over the \mathbb{R}-algebra

Ω Σ 0,0(E)=C (J Σ (E)) \Omega^{0,0}_\Sigma(E) \;=\; C^\infty\left( J^\infty_\Sigma(E) \right)

of smooth functions on the jet bundle.

An evolutionary vector field (def. ) describes an infinitesimal change of field values depending on, possibly, the point in spacetime and the values of the field and all its derivatives (locally to finite order, by prop. ).

This induces a corresponding infinitesimal change of the derivatives of the fields, called the prolongation of the evolutionary vector field:


(prolongation of evolutionary vector field)

Let EfbΣE \overset{fb}{\to} \Sigma be a fiber bundle.

Given an evolutionary vector field vv on EE (def. ) there is a unique tangent vector field v^\hat v on the jet bundle J Σ (E)J^\infty_\Sigma(E) such that

  1. v^\hat v agrees on field coordinates (as opposed to jet coordinates) with vv:

    (jb ,0) *(v^)=v, (jb_{\infty,0})_\ast(\hat v) = v \,,

    which means in the special case that EfbΣE \overset{fb}{\to} \Sigma is a trivial vector bundle over Minkowski spacetime that v^\hat v is of the form

    (1)v^=v a ϕ a=v+v^ μ a ϕ ,μ a+v^ μ 1μ 2 a ϕ ,μ 1μ 2 a+ \hat v \;=\; \underset{ = v }{ \underbrace{ v^a \partial_{\phi^a} }} \,+\, \hat v^a_{\mu} \partial_{\phi^a_{,\mu}} + \hat v^a_{\mu_1 \mu_2} \partial_{\phi^a_{,\mu_1 \mu_2}} + \cdots
  2. contraction with v^\hat v (def. ) anti-commutes with the total spacetime derivative (def. ):

    (2)ι v^d+dι v^=0. \iota_{\hat v} \circ d + d \circ \iota_{\hat v} = 0 \,.

In particular Cartan's homotopy formula (prop. ) for the Lie derivative v^\mathcal{L}_{\hat v} holds with respect to the variational derivative δ\delta:

(3) v^=δι v^+ι v^δ \mathcal{L}_{\hat v} = \delta \circ \iota_{\hat v} + \iota_{\hat v} \circ \delta

Explicitly, in the special case that the field bundle is a trivial vector bundle over Minkowski spacetime (example ) v^\hat v is given by

(4)v^=n=0d nv adx μ 1dx μ n ϕ μ 1μ n a. \hat v = \underoverset{n = 0}{\infty}{\sum} \frac{d^n v^a}{ d x^{\mu_1} \cdots d x^{\mu_n} } \partial_{\phi^a_{\mu_1 \cdots \mu_n}} \,.

It is sufficient to prove the coordinate version of the statement. We prove this by induction over the maximal jet order kk. Notice that the coefficient of ϕ μ 1μ k a\partial_{\phi^a_{\mu_1 \cdots \mu_k}} in v^\hat v is given by the contraction ι v^δϕ μ 1μ k a\iota_{\hat v} \delta \phi^a_{\mu_1 \cdots \mu_k} (def. ).

Similarly (at “k=1k = -1”) the component of μ 1\partial_{\mu_1} is given by ι v^dx μ\iota_{\hat v} d x^{\mu}. But by the second condition above this vanishes:

ι v^dx μ =dι v^x μ =0 \begin{aligned} \iota_{\hat v} d x^\mu & = d \iota_{\hat v} x^\mu \\ & = 0 \end{aligned}

Moreover, the coefficient of ϕ a\partial_{\phi^a} in v^\hat v is fixed by the first condition above to be

ι v^δϕ a=v a. \iota_{\hat v} \delta \phi^a = v^a \,.

This shows the statement for k=0k = 0. Now assume that the statement is true up to some kk \in \mathbb{N}. Observe that the coefficients of all ϕ μ 1μ k+1 a\partial_{\phi^a_{\mu_1 \cdots \mu_{k+1}}} are fixed by the contractions with δϕ μ 1μ kμ k+1 adx μ k+1\delta \phi^a_{\mu_1 \cdots \mu_{k} \mu_{k+1}} \wedge d x^{\mu_{k+1}}. For this we find again from the second condition and using δd+dδ=0\delta \circ d + d \circ \delta = 0 as well as the induction assumption that

ι v^δϕ μ 1μ k+1 adx μ k+1 =ι v^δdϕ μ 1μ k a =dι v^δϕ μ 1μ k a =dd kv adx μ 1dx μ k =d k+1v adx μ 1dx μ k+1dx μ k+1. \begin{aligned} \iota_{\hat v} \delta \phi^a_{\mu_1 \cdots \mu_{k+1}} \wedge d x^{\mu_{k+1}} & = \iota_{\hat v} \delta d \phi^a_{\mu_1 \cdots \mu_k} \\ & = d \iota_{\hat v} \delta \phi^a_{\mu_1 \cdots \mu_k} \\ & = d \frac{d^k v^a}{d x^{\mu_1} \cdots d x^{\mu_k}} \\ & = \frac{d^{k+1}v^a }{d x^{\mu_1} \cdots d x^{\mu_{k+1}}} d x^{\mu_{k+1}} \,. \end{aligned}

This shows that v^\hat v satisfying the two conditions given exists uniquely.

Finally formula (3) for the Lie derivative follows from the second of the two conditions with Cartan's homotopy formula v^=dι v^+ι v^d\mathcal{L}_{\hat v} = \mathbf{d} \circ \iota_{\hat v} + \iota_{\hat v} \circ \mathbf{d} (prop. ) together with d=δ+d\mathbf{d} = \delta + d.


(evolutionary vector fields form a Lie algebra)

Let EfbΣE \overset{fb}{\to} \Sigma be a fiber bundle. For any two evolutionary vector fields v 1v_1, v 2v_2 on EE (def. ) the Lie bracket of tangent vector fields of their prolongations v^ 1\hat v_1, v^ 2\hat v_2 (def. ) is itself the prolongation [v 1,v 2]^\widehat{[v_1, v_2]} of a unique evolutionary vector field [v 1,v 2][v_1,v_2].

This defines the structure of a Lie algebra on evolutionary vector fields.


It is clear that [v^ 1,v^ 2][\hat v_1, \hat v_2] is still vertical, therefore, by prop. , it is sufficient to show that contraction ι [v 1,v 2]\iota_{[v_1, v_2]} with this vector field (def. ) anti-commutes with the horizontal derivative dd, hence that [d,ι [v^ 1,v^ 2]]=0[d, \iota_{[\hat v_1, \hat v_2]}] = 0.

Now [d,ι [v^ 1,v^ 2]][d, \iota_{[\hat v_1, \hat v_2]}] is an operator that sends vertical 1-forms to horizontal 1-forms and vanishes on horizontal 1-forms. Therefore it is sufficient to see that this operator in fact also vanishes on all vertical 1-forms. But for this it is sufficient that it commutes with the vertical derivative. This we check by Cartan calculus, using [d,δ]=0[d,\delta] = 0 and [d,ι v^ i][d, \iota_{\hat v_i}], by assumption:

[δ,[d,ι [v^ 1,v^ 2]]] =[d,[δ,ι [v^ 1,v^ 2]]] =[d, [v^ 1,v^ 2]] =[d,[ v^ 1,ι v^ 2]] =[d,[[δ,ι v^ 1],ι v^ 2]] =0. \begin{aligned} {[ \delta, [ d,\iota_{[\hat v_1, \hat v_2]}] ]} & = - [d, [\delta, \iota_{[\hat v_1, \hat v_2]}]] \\ & = - [d, \mathcal{L}_{[\hat v_1, \hat v_2]}] \\ & = -[d, [\mathcal{L}_{\hat v_1}, \iota_{\hat v_2}] ] \\ & = - [d, [ [\delta, \iota_{\hat v_1}], \iota_{\hat v_2} ]] \\ & = 0 \,. \end{aligned}

Now given an evolutionary vector field, we want to consider the flow that it induces on the space of field histories:


(flow of field histories along evolutionary vector field)

Let EfbΣE \overset{fb}{\to} \Sigma be a field bundle and let vv be an evolutionary vector field (def. ) such that the ordinary flow of its prolongation v^\hat v (prop. )

exp(tv^):J Σ (E)J Σ (E) \exp(t \hat v) \;\colon\; J^\infty_\Sigma(E) \longrightarrow J^\infty_\Sigma(E)

exists on the jet bundle (e.g. if the order of derivatives of field coordinates that it depends on is bounded).

For Φ ():U 1Γ Σ(E)\Phi_{(-)} \colon U_1 \to \Gamma_\Sigma(E) a collection of field histories (hence a plot of the space of field histories) the flow of vv through Φ ()\Phi_{(-)} is the smooth function

U 1× 1exp(v)(Φ ())Γ Σ(E) U_1 \times \mathbb{R}^1 \overset{\exp(v)(\Phi_{(-)})}{\longrightarrow} \Gamma_\Sigma(E)

whose unique factorization exp(v)^(Φ ())\widehat{\exp(v)}(\Phi_{(-)}) through the space of jets of field histories (i.e. the image im(j Σ )im(j^\infty_\Sigma) of jet prolongation)

im(j Σ ) Γ Σ(J Σ (E)) exp(v)^(Φ ()) U 1× 1 exp(v)(Φ) Γ Σ(E) \array{ && im(j^\infty_\Sigma) &\hookrightarrow& \Gamma_\Sigma(J^\infty_\Sigma(E)) \\ & {}^{\mathllap{\widehat{\exp(v)}(\Phi_{(-)})}} \nearrow & \Big\downarrow^{\mathrlap{\simeq}} \\ U_1 \times \mathbb{R}^1 &\underset{ \exp(v)(\Phi) }{\longrightarrow}& \Gamma_{\Sigma}(E)_{} }

takes a plot t ():U 2 1t_{(-)} \;\colon\; U_2 \to \mathbb{R}^1 of the real line (regarded as a super smooth set), to the plot

(5)(exp(t()v^)j Σ (Φ ()):U 1×U 2Γ Σ(J Σ (E)) (\exp(t(-) \hat v) \circ j^\infty_\Sigma(\Phi_{(-)}) \;\colon\: U_1 \times U_2 \longrightarrow \Gamma_\Sigma\left( J^\infty_\Sigma(E) \right)

of the smooth space of sections of the jet bundle.

(That exp(t()v^)\exp(t(-) \hat v) indeed flows jet prolongations j Σ (Φ())j^\infty_\Sigma(\Phi(-)) again to jet prolongations is due to its defining relation to the evolutionary vector field vv from prop. .)


(infinitesimal symmetries of the Lagrangian and conserved currents)

Let (E,L)(E,\mathbf{L}) be a Lagrangian field theory (def. ).


  1. an infinitesimal symmetry of the Lagrangian is a variation vv (def. ) which arises as the prolongation v^\hat v (prop. ) of an evolutionary vector field vv (def. ) such that the Lie derivative v\mathcal{L}_v of the Lagrangian density along v^\hat v is a total spacetime derivative

    vL=dJ˜ v \mathcal{L}_v \mathbf{L} = d \tilde J_v
  2. an on-shell conserved current is a horizontal pp-form JΩ Σ p,0(E)J \in \Omega^{p,0}_\Sigma(E) whose total spacetime derivative vanishes on the prolonged shell (?)

    dJ| =0. d J\vert_{\mathcal{E}^\infty} = 0 \,.

(Noether's theorem I)

Let (E,L)(E,\mathbf{L}) be a Lagrangian field theory (def. ).

If vv is an infinitesimal symmetry of the Lagrangian (def. ) with vL=dJ˜ v\mathcal{L}_v \mathbf{L} = d \tilde J_v, then

J vJ˜ vι vΘ BFV J_v \coloneqq \tilde J_v - \iota_v \Theta_{BFV}

is an on-shell conserved current (def. ), for Θ BFV\Theta_{BFV} a presymplectic potential (?) from def. .


By Cartan's homotopy formula for the Lie derivative (prop. ) and the decomposition of the variational derivative δL\delta \mathbf{L} (?) and the fact that contraction ι v^\iota_{\hat v} with the prolongtion of an evolutionary vector field vanishes on horizontal differential forms (1) and anti-commutes with the horizontal differential (2), by def. , we may re-express the defining equation for the symmetry as follows:

dJ˜ v = vL =ι vdL=δ ELLdΘ BFV+dι vL=0 =ι vδ ELL+dι vΘ BFV \begin{aligned} d \tilde J_v & = \mathcal{L}_v \mathbf{L} \\ & = \iota_v \underset{= \delta_{EL}\mathbf{L} - d \Theta_{BFV}}{\underbrace{\mathbf{d} \mathbf{L}}} + \mathbf{d} \underset{= 0}{\underbrace{\iota_v \mathbf{L}}} \\ & = \iota_v \delta_{EL} \mathbf{L} + d \iota_v \Theta_{BFV} \end{aligned}

which is equivalent to

d(J˜ vι vΘ BFV=J v)=ι vδ ELL d(\underset{= J_v}{\underbrace{\tilde J_v - \iota_v \Theta_{BFV}}}) = \iota_v \delta_{EL}\mathbf{L}

Since, by definition of the shell \mathcal{E}, the form δ ELLδv\frac{\delta_{EL} \mathbf{L}}{\delta v} vanishes on \mathcal{E} this yields the claim.


(flow along infinitesimal symmetry of the Lagrangian preserves on-shell space of field histories)

Let (E,L)(E,\mathbf{L}) be a Lagrangian field theory (def. ).

For vv an infinitesimal symmetry of the Lagrangian (def. ) the flow on the space of field histories (example ) that it induces by def. preserves the space of on-shell field histories (from prop. ):

Γ Σ(E) δ ELL=0 Γ Σ(E) exp(v^)| δ ELL=0 exp(v^) Γ Σ(E) δ ELL=0 Γ Σ(E) \array{ \Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0} &\hookrightarrow& \Gamma_\Sigma(E) \\ {\mathllap{\exp(\hat v)\vert_{\delta_{EL}\mathbf{L} = 0} }} \uparrow && \uparrow {\mathrlap{\exp(\hat v)}} \\ \Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0} &\hookrightarrow& \Gamma_\Sigma(E) }

By def. a field history ΦΓ Σ(E)\Phi \in \Gamma_\Sigma(E) is on-shell precisely if its jet prolongation j Σ (E)j^\infty_\Sigma(E) (def. ) factors through the shell J Σ (E)\mathcal{E} \hookrightarrow J^\infty_\Sigma(E) (?). Hence by def. the statement is equivalently that the ordinary flow (prop. ) of v^\hat v (def. ) on the jet bundle J Σ (E)J^\infty_\Sigma(E) preserves the shell. This in turn means that it preserves the vanishing locus of the Euler-Lagrange form δ ELL\delta_{EL} \mathbf{L}.

For this it is sufficient to show that the derivative of the components of the Euler-Lagrange form along v^\hat v vanish on the prolonged shell

v^(δ ELLδϕ a)=0AAon J Σ (E). \hat v \left( \frac{\delta_{EL} L}{\delta \phi^a} \right) \;=\; 0 \phantom{AA} on \mathcal{E}^\infty \hookrightarrow J^\infty_\Sigma(E) \,.

This is the statement of Olver 95, theorem 5.53.


  • Peter Olver, Applications of Lie groups to differential equations, Springer; Equivalence, invariants, and symmetry, Cambridge Univ. Press 1995.

Last revised on October 9, 2023 at 16:21:36. See the history of this page for a list of all contributions to it.