Variational calculus


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In physics the critical locus of the action functional, hence the solution space to the partial differential equations of motion is often called the shell, or the physical shell (the diffiety corresponding to the PDE of motion).

This terminology derives from the case of the free relativistic particle propagating in Minkowski spacetime, for which the space of solutions is the hyperbola of vectors whose Minkowski norm square is the mass of the particle. This hyperbola is naturally called the mass shell.

It is common to use the adjective “on-shell” for statements that apply after restriction to the shell, and “off-shell” for statements that apply generally.

For instance Noether's theorem gives for each symmetry of a local Lagrangian an on-shell conserved current, namely a differential form on spacetime, depending on the field configurations, which is closed (only, in general) if the given field configuration satisfies its equations of motion. Such a differential form which is closed even if the equations of motion do not hold would be called an off-shell conserved current.


In terms of variational calculus the field configurations are encoded by a field bundle EE over spacetime/worldvolume Σ\Sigma as being the sections of this bundle. The shell then is encoded by a sub-bundle

J Σ E Σ \array{ \mathcal{E} &&\hookrightarrow && J^\infty_\Sigma E \\ & \searrow && \swarrow \\ && \Sigma }

of the jet bundle J Σ EJ^\infty_\Sigma E of EE, namely the subbundle of those pointwise field configurations with those jets (derivatives) that do solve the equations of motion (see at diffiety).

This way a field configuration given by a section ϕ\phi of EE is a solution to the equations of motion precisely if its jet prolongation, being a section of J Σ EJ^\infty_\Sigma E comes from a section of \mathcal{E} under this inclusion.

If one thinks of all this not as happening in the category of bundles over Σ\Sigma but in the category PDE ΣPDE_\Sigma of partial differential equations over Σ\Sigma, then the above inclusion becomes simply ι:E\iota \colon \mathcal{E} \hookrightarrow E, where now EE is thought of as the trivial partial differential equation on its sections, the one for which each section is a solution. Then a field configuration is just a morphism ϕ:ΣE\phi : \Sigma \longrightarrow E in PDE ΣPDE_\Sigma, and this is a solution precisely if it factors through the inclusion of the shell:

ϕ sol ι Σ ϕ E \array{ && \mathcal{E} \\ & {}^\mathllap{\phi_{sol}}{\nearrow} & \downarrow^{\mathrlap{\iota}} \\ \Sigma &\underset{\phi}{\longrightarrow}& E }

Accordingly, a horizontal form JΩ H p(E)Ω p+1(J Σ E)J \in \Omega^{p}_H(E) \hookrightarrow \Omega^{p+1}(J^\infty_\Sigma E) is an on-shell conserved current if it becomes horizontally closed after restriction to the shell:

ι *d HJ=0. \iota^\ast d_H J = 0 \,.


For (E,L)(E,\mathbf{L}) a Lagrangian field theory, let J Σ (E)\mathcal{E}^\infty \hookrightarrow J^\infty_\Sigma(E) be the prolonged shell.

The following is intuitively obvious but not entirely trivial to prove:


(infinitesimal symmetry of the Lagrangian is also symmetry of the shell)

Let (E,L)(E,\mathbf{L}) be a Lagrangian field theory. Then if v^\hat v is the prolongation of an evolutionary vector field which is an infinitesimal symmetry of the Lagrangian in that the Lie derivative of L\mathbf{L} along v^\hat v is horizontally exact, then the flow of v^\hat v preserves the prolonged shell.

(Olver 95, theorem 5.53)


Last revised on December 11, 2017 at 06:35:02. See the history of this page for a list of all contributions to it.