Euler-Lagrange Forms and Boundary Decomposition

The first variation of a local action has two geometrically different parts. One multiplies arbitrary interior variations and therefore gives the Euler–Lagrange equations; the other is an exact spacetime form and records what reaches the boundary. Keeping both parts is essential: the same bulk equation can belong to inequivalent variational problems, while the boundary term later becomes the seed of the covariant presymplectic structure.

Required background. Exterior Algebra and Differential Forms supplies forms and graded products; the Exterior Derivative and Poincaré Lemma supplies exact forms, and the generalized Stokes theorem turns them into boundary terms. Tensor Fields and Natural Operations supplies smooth tensor fields and coordinate-independent tensor operations; this page defines their field-space variations.

Helpful background. Calculus and Linear Algebra supplies integration by parts and linearization; Variational Mechanics and Symmetries supplies endpoint variations and ordinary Euler–Lagrange equations.

The boundary decomposition in mechanics

For a curve $q^i(t)$ with Lagrangian $L(q,\dot q,t)$, vary $q^i$ by $\delta q^i$. Before imposing endpoint conditions,

$$\delta L =\frac{\partial L}{\partial q^i}\delta q^i +\frac{\partial L}{\partial\dot q^i}\frac{d}{dt}\delta q^i.$$

Writing $p_i=\partial L/\partial\dot q^i$ and integrating the second term by parts gives the local identity

$$\delta(L\,dt) =\left(\frac{\partial L}{\partial q^i}-\dot p_i\right) \delta q^i\,dt+d(p_i\delta q^i).$$

The coefficient of $\delta q^i$ is the Euler–Lagrange expression; the endpoint form is $\Theta=p_i\delta q^i$. After integration over $[t_1,t_2]$, fixed endpoints remove $[p_i\delta q^i]_{t_1}^{t_2}$ and arbitrary interior variations imply $\dot p_i=\partial L/\partial q^i$. If endpoints are not fixed, stationarity instead requires boundary conditions or an added boundary action. The bulk equation alone does not decide which choice is intended.

For a local Lagrangian $n$-form $\mathbf L$ depending on fields $\phi^A$ and finitely many derivatives, repeated integration by parts yields

$$\boldsymbol\delta\mathbf L =\mathbf E_A(\phi)\,\boldsymbol\delta\phi^A+d\boldsymbol\Theta(\phi;\boldsymbol\delta\phi).$$

Here $d$ is the spacetime exterior derivative, while $\boldsymbol\delta$ is the field-space exterior derivative. The $n$-forms $\mathbf E_A\boldsymbol\delta\phi^A$ define the equations of motion $\mathbf E_A=0$; $\boldsymbol\Theta$ is a spacetime $(n-1)$-form and a field-space one-form. This standard covariant decomposition is stated, with degrees and ambiguities, in Iyer and Wald 1994, §3, Eq. (20), p. 6 of the preprint, PDF.

Scalar-field calculation

Take a real scalar on a fixed oriented Lorentzian spacetime with Lagrangian form

$$\mathbf L=-\frac12 d\phi\wedge\star d\phi-V(\phi)\,\varepsilon,$$

where $\varepsilon$ is the metric volume form and the metric is not varied. Because the Hodge operator is fixed,

$$\boldsymbol\delta\mathbf L =-d(\boldsymbol\delta\phi)\wedge\star d\phi -V'(\phi)\boldsymbol\delta\phi\,\varepsilon.$$

The product rule

$$d(\boldsymbol\delta\phi\,\star d\phi) =d(\boldsymbol\delta\phi)\wedge\star d\phi +\boldsymbol\delta\phi\,d\star d\phi$$

then gives

$$\boldsymbol\delta\mathbf L =\boldsymbol\delta\phi\,[d\star d\phi-V'(\phi)\varepsilon] +d[-\boldsymbol\delta\phi\,\star d\phi].$$

Thus $\mathbf E=d\star d\phi-V'(\phi)\varepsilon$ and one valid potential is $\boldsymbol\Theta=-\boldsymbol\delta\phi\,\star d\phi$. With the site signature, $d\star d\phi=(\Box\phi)\varepsilon$, so the bulk equation is $\Box\phi-V'(\phi)=0$. The result has also passed a component check: varying $-\sqrt{-g}\,g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi/2$ and integrating the ordinary divergence reproduces the same sign.

What boundary stationarity requires

On a region $M$ with boundary, Stokes’ theorem gives

$$\boldsymbol\delta S =\int_M\mathbf E_A\boldsymbol\delta\phi^A +\int_{\partial M}\boldsymbol\Theta.$$

For the scalar example, Dirichlet conditions set $\boldsymbol\delta\phi=0$ on the boundary. Neumann or mixed conditions require a different admissible variation space, often together with a boundary term chosen so that the remaining boundary variation vanishes. Which data may vary must be stated before declaring the action differentiable. Harlow and Wu emphasize this boundary-first formulation in Harlow and Wu 2020, §2.2, Eqs. (2.15)–(2.20), pp. 11–13.

Integration by parts is local, so corners require care: applying Stokes on adjacent boundary pieces can leave codimension-two contributions. Moving boundaries also contribute variations of the embedding, not merely of the fields. Neither issue changes the interior Euler–Lagrange form, but both can change the admissible boundary data and the potential.

Nonuniqueness and limits

The decomposition is not unique. If $\mathbf L\mapsto\mathbf L+d\boldsymbol\mu$, then

$$\boldsymbol\Theta\mapsto \boldsymbol\Theta+\boldsymbol\delta\boldsymbol\mu+d\mathbf Y$$

is allowed, where $\mathbf Y$ is linear in the variation and has spacetime degree $n-2$. These terms do not alter the bulk Euler–Lagrange equations, but they can alter corner terms and later surface structures. One must therefore carry them rather than dismissing every exact term as physically irrelevant.

This page assumes local finite-order Lagrangians and smooth fields. Nonlocal actions, distributional configurations, and theories with unresolved functional domains need additional analysis. It also establishes only the geometric decomposition; it does not introduce the Einstein–Hilbert action, gravitational charges, or quantization.

Exercise: oscillator endpoints

For $L=m\dot q^2/2-m\omega^2q^2/2$, derive the bulk and endpoint pieces of $\delta S$. State two different ways to make the endpoint contribution vanish.

Solution

Direct variation and one integration by parts give

$$\delta S=\int_{t_1}^{t_2}[-m\ddot q-m\omega^2q]\delta q\,dt +[m\dot q\,\delta q]_{t_1}^{t_2}.$$

The bulk equation is $\ddot q+\omega^2q=0$. Dirichlet endpoint data impose $\delta q(t_1)=\delta q(t_2)=0$. Alternatively, allowing free endpoint positions makes stationarity require the natural condition $p=m\dot q=0$ at each free endpoint. These are different variational problems with the same interior equation.

References

• Harlow, Daniel, and Jie-qiang Wu. “Covariant Phase Space with Boundaries.” Journal of High Energy Physics 10 (2020), 146. DOI; Open PDF.
• Iyer, Vivek, and Robert M. Wald. “Some Properties of Noether Charge and a Proposal for Dynamical Black Hole Entropy.” Physical Review D 50 (1994): 846–864. DOI.