Variational and Symplectic Geometry

Variational geometry explains how an action responds when an entire field is displaced; symplectic geometry explains how the resulting data organize motion, constraints, and redundancy. The chapter builds one continuous chain: field-space tangent vectors lead to the Euler–Lagrange boundary decomposition, its boundary potential leads to a presymplectic current, and reduction turns appropriate null directions into a phase space of distinct states. The emphasis is geometric and convention-controlled. Specific gravitational actions, charges, and quantization belong elsewhere.

Helpful background. Calculus and Linear Algebra supplies multivariable differentiation, dual spaces, and antisymmetric bilinear forms; Exterior Algebra and Differential Forms supplies forms, wedge products, pullbacks, and contraction; Tensor Fields and Natural Operations supplies smooth tensor fields and their metric-free operations; Variational Mechanics and Symmetries supplies the finite-dimensional action, endpoint-variation, and Noether model.

The chapter’s central distinction

Two exterior derivatives appear throughout. The spacetime derivative $d$ changes form degree on the base manifold. The field-space derivative $\boldsymbol\delta$ changes degree on the space of configurations. For a local Lagrangian $n$-form,

$$\boldsymbol\delta\mathbf L =\mathbf E_A\boldsymbol\delta\phi^A+d\boldsymbol\Theta,$$

and an antisymmetrized second variation defines a spacetime $(n-1)$-form. In the Iyer–Wald ordering used throughout this chapter,

$$\boldsymbol\omega(\delta_1,\delta_2) =\delta_2\boldsymbol\Theta(\delta_1) -\delta_1\boldsymbol\Theta(\delta_2)$$

for commuting variations. With the standard field-space exterior derivative $\boldsymbol\delta\boldsymbol\Theta(\delta_1,\delta_2) =\delta_1\boldsymbol\Theta(\delta_2) -\delta_2\boldsymbol\Theta(\delta_1) -\boldsymbol\Theta([\delta_1,\delta_2])$, this convention is $\boldsymbol\omega=-\boldsymbol\delta\boldsymbol\Theta$; field-dependent variation vectors require the corresponding bracket term. The spacetime and field-space differentials commute only when the base manifold and the identification of nearby fields are fixed.

The integrated form $\Omega_\Sigma=\int_\Sigma\boldsymbol\omega$ is conserved only when the background and tangent vectors satisfy their equations and boundary flux vanishes. It is symplectic only when it has no kernel. Gauge directions often lie in that kernel, but a transformation that acts at a boundary need not do so. Iyer and Wald give the Lagrangian decomposition and symplectic current in Iyer and Wald 1994, §3, Eqs. (20) and (44), pp. 6 and 9 of the preprint, PDF, while the boundary-sensitive variational principle is developed in Harlow and Wu 2020, §§2.1–2.2, pp. 6–15.

Preparation and shortest repair routes

Readers should be comfortable with smooth manifolds, tangent vectors, pullbacks, differential forms, Stokes’ theorem, and elementary variational mechanics. If field-space notation is unfamiliar, begin with Variations and Tangent Vectors on Field Space. If integration by parts in an action is the weak point, review Variational Mechanics and Symmetries and The Generalized Stokes Theorem before continuing.

Linear algebra supplies an important diagnostic: a nondegenerate antisymmetric form identifies vectors with covectors; a degenerate one does not. Familiarity with group actions and regular values is helpful for the final two pages, but each page develops the bounded definitions it needs. No functional-analysis course is assumed. Infinite-dimensional field spaces are treated formally, and every conclusion that requires stronger topology or regularity is labeled accordingly.

A route through variations and currents

The first route follows a local action from variation to hypersurface data:

1. Variations and Tangent Vectors on Field Space distinguishes $d$, $\boldsymbol\delta$, parameter derivatives, and admissible tangent vectors.
2. Euler–Lagrange Forms and Boundary Decomposition derives the bulk equation and exact boundary term in mechanics and scalar field theory.
3. Presymplectic Potentials and Currents antisymmetrizes a second variation, checks on-shell closure, and tracks exact-form ambiguities and boundary flux.

This is the shortest route for readers preparing for covariant phase-space methods. It deliberately stops before choosing the Einstein–Hilbert action or defining a gravitational surface charge. Its recurring check is a scalar field: the linearized equation makes the current closed, and Stokes’ theorem shows exactly what additional boundary condition makes its hypersurface integral independent of $\Sigma$.

A route through Hamiltonian geometry and reduction

The second route begins with a finite-dimensional model where every hypothesis can be seen directly:

1. Symplectic Forms and Poisson Brackets fixes the sign $\iota_{X_F}\Omega=dF$, derives Hamilton’s equations, and relates $d\Omega=0$ to the Jacobi identity.
2. Constraints, Moment Maps, and Gauge Orbits shows when constraint-generated vectors are null directions and computes the $SO(2)$ moment map on $T^*\mathbb R^2$.
3. Symplectic Reduction and Boundary Degrees of Freedom states the regular-value, free-action, and proper-action assumptions under which restriction and quotient produce a symplectic manifold.

Moment maps and regular symplectic reduction are developed in Cannas da Silva 2006, Lectures 22–24, pp. 133–154. When stabilizers change, the quotient can instead be stratified; this is a geometric limitation, not a defect repaired by a formal dimension count.

Convention-sensitive ideas

Three pairs should remain separate. First, an exact spacetime form need not be irrelevant when the integration surface has a boundary. Second, a first-class constraint is not by itself proof that every generated transformation is redundant; admissibility and boundary action matter. Third, adding boundary variables is one possible representation of subregion symmetries, not a universal requirement or a unique construction.

Sign conventions also travel as a package. This chapter uses $\iota_{X_F}\Omega=dF$ and $\{F,G\}=\Omega(X_F,X_G)$. With $\Omega=dq^i\wedge dp_i$, these definitions yield $\dot q^i=\partial H/\partial p_i$, $\dot p_i=-\partial H/\partial q^i$, and $\dot F=\{F,H\}$. Translating only one of these equations would introduce a silent sign inconsistency.

Chapter diagnostic

On $P=T^*\mathbb R$ with $\Omega=dq\wedge dp$, impose the constraint $p=0$. Find the kernel of the pulled-back form, reduce by translations in $q$, and explain why the same quotient is invalid if allowed transformations must fix two finite endpoints of the $q$ interval.

Solution

The constraint surface is the line $\mathcal C=\{p=0\}$. Its tangent vectors are multiples of $\partial_q$, and $i^*\Omega=0$, so the entire tangent space is the kernel. The Hamiltonian vector field of $p$ is $X_p=\partial_q$; on the full real line, translations act freely and properly, and $\mathcal C/\mathbb R$ is one point with the unique zero two-form.

If $q$ is restricted to an interval whose endpoints are fixed, a nonzero translation moves those endpoints and does not preserve the allowed configurations. It is therefore not an admissible gauge action for that variational problem. The local constraint calculation is unchanged, but the proposed global quotient has lost one of its hypotheses. This is the same logical test required for boundary gauge transformations in field theory.

What this chapter enables

After the first route, a reader can vary a local Lagrangian while preserving spacetime and field-space degrees, identify bulk equations and boundary potentials, build a presymplectic current, and state the conditions for closure and hypersurface independence. After the second, a reader can derive Hamiltonian vector fields and Poisson brackets, recognize moment-map constraints, distinguish restriction from quotient, and test whether a reduced space is smooth or stratified. Together these skills prepare the geometric side of constrained and covariant formulations without predetermining their physical boundary conditions.

References

• Cannas da Silva, Ana. Lectures on Symplectic Geometry. Lecture Notes in Mathematics 1764, revised edition, Springer, 2006. Lectures 1–5 and 22–24, pp. 3–29 and 133–154.
• 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.