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Second variation formula for energy

Statement

Assume ${A C}_{ω}$ through the declared dependencies, as required by Algebraic symmetries of the Riemann tensor. Let $a < b$ and let $F : (- ε, ε)^{2} \times [a, b] ⟶ M$ be continuous and smooth in $(s, r)$ , and smooth on each strip of one common finite subdivision $a = t_{0} < \dots < t_{m} = b$ , with smooth parameter derivatives at the seams. Suppose the central curve $γ (t) = F (0, 0, t)$ is an affinely parametrized geodesic. Put $E (s, r) = \frac{1}{2} \sum_{k = 1}^{m} \int_{t_{k - 1}}^{t_{k}} ∣ \partial_{t} F (s, r, t) ∣_{g}^{2} d t,$ and at $(s, r) = (0, 0)$ write $V = \partial_{s} F$ , $W = \partial_{r} F$ , and $T = \dot{γ}$ . Then ${\partial_{s} \partial_{r} E (s, r) ∣}_{(0, 0)} = \int_{a}^{b} (g (D_{t} V, D_{t} W) - g (R (V, T) T, W)) d t + {[g (D_{s} W, T)]}_{a}^{b} .$ Here $[h]_{a}^{b} = h (b) - h (a)$ , and $D_{s} W$ is the covariant mixed acceleration at the outer endpoint. The curvature convention is $R (X, Y) Z = \nabla_{X} \nabla_{Y} Z - \nabla_{Y} \nabla_{X} Z - \nabla_{[X, Y]} Z .$

Equivalently, integrating the derivative-product term by parts on each subinterval gives the full jump form ${[g (D_{s} W, T)]}_{a}^{b} + {[g (D_{t} V, W)]}_{a}^{b} - \sum_{j = 1}^{m - 1} g (Δ_{j} D_{t} V, W (t_{j})) - \int_{a}^{b} g (D_{t}^{2} V + R (V, T) T, W) d t,$ where $Δ_{j} D_{t} V = (D_{t} V) (t_{j}^{+}) - (D_{t} V) (t_{j}^{-})$ . Thus the corner term is present in the integrated form; it cancels with the corresponding jump created when that form is integrated back to the derivative-product expression. If the endpoints are fixed for every $(s, r)$ , then $D_{s} W (a) = D_{s} W (b) = 0$ , so the acceleration boundary term vanishes; also $W (a) = W (b) = 0$ in the integrated form.

Facts & Assumptions

Given: The variation, its common finite subdivision, the Levi-Civita connection, and the energy convention above.

[A1]

${A C}_{ω}$ is required here through The Axiom of Countable Choice ( ${A C}_{ω}$ ) and the declared algebraic-symmetry dependency; its local use is to supply pair interchange for the curvature bilinear form. The remaining differentiation and integration use no choice.

[F1]

Energy is the half-integral of squared speed, summed over the supplied finite subdivision, by Energy of a piecewise smooth curve.

[F2]

A piecewise smooth variation is smooth on every strip of one common finite subdivision and has a continuous variation field, by Smooth variation and variation field of a curve.

[F3]

The first variation of energy includes the outer endpoint terms, the internal corner jumps, and the integral against $D_{t} T$ , by First variation formula for energy.

[F4]

Variation covariant derivatives satisfy $D_{s} D_{t} U - D_{t} D_{s} U = R (F_{s}, F_{t}) U$ by Covariant derivatives commute up to curvature in a two parameter variation.

[F5]

The Levi-Civita connection is torsion free, so covariant derivatives of the two parameter directions commute; in particular $D_{s} T = D_{t} V$ and $D_{s} W = D_{r} V$ , by Levi civita connection.

[F6]

The Riemannian curvature tensor is skew in each pair and invariant under pair interchange, by Algebraic symmetries of the Riemann tensor.

[F9]

The four-tensor is defined by $Rm (X, Y, Z, W) = g (R (X, Y) Z, W)$ , by Riemann curvature four-tensor.

[F7]

$D_{t}$ is covariant differentiation along the curve, by Covariant derivative along a curve.

[F8]

A continuous parameter derivative of the scalar integrand passes through each compact-interval Riemann integral by Leibniz's rule on a compact rectangle: an interior parameter derivative with a continuous extension may be passed through a Riemann integral.

Proof

Proof technique: Differentiate the full first-variation formula and then integrate by parts piecewise.

1.1

Fix $s$ and apply [F3] to the $r$ -variation $t \mapsto F (s, r, t)$ at $r = 0$ . [F1, F2, F3, F7] With $T_{s} = \partial_{t} F (s, 0, \cdot)$ and $W_{s} = \partial_{r} F (s, 0, \cdot)$ this gives $\partial_{r} E (s, 0) = g (W_{s} (b), T_{s} (b^{-})) - g (W_{s} (a), T_{s} (a^{+})) - \sum_{j = 1}^{m - 1} g (W_{s} (t_{j}), Δ_{j} T_{s}) - \sum_{k = 1}^{m} \int_{t_{k - 1}}^{t_{k}} g (W_{s}, D_{t} T_{s}) d t .$ The energy normalization in [F1] is the one used in this first variation.

2.1

Differentiate this identity in $s$ at zero. [F2, F3, F5, F7, F8, step 1.1] Since the central path is a smooth affine geodesic, $D_{t} T = 0$ on every strip and $Δ_{j} T = 0$ at each seam. The derivative therefore reduces to ${[g (D_{s} W, T) + g (W, D_{s} T)]}_{a}^{b} - \sum_{j = 1}^{m - 1} g (W (t_{j}), Δ_{j} D_{s} T) - \sum_{k = 1}^{m} \int_{t_{k - 1}}^{t_{k}} g (W, D_{s} D_{t} T) d t .$ The differentiation-under-the-integral result [F8] applies on each compact strip. Torsion freeness gives $D_{s} T = D_{t} V$ on each strip: in target coordinates the difference is the mixed-partial difference plus $Γ^{k}_{i j} (F_{s}^{i} F_{t}^{j} - F_{t}^{i} F_{s}^{j})$ , both zero by equality of mixed partials and symmetry of the Levi-Civita symbols. Parameter smoothness makes $W$ continuous at every seam.

3.1

Apply [F4] to the field $T$ along the $(s, t)$ parameter surface and substitute into [2.1]. [F3, F4, F5, F7, step 2.1] $D_{s} D_{t} T = D_{t} D_{s} T + R (V, T) T = D_{t}^{2} V + R (V, T) T .$ This yields the piecewise integrated form ${[g (D_{s} W, T) + g (D_{t} V, W)]}_{a}^{b} - \sum_{j = 1}^{m - 1} g (W (t_{j}), Δ_{j} D_{t} V) - \int_{a}^{b} g (D_{t}^{2} V + R (V, T) T, W) d t .$ The jump is right trace minus left trace, exactly as in [F3].

4.1

Integrating $- g (W, D_{t}^{2} V)$ by parts on every strip gives the identity below. [F2, F3, F7, step 3.1] $- \int_{a}^{b} g (W, D_{t}^{2} V) d t = - {[g (W, D_{t} V)]}_{a}^{b} + \sum_{j = 1}^{m - 1} g (W (t_{j}), Δ_{j} D_{t} V) + \int_{a}^{b} g (D_{t} W, D_{t} V) d t .$ The interior jump terms in [3.1] cancel these seam terms, and its $[g (D_{t} V, W)]_{a}^{b}$ cancels the displayed outer derivative term. What remains is the stated half-energy second-variation formula, including the mixed acceleration $[g (D_{s} W, T)]_{a}^{b}$ . The same calculation proves the equivalent corner form in the Statement.

5.1

The curvature term and endpoint acceleration are symmetric under exchanging the two variation directions. [F5, F6, F9, step 2.1, step 4.1] For $V, W$ as curvature inputs, $g (R (V, T) T, W) = Rm (V, T, T, W) = Rm (W, T, T, V) = g (R (W, T) T, V)$ by [F9] and pair interchange and the two pair skews in [F6]. The same coordinate calculation as in step 2.1, now in the $s, r$ directions, gives $D_{s} W = D_{r} V$ by [F5]. Thus both the curvature term and the endpoint acceleration are symmetric; the derivative-product term in step 4.1 is symmetric by symmetry of $g$ .

6.1

The endpoint, corner, dimension, and choice cases are as follows. [A1, F2, F3, F5, F6, F7, step 3.1, step 4.1, step 5.1] If endpoints are fixed, $W (a) = W (b) = 0$ for all parameters, hence $D_{s} W (a) = D_{s} W (b) = 0$ and the acceleration term vanishes; the integrated-form endpoint term vanishes as well. For moving endpoints retain the acceleration term. With a piecewise smooth variation the jump in [3.1] is required in the integrated form, while [4.1] shows why no separate seam term remains in the derivative-product formula. If $γ$ is constant, $T = 0$ and the curvature and acceleration terms vanish, but the integral of $g (D_{t} V, D_{t} W)$ need not vanish. In dimension zero all variation fields vanish; in dimension one the curvature term vanishes by first-pair skewness. The empty manifold admits no given variation, and $a = b$ is excluded because the covariant derivative convention needs a nondegenerate interval. If one first-variation field is zero, the bulk and curvature terms vanish, but a moving-endpoint mixed acceleration may remain; under fixed endpoints it is zero. The exact choice assumption is [A1], through the declared algebraic-symmetry dependency, with no full AC or additional choice. The claim is an equality, not a biconditional. $□$

Depends on

Used by

Index form of a geodesic segment Definition
A geodesic does not minimize past its first conjugate point Theorem

Dependency tree · two levels

39 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John M. Lee, Riemannian Manifolds: An Introduction to Curvature (1997), Theorem 10.12 (standard reference, not scraped)
Ved Datar, Lectures on Riemannian Geometry (2025), Theorem 21.1.2 (standard reference, not scraped)