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--- dirichlet_energy [2022/09/01 05:10] – [Euler-Lagrange equations] spencer
+++ dirichlet_energy [2022/09/05 05:44] (current) – [Euler-Lagrange equations] spencer
@@ Line 13: / Line 13: @@
 ===== Euler-Lagrange equations =====
+These computations are done in Eells--Lemaire (2.4).
 Consider a family of maps $u_t \in C^2(M,N)$ such that $u_0 = u$.
 Let $U : \mathbb{R} \times M \to N$ be the map defined by $U(t,x) = u_t(x)$.
+Let $\nabla_{\mathrm{d}/\mathrm{d}t}$ be the covariant derivative, and then compute on a vector field $X$ on $M$:
+\begin{align*}
+(\nabla_{\mathrm{d}/\mathrm{d} t} \mathrm{d} u_t)X &= \nabla_{\mathrm{d}/\mathrm{d}t}(\mathrm{d}u_t \cdot X) - \mathrm{d}u_t \cdot \nabla_{\mathrm{d}/\mathrm{d} t} X \\
+&= \nabla_{\mathrm{d}/\mathrm{d}t}(\mathrm{d}U \cdot X) - 0\\
+&= \nabla_X (\mathrm{d}U \cdot \mathrm{d}/\mathrm{d} t) + \mathrm{d}U\left[ \frac{\mathrm{d}}{\mathrm{d}t}, X\right] \\
+&= \nabla_X \frac{\partial U}{\partial t} + 0.
+\end{align*}
 The family of maps generates a section $\frac{\partial U}{\partial t}$ of the bundle $u^*TN$ over $M$ by \[ \left(\frac{\partial U}{\partial t}\right)_p = \left. \frac{\mathrm{d}}{\mathrm{d}t} u_t(p)\right\rvert_{t=0} \in T_{u(p)} N. \]
-Write $\partial_t U$ for this section.
-The bundle $u^*TN$ has a natural connection $\nabla$, which is the pullback by $u$ of the Levi-Civita connection on $TN$. Thus $\nabla (\partial_t U) \in \Gamma(T^* M \otimes u^* TN)$.
-Since on ($u^*TN$-valued) $0$-forms the connection agrees with the exterior covariant derivative it induces, one may also write $\mathrm{d} (\partial_t U)$ for this section of $T^* M \otimes u^* TN$, or $\mathrm{d}_{u^* TN} (\partial_t U)$ to emphasize the bundle over with respect to which the exterior derivative is being taken.
-For fixed $p \in M$, pullback by the constant map $\psi_p : \mathbb{R} \to M$ defined by $\psi_p(t) = p$ allows one to pull back the bundle $T^* M  \otimes u^* TN$ over $M$ to one on $\mathbb{R}$; then the covariant derivative $\nabla_{\frac{\mathrm{d}}{\mathrm{d}t}} \psi_p^* \mathrm{d}u_t$ evaluated at $t=0$ is an element of $\psi_p^*(T^*M \otimes u^*TN)(0) = T_p^*M \otimes T_{u(p)} N$.
-On the other hand, $\psi_p^* \mathrm{d} u_t = \mathrm{d} u_t(p)$. **TODO**
 The Euler-Lagrange equations for the Dirichlet energy functional are derived from
@@ Line 33: / Line 35: @@
 and so it is necessary and sufficient for $u$ to be a critical point of the functional $E$ that $\mathrm{d}^*_{u^* TN} (\mathrm{d} u) = 0$, or more compactly that $\mathrm{d}^* (\mathrm{d}u) = 0$.
 A map that is a critical point of the functional $E$ is called //harmonic//.
-Write $\tau(u) := \mathrm{d}^* (\mathrm{d}u)$ for the //torsion field of $u$//.
+Write $\tau(u) := -\mathrm{d}^* (\mathrm{d}u) = \mathrm{tr} \nabla \mathrm{d}u$ for the //torsion field of $u$//.
+See the page [[weitzenboeck_identity|Weitzenböck identity]] for why these two expressions are equal.
-**Claim:** Another expression for the torsion field is
-\[ \tau(u) = \mathrm{tr}_g (d^\nabla (\mathrm{d}u)). \]
-In this expression, $\nabla$ is the [[exterior_derivative|exterior derivative]] on $u^*TN$-valued 1-forms on $M$ and so $\nabla (\mathrm{d}u) \in \Gamma(\Lambda^2T^*M \otimes u^*TN)$. In a local orthonormal frame $\{e_1,\ldots,e_n\}$ for $TM$, the trace on an element $\omega \otimes \sigma$ is then $\sum_i \omega(e_i,e_i)\sigma \in \Gamma(u^*TN)$.
-**TODO:** This trace can't be correct, because $\omega(e_i,e_i) = 0$ for each 2-form $\omega$.
-**Proof:** I claim more generally that on $u^*TN$-valued 1-forms, $\mathrm{tr}_g \circ \mathrm{d}^\nabla= \nabla^* = -e(\omega^i)^* \nabla_{e_i}$.
-It suffices to work on a decomposable $\omega \otimes \sigma$ where $\omega$ is a 1-form and $\sigma$ is a section of $u^* TN$.
-Then
-\[ \mathrm{d}^\nabla (\omega \otimes \sigma) = \mathrm{d}\omega \otimes \sigma - \omega \otimes \nabla \sigma. \]
 ===== Properties of the Dirichlet energy =====