I'm reading a paper on Perelman's solution to the Poincare conjecture. The paper derives a geodesic equation by minimizing the following $\mathcal{L}$-length of each curve $\gamma(\tau)$, $0<\tau_1 \le \tau \le \tau_2$ on a manifold $M$ where $\tau=-t$, $$\mathcal{L}(\gamma)=\int_{\tau_1}^{\tau_2}(R(\gamma(\tau),\tau)+|\dot{\gamma}(\tau)|^2)d\tau.$$ Let $\textbf{X}=d\gamma_s/d\tau=\dot{\gamma_s}(\tau)$, $\textbf{Y}=d\gamma_s/ds$, and $\delta_{\textbf{Y}}(\mathcal{L})=d\mathcal{L}/ds$ where $s= \in(-\epsilon,\epsilon)$ parametrizes a family of curves $\gamma_s(\tau)$. The first variation formula of $\mathcal{L}$ reads
$$\delta_{\textbf{Y}}(\mathcal{L})=\int_{\tau_1}^{\tau_2}\sqrt{\tau}(\nabla_{\textbf{Y}} R+2 \langle \textbf{X},\nabla_{\textbf{Y}}\textbf{X}\rangle)d\tau$$
$$=\int_{\tau_1}^{\tau_2}\sqrt{\tau}(\langle \nabla R, \textbf{Y}\rangle+2 \langle \textbf{X},\nabla_{\textbf{X}}\textbf{Y}\rangle)d\tau$$
$$=\int_{\tau_1}^{\tau_2}\sqrt{\tau}\bigg(\langle \nabla R, \textbf{Y}\rangle +2\frac{d}{d\tau}\langle\textbf{X},\textbf{Y}\rangle -2 \langle \nabla_{\textbf{X}}\textbf{X},\textbf{Y}\rangle)-4Ric(\textbf{X},\textbf{Y})\bigg)d\tau$$
$$=2\sqrt{\tau}\langle\textbf{X},\textbf{Y}\rangle+\int_{\tau_1}^{\tau_2}\sqrt{\tau}\langle \textbf{Y} ,\nabla R -2\nabla_{\textbf{X}}\textbf{X} -4Ric(\cdot,\textbf{X}) -\frac{1}{\tau}\textbf{X} \rangle d\tau.$$
where $\langle\cdot,\cdot\rangle$ denotes the inner product wrt the metric. From this, the $\mathcal{L}$-geodesic equation is derived by halving the right side of the inner product above and switching the signs, giving
$$\nabla_{\textbf{X}}\textbf{X} - \frac{1}{2}\nabla R +\frac{1}{2\tau}\textbf{X} +2Ric(\cdot,\textbf{X})=0.$$
My question is:
Just like the standard geodesic equation, $$\frac{d^2 \gamma^k}{d\tau^2}+\Gamma_{ij}^k\frac{d\gamma^i}{d\tau}\frac{d\gamma^j}{d\tau}=0,$$
is the LHS of Perelman's version also the tangential component of the curve's acceleration vectors? If so, how can one deduce that based on the first variation equation? Or is the equation more analogous to the geodesic equation $\nabla_{\textbf{v}}\textbf{v}=0$ given the $\nabla_{\textbf{X}}\textbf{X}$ term?
Thanks in advance.