Finding the instantaneous curvature of a trajectory on a 2D surface embedded in ordinary Euclidean 3D space I have an unknown trajectory of a particle, ${\bf r}(t)$ constrained to follow a 2D surface parameterised in $\theta$ and $\varphi$: ${\bf r}(\theta(t),\varphi(t))$.
I do know the surface completely (i.e. I have explicit expressions for $r_x(\theta,\phi)$, $r_y(\theta,\phi)$, and $r_z(\theta,\phi)$), but I don't know the trajectory $\theta(t)$, $\varphi(t)$. However, at a particular point $P$ in this trajectory, say, at $t_1$, I know both (A) its instantaneous velocity $\dot{\bf r}(\theta(t_1),\varphi(t_1))$, and (B) its "parametric" accelerations $\ddot{\theta}(t_1)$ and $\ddot{\varphi}(t_1)$.
What I ultimately want to find is the instantaneous acceleration, $\ddot{\bf r}(\theta(t_1),\varphi(t_1))$, and in particular, the instantaneous curvature of the trajectory, $\kappa(t_1)$. I intuitively feel that I have enough information to compute these, but I'm new enough to differential geometry that I'm struggling to find/derive the right formulae.
For instance, for the curvature, I have found formulae for both the normal curvature and the geodesic curvature, but the formulae I found are given in terms of the arc length parameterised curve. For example, in the formula for the geodesic curvature:
$\kappa_g = \left[\Gamma^2_{11}\left(\frac{d\theta}{ds}\right)^3 + (2\Gamma^2_{12} - \Gamma^1_{11})\left(\frac{d\theta}{ds}\right)^2\frac{d\varphi}{ds} + (\Gamma^2_{22} - 2\Gamma^1_{12})\frac{d\theta}{ds}\left(\frac{d\varphi}{ds}\right)^2\right.$
$\left.\qquad -\Gamma^1_{22}\left(\frac{d\varphi}{ds}\right)^3 + \frac{d\theta}{ds}\frac{d^2\varphi}{ds^2} - \frac{d^2\theta}{ds^2}\frac{d\varphi}{ds}\right]\sqrt{EG-F^2}$,
I know how to compute everything in this formulae (at point $P$) except the second derivatives $\frac{d^2\theta}{ds^2}$ and $\frac{d^2\varphi}{ds^2}$, but I don't see how to calculate those when I only know the behaviour of the particle at the point $P$.
So my questions are:


*

*Do I really have enough information to calculate both the instantaneous acceleration and the instantaneous curvature at point $P$ (I don't care about other points on the trajectory)?

*If so, what are their formulae in terms of my known boundary conditions $\dot{\bf r}(\theta(t_1),\varphi(t_1)) = {\bf v}$, $\ddot{\theta}(t_1) = a_{\theta}$, and $\ddot{\varphi}(t_1) = a_{\varphi}$?


Extra information, in case it's relevant


*

*My surface is a surface of revolution about the $z$-axis, but my parameterisation is not orthogonal (i.e. in the first fundamental form, $F \ne 0$). The parameter $\varphi$ is, however, the "parameter of revolution".

*$a_{\varphi}$ happens to be zero in my particular case. That is, the particle is free to move arbitrarily along the iso-parametric curve corresponding to variable $\theta$, but $\varphi(t)$ is strictly of the form $\varphi(t) = At+B$, for some constants $A$ and $B$.


I can translate all the unknown derivatives in $s$ into derivatives in $t$ by an application of the chain rule.
$$\frac{d\theta}{ds} = \frac{d\theta}{dt}\frac{dt}{ds}$$
and
$$\begin{aligned}
\frac{d^2\theta}{ds^2}
    &= \frac{d}{ds}\frac{d\theta}{ds} \\
    &= \frac{dt}{ds} \frac{d}{dt}
       \left( \frac{d\theta}{dt}\frac{dt}{ds} \right) \\
    &= \frac{1}{ds/dt}
       \left( \frac{d^2\theta}{dt^2}\frac{1}{ds/dt} +
              \frac{d\theta}{dt} \frac{d}{dt} \left(\frac{1}{ds/dt}
       \right)\right) \\
    &= \frac{1}{ds/dt}
       \left( \frac{d^2\theta}{dt^2}\frac{1}{ds/dt} -
              \frac{d\theta}{dt} \left(\frac{1}{ds/dt} \right)^2
              \frac{d^2s}{dt^2}
       \right),
\end{aligned}$$
and similarly for $\varphi$. When I substitute these back into the equation for geodesic curvature, for example, I get
$$
\begin{aligned}
\kappa_g
    &= \left(\frac{1}{ds/dt}\right)^3
       \left[\Gamma^2_{11}\left(\frac{d\theta}{dt}\right)^3 +
       (2\Gamma^2_{12} - \Gamma^1_{11})\left(\frac{d\theta}{dt}\right)^2
       \frac{d\varphi}{dt} + (\Gamma^2_{22} -
       2\Gamma^1_{12})\frac{d\theta}{dt}\left(\frac{d\varphi}{dt}
       \right)^2\right. \\
    &\qquad \left. -\Gamma^1_{22}\left(\frac{d\varphi}{dt}\right)^3 +
       \frac{d\theta}{dt}\frac{d^2\varphi}{dt^2} - \frac{d^2\theta}{dt^2}
       \frac{d\varphi}{dt}\right]\sqrt{EG-F^2},
\end{aligned}
$$
which resembles the original, except that all the derivatives are now taken with respect to $t$, and there is the extra factor of $1/(ds/dt)^3$ out the front.
