Geometry / dynamics analogues In 3-space geometry we have curvatures when a point is proceeding along a curved arc. Similarly when particle motion occurs with respect to time we have accelerations.
Is there a one to one correspondence or combination relation among curvatures/ tangent rotations on the one hand and accelerations/velocities on the other?
EDIT1:
I hope to be able to see a table of dynamic/geometric analogues.
e.g., for plane curves
Centripetal acceleration ~ normal curvature 
Coreolis acceleration ~  ?
tangential acceleration ~ ? 
Angular momentum ~ ?
~ geodesic torsion  ?
Most importantly a query remains with me longtime:
What are the characteristic accelerations or velocities of a particle moving along a curved asymptotic line of a ( negative curved ) surface?
EDIT2
added tags: curves and differential geometry
 A: $\newcommand{\Reals}{\mathbf{R}}\DeclareMathOperator{\Tgt}{\mathbf{T}}\DeclareMathOperator{\Nml}{\mathbf{N}}$Your question is qualitative and open-ended, but perhaps the following observations get at what you're asking:
Let $I$ be an open interval of real numbers, and suppose $\gamma:I \to \Reals^{n}$ is a twice continuously-differentiable curve with non-vanishing velocity. For each $t$ in $I$, the velocity $\gamma'(t)$ is non-zero, so we may write
$$
\gamma'(t) = \|\gamma'(t)\|\, \frac{\gamma'(t)}{\|\gamma'(t)\|}
  = v(t) \Tgt(t),
$$
with $v(t) = \|\gamma'(t)\|$ the speed at $t$ and $\Tgt(t)$ the unit tangent vector.
By the product rule, the acceleration decomposes as
$$
\gamma''(t) = v'(t) \Tgt(t) + v(t) \Tgt'(t).
\tag{1}
$$
Remarkably, this is the decomposition of the acceleration $\gamma''(t)$ into parallel and orthogonal components with respect to $\Tgt(t)$, i.e., into tangential and normal components: Since $\Tgt(t) \cdot \Tgt(t) = 1$ for all $t$, we have $2\Tgt(t) \cdot \Tgt'(t) = 0$ for all $t$.
If $\Tgt'(t) \neq 0$, we may write
$$
\Tgt'(t) = \|\Tgt'(t)\| \frac{\Tgt'(t)}{\|\Tgt'(t)\|}
  = \frac{\kappa(t)}{v(t)} \Nml(t),
$$
with $\kappa(t)/v(t) = \|\Tgt'(t)\|$. Qualitatively, (1) says acceleration can be resolved into a term $v'(t) \Tgt(t)$ measuring how the speed changes at each instant, and a term $v(t) \Tgt'(t) = \kappa(t) \Nml(t)$ measuring how the direction changes at each instant.
The function $\kappa(t)$, the "rate of rotation of $\Tgt(t)$, normalized according to the speed $v(t)$", has a pleasant geometric interpretation as the reciprocal radius of curvature of $\gamma$ at $t_{0}$. Let $s:I \to \Reals$ be the arc length measured along the curve starting at $\gamma(t_{0})$, i.e., 
$$
s(t) = \int_{t_{0}}^{t} \|\gamma'(\tau)\|\, d\tau,\quad
\text{so that $s'(t) = v(t)$ and $s(t_{0}) = 0$.}
$$
It's straightforward to check that the curve
$$
C_{t_{0}}(t)
  = \left[\gamma(t_{0}) + \frac{1}{\kappa(t_{0})} \Nml(t_{0})\right]
  - \frac{1}{\kappa(t_{0})} \cos \bigl(\kappa(t_{0}) s(t)\bigr) \Nml(t_{0})
  + \frac{1}{\kappa(t_{0})} \sin \bigl(\kappa(t_{0}) s(t)\bigr) \Tgt(t_{0}),
$$
which parametrizes the osculating circle, of radius $r(t_{0}) = 1/\kappa(t_{0})$ lying in the plane through $\gamma(t_{0})$ and spanned by $\Tgt(t_{0})$ and $\Nml(t_{0})$, agrees with $\gamma$ to second order at $t_{0}$, in the sense that
$$
C_{t_{0}}(t_{0}) = \gamma(t_{0}),\qquad
C_{t_{0}}'(t_{0}) = \gamma'(t_{0}), \qquad
C_{t_{0}}''(t_{0}) = \gamma''(t_{0}).
$$
In words, if $\gamma'(t_{0})$ and $\gamma''(t_{0})$ are non-zero, then so far as velocity and acceleration (at that instant) are concerned, a particle may as well be moving along a circle of radius $1/\kappa(t_{0})$.
Particularly, if a particle moves at unit speed, i.e., $\gamma$ is "parametrized by arc length", then $\Tgt'(t) = \kappa(t) \Nml(t)$, and the rate of rotation of $\Tgt$ at $t_{0}$ is precisely the reciprocal radius of the osculating circle to $\gamma$ at $t_{0}$.
A: As user86418 pointed out, your question is too vast. Here is a table with analogous concepts in mechanics and geometry that I can remember now. You can see more details about each one by a simple search on google.
Geometry | Mechanics
tangent vector$\quad\quad$ velocity vector
curvature of a curve$\quad\quad$ acceleration intensity
geodesic curvature$\quad\quad$ tangential acceleration intensity
normal curvature$\quad\quad$ normal acceleration intensity
geodesics$\quad\quad$ path taken by a free particle
Clairaut's relation$\quad\quad$ angular momentum
