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Many people with long hair tie their hair to ponytail-style:
Closely observing the movement of their hair when they are running, I have noticed that the ponytail oscillates only horizontally, that is, in "left-right direction". Never I have seen movement in vertical "up-down" direction or the third direction (away-and-back from the jogger's back). Why is the horizontal direction the only oscillation?
The human gait has a natural bobbing motion, with the head moving slightly up-and-down and side-to-side. The side-to-side motion (swinging on an axis parallel to the nose) turns the ponytail into a natural pendulum which swings back and forth, since this plane of motion is gravitationally symmetric and has nothing to stop the swing. Small driving forces can build up over time, causing a noticeable swing, very similar to how one would use a swing on a swingset.
The up-and-down motion (swinging on an axis parallel to the shoulders) does not turn the ponytail into a pendulum, because the hair cannot swing freely on this axis. The problem is, there no mechanism to conserve energy at the bottom of the up-down swing, since the ponytail hits the back of the runner's head and loses all its energy. For the side-to-side swing, there's a constant oscillation of gravitational potential and kinetic energy in the ponytail, which isn't so in an up-and-down swing - when the ponytail reaches the bottom of an up-down swing, it has lost all its potential and kinetic energy, so you can't keep imparting small forces which will grow over time and produce a repeating oscillation.
The front-to-back oscillations described in the question are the same as the up-and-down oscillations described in the previous paragraph (swinging along an axis parallel to the shoulders). The third axis of oscillation would be swinging on an axis parallel to the spine, which I think does happen to an extent. But since this axis is parallel to gravity, the ponytail hangs down very close to the axis, and rotations at this small radius tend to be lost in the much larger side-to-side swing. I suspect that the ponytail doesn't swing perfectly in a flat plane along only one axis, but actually wraps "around" the head slightly as it swings side-to-side - there may be a major swing along the axis of the nose, and a minor one along the axis of the spine.
In the end, the most noticeable swing is side-to-side along the axis of the nose. Up-and-down oscillations on the axis of the shoulders cannot build up over time with small driving forces. And since the ponytail hangs very close to the third axis of rotation (along the axis of the spine), these are of much smaller magnitude than the obvious swing along the axis of the nose.
I think the longitudinal oscillations of a ponytail are quickly damped (more precisely overdamped), since they involve layers of hair sliding along each other, as well as inelastic collisions of the back of the neck. On the other hand, the transversal oscillations require merely twisting the ponytail near the elastic band holding it together.
I think it would be harder to make an argument based on the forces causing the oscillations, since stepping from one foot onto another involves displacement in all directions: forward/backward, up/down, left/right.
Update
Those interested in exploring this subject deeper may be interested in article The shape of a ponytail and the statistical physics of hair fiber bundles, as well as in a more extended abstract discussion in this thesis (open access).
Not a complete answer, but a few notes.
There is a significant amount of rotational force going on in the head and neck during running, so much that humans (as well as other running animals) have specific adaptations to help manage them. Cf "that pig can't hold its head still"[1].
I can at least find one source[2] finding that the head has a yaw (side-to-side rotation) frequency of about 1.6 Hz in running (read the abstract carefully); I'll assert that this is definitely synchronized to rotational forces from arm swings, i.e. the period of the stride.
At about 1:47 in this video[3] on the left, you see a very short ponytail that is bouncing vertically and forward-back, so now you've seen it happen at least once. Note the vertical bounce occurs every step when she pushes off from the ground, probably about 3.2 Hz as found in [2], whereas the side-to-side swing of long ponytails (several of which can be seen in the video) has a period of every other step or probably ~1.6 Hz, i.e. swings left after the right foot pushes off and vice versa.
Each push-off, the head accelerates a bit forward with the body, so the hair's center of mass has to move backward relative to the head until it is pulled forward again. See time 1:34 in the video. So to some extent it's an optical illusion that motion is only side-to-side, possibly because different parts of the hair are moving in different directions at the same time.
So again, this isn't a complete answer, but it may help to summarize that the main force acting on the hair from the head attachment comes at a very constant rate (about 3 times per second usually) during toe-off, and includes some forward acceleration as well as a sideways component that alternates between right and left.
[1] https://www.nytimes.com/2011/08/23/science/23conversation.html?pagewanted=all
As @nuclear-hoagie said, the ponytail is basically a pendulum, so vibrations up and down are not really possible (or are much more complicated) as the hair would have to elongate itself in order to store the kinetic energy from the hair falling down so it cannot release it later. However the sideways and back and forth vibrations can be maintained almost for free as they do not involve stretching the hair. I guess that you don't see the back and forth vibration for a reason somewhat similar to the up and down case, when the hair stops agains the back it loses all its kinetic energy and cannot continue the vibration.
You can look for parametric resonance in vertically vibrated pendulums. Basically when the point from which the pendulum is hanging is forced to oscillate vertically at double the natural frequency of the pendulum, an bifurcation occurs. The configuration of equilibrium (the hair hanging down vertically) is no longer stable. It is still a point of equilibrium however any slight missaligment makes the pendulum deviate from its resting point and oscillate at the natural frequency.
If you do it with a solid pendulum you can even reach a limit where the normally unstable equilibrium (the pendulum pointing upwards) becomes stable.
There are demonstrations of this dynamic stabilization on youtube, and basically is the same process but in the extreme case:
This phenomenon happens with ear plugs too. I like to think of it by considering what each step of the run does to the motion of the hair. Hair in a ponytail has a fulcrum where it is attached to the head, so it can freely move like a pendulum in its plane.
When someone takes a step forward with their right foot and pushes off, it pushes the body to the left which then provides an equal an opposite force to the right to the hair higher up the body. I am unsure if this exact force exchange process described is correct. Small impulses build up to oscillate the hair at its natural frequency (dependent on the material properties of the human hair and head pendulum). If the human drives this pendulum at different driving forces, it will oscillate at different amplitudes and frequencies, where there will be a maximum amplitude at its natural frequency for a particular driving force of the human's run or walk.
Very simple: The frequency for the up and down movement is twice that of the left and right swing.
Plus the up and down excitation (acceleration) is not symmetric, a good part of it consists of overtones of even higher frequency. Actually the hair and attached head are in free fall between steps.
So to get to the same amplitude, the hair would need to move twice as fast in up and down direction, and reach even higher accelerations.
That's only possible for a very short bun (see comment about the British marathon runner above), but not with a ponytail. The damping by air and internal friction is much too large.
