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I'm trying to understand the physics of tilt twisting in gymnastics / trampolining, and I've run into a conundrum where theory doesn't seem to agree with practice, so I'm looking for where my mistake could be.

In tilt twisting, the gymnast performs a flip, and, during the flip, also starts turning. The motion is described by an Olympic champion in this video, and at 2:38, it says that, for a back flip, LEFT twist = LEFT arm drops first.

Here is how I formalized the problem: Let the gymnast be standing, with the positive X-axis pointing towards his right side, the positive Y-axis pointing towards his head, and the positive Z-axis pointing towards his front. Assume the gymnast is in the air doing a backflip; this would mean that the angular momentum only has a positive X component, and zero Y Z components (assuming I've correctly used the right hand rule; a backflip is the same rotation motion as when falling backwards from standing).

I also assume that the gymnast, when standing aligned with the axes, has a moment of inertia that is a diagonal matrix, and that a gymnast is much taller than either wide or long (fat?), meaning that the Y component of the moment of inertia matrix is lower than the other two non-zero entries.

Now, assume the gymnast has jumped into the air with his arms up, with a moment of inertia as described above, and, once in the air, the gymnast drops his left arm completely straight tracing a path in the X-Y plane, and so rotating around an axis parallel to the Z-axis. This would cause his entire body to tilt to the right, let's make a simple assumption and say less than $45^{\circ}$. A slight tilting to the right can be represented by a rotation vector with a negative Z component and zero X and Y components, or a rotation matrix of the form $$\mathbf{R}=\left[\begin{array}{l}a&b&0\\-b&a&0\\0&0&1\end{array}\right],$$ with $a, b > 0$, and we can use this to compute the updated moment of inertia using $\mathbf{I}_\text{new} = \mathbf{RIR^T}$, which, given the gymnast's height is much greater than his other dimensions, will yield a matrix of all non-negative numbers.

Computing the angular velocity based on the formula $\mathbf{h} = \mathbf{I}_\text{new}\mathbf{\omega}$ will yield a vector with negative values and zeros for the entries of $\omega$, as $\mathbf{h}$ is composed of all negative values and zeros, and $\mathbf{I}_\text{new}$ of all positive values and zeros.

So the angular velocity in the Y-axis is negative, which according to the right-hand rule implies spinning to the right, yet contradicting the video that dropping the left arm during a backflip results in a left twist.

What is my error here?

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    – Gabi
    Commented Jan 7 at 9:12

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I think it was an error in the right-hand rule for tilting; the statement above should read: A slight tilting to the right can be represented by a rotation vector with a negative positive Z component.

All the signs after that statement are therefore flipped, and the result is as expected.

I found the error while checking everything for the 5th time, apologies for bothering people; hopefully, this is the only error, and I haven't made two errors that cancel each other out.

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