Your choice of Amperian loop is inappropriate for this situation. In these types of problems, you need to choose an Amperian loop where the length of the loop is either parallel or perpendicular to the field that you expect to arise from the situation (try to, its not always possible to choose something that has a convenient shape, but the symmetry of this situation allows it). The real equation for ampere's law is:

$$\oint_L \vec{B}\cdot d\vec{l} =\mu_0 I_{enc}$$

where $L$ is the path of your Amperian loop. The dot product here signifies that only the component of $\vec{B}$ along the path contributes, so choosing a path along the field lines will simplify things greatly (or if its perpendicular, it contributes zero).

Given the axial symmetry of the case you have here, you expect the field lines inside of the cylinder to point along the axis of the cylinder, much like in the case of the long solenoid. Therefore, it would be better to choose a square Amperian loop, much like that of the linked example in the last sentence.

Indeed the problem with boundary conditions, generally speaking, is not well-posed.

*There are boundary conditions admitting no curves or admitting many curves, satisfying both these conditions and Euler-Lagrange equations.*

**Examples.**

(1) Think of a particle constrained to stay on a smooth sphere where it can freely move. If you assign the North and the South pole of the sphere as boundary conditions, you get infinitely many solutions as the motion always describes a geodesic.

(2) Similarly, if you remove an open ball in $\mathbb R^3$ you do not have solutions when assigning boundary conditions on the opposite sides of the ball for a free particle in $\mathbb R^3$.

If $L$ is *quadratic* with respect to the $\dot{q}$ variables and this quadratic form is *strictly positively defined*, as is the case for systems of classical particles (also satisfying ideal holonomic constraints), the problem with initial conditions is always well-posed provided $L$ is sufficiently regular. There is exactly one maximal solution satisfying both Euler-Lagrange equations and initial conditions.

With these hypotheses also the problem with boundary conditions is well-posed with the additional condition that the two boundary conditions are sufficiently close to each other (this is evident from the two examples I presented above).

For these reasons a safer (mathematically minded) viewpoint is assuming that the variational principle determines the equation of motion, but not the solutions themselves.

## Best Answer

Before developing the theory, I decided to first make an experiment in order to understand, what we are dealing with. A cylinder with a diameter of 11.5 cm is mounted on the motor shaft (I used an old popcorn machine). I attached a 12.5 cm length of clothesline with a screw, so that exactly 11.5 cm leaves the cylinder. When the rope hangs freely, it forms a specific figure, which must be described first of all in order to find the parameters of the model (see Figure 1 left, center). When the cylinder rotated with an angular velocity of $\omega = 8 \pi$, the rope became almost horizontal. In this case, the rope was slightly bent in the horizontal plane due to the aerodynamic drag (See fig.1 right).

I used the theory of elastic rods described in the book L.D. Landau, E.M. Lifshitz, Theory of Elasticity. From this theory, I derived a system of equations: $$-\frac {EIL}{M}\theta ''=F_x\cos (\theta (l))-F_y\sin(\theta (l))$$ $$F_x'=-x\omega^2, F_y'=-g$$ $$x'=\sin(\theta (l)), y'=\cos (\theta (l))$$ This system of equations describes the equilibrium of a round rod under the action of distributed forces and torques. Here $E$ is Young's modulus, $I$ is the moment of inertia, $L$ is the length of the rope, $M$ is the mass of the rope. All derivatives are calculated by the length parameter $l$. The $\theta $ angle is measured from the vertical axis $y$. Boundary conditions are as following: $$\theta (0)=\frac {\pi}{2}, \theta '(L)=0, x(0)=R, y(0)=0, F_x(L)=0, F_y(L)=0$$ Here $R$ is the radius of cylinder. We set $g = 9.81, \omega = 0, \frac {EIL}{M} = 0.00012, L=0.115 m$, then the calculation curve in Fig. 2 above, qualitatively corresponds to the free-hanging rope in Fig. 1 on the left. We set $\omega = 8 \pi $, then the calculated curve in Fig.2 below, qualitatively corresponds to the rotating rope in Fig.1 on the right. Some intermediate cases we will consider as the implementation of experiments. I took a short rope of 6 cm outside the cylinder. This rope (green) has a different texture and thickness. However, in the free state with $\omega = 0$, it takes the form as the first rope, and with $\omega = 40 rad / s$ rises horizontally as the first long rope - see Figure 3 at the top. In this case, at $\frac {EIL}{M} =6*10^{-6}$, the model describes both states of the rope - see Figure 3 below.

In the third experiment, I took a 16 cm long rope that was more rigid than the first two. In the absence of rotation, the rope had an incline of about 34 degrees to horizon line at the free end, see fig.4 at the top left. In the presence of rotation with the speed of $\omega = 50 rad / s$, the rope takes a horizontal position and even slightly above the horizon line - fig.4 in the upper right. In this case, at $\frac {EIL}{M} =0.01$, the model describes both states of the rope - see Figure 4 below.

In the fourth experiment, I took a rope the same as in the third, but 20 cm long. I wanted to check whether standing waves with an amplitude of 1-2 millimeters were formed on the rope. These waves are clearly visible in Figure 2-4 on the calculated curves with $\omega > 0$. I photographed with a flash a rotating rope with $\omega = 125$ opposite the screen, so that the shadow from the rope was visible. In Figure 5, this photo is shown at the top right. Top left is the exact same rope in a free state, and below are calculated curves for two states (rest and rotation).

Finally, in Figure 6 shows photographs of various ropes at a rotation speed of $\omega=3-6 rad / s$. The right photo shows the general view of the ropes used in the experiments. It can be seen that the shape of the ropes is not similar to that obtained in the calculations on the theory of chains. This is due to the fact that all the ropes start from a horizontal surface to which they are attached with a screw. In the lower part of Fig. 6 shows the calculated curves describing the experimental data.

Experiment with soft rope. I took two soft ropes 20 cm and 30 cm long and rotated them at high and low speed. As it turned out, a soft rope of such length at a low speed of rotation bends like a chain. Figures 7 and 8 show the shape of the rope with a length of 20 and 30 cm, respectively, at different speeds of rotation. The calculations are made on the model presented above with various parameters of the stiffness and the angle of contact of the rope with the cylinder.

Soft rope 20 cm long.

Soft rope 30 cm long at different speeds of rotation.