^{1}

### Abstract

We propose a new method of attacking problems in rigid body rotation, focusing on the heavy symmetric top. The technique is a direct extension of the method traditionally applied to the free symmetric top. We write Euler's equations in a frame which is attached to the top and thus shares its entire angular velocity. The structure of the resulting equations is such that it is advantageous to cast and solve them in terms of complex variables (space phasors). Through this formalism, we obtain a direct link between the initial conditions at the time of launch and the subsequent behavior of the top. The insertion of a damping term allows us to further explain the behavior of a top where the pivot is non-ideal and has friction. Finally, we make some suggestions regarding experimental verification of our results.

The author is grateful to Kishore Vaigyanik Protsahan Yojana (KVPY), Government of India, for a generous fellowship.

I. INTRODUCTION

II. EULER'S EQUATIONS IN SPACE PHASOR VARIABLES

III. ILLUSTRATIVE EXAMPLES

A. The free symmetric top (FST)

B. Heavy symmetric top (HST)

C. Damped compensated top (DCT)

D. DCHST with horizontal acceleration

IV. RECAPITULATION AND CONCLUSION

### Key Topics

- Lagrangian mechanics
- 15.0
- Torque
- 14.0
- Friction
- 13.0
- Differential equations
- 3.0
- Gyroscope motion
- 3.0

## Figures

Orthographic views of orthogonal transformations. In each view the axis of rotation comes out of the plane of the paper and is shown as a circle in the relevant style. The xyz-system is fixed in the lab frame with gravity along the −z direction. The origin is at the top's pivot. The first rotation (left panel) is through angle about the z-axis to form the -axes (these axes are not used in the analysis that follows). The next step (middle panel) consists of a rotation through angle θ about the -axis and the set of axes thus formed is denoted by abc. The axis of symmetry of the top lies along the c-axis and the top can be seen in profile view. The final step (right panel) is a rotation through angle ψ about the c-axis to get the dqo-axes. The gray background highlights the fact that the rotation takes place in the plane of the top's spin. The light gray color has been consistently used to indicate the top here and in Fig. 2 ; moreover the d, q, and o axes are shown in a darker shade of gray to highlight the fact that they are fixed in the top's frame. We note that abc and dqo are both principal axes for the top; however, the latter frame carries the entire angular velocity of the top while the former has only a portion of it.

Orthographic views of orthogonal transformations. In each view the axis of rotation comes out of the plane of the paper and is shown as a circle in the relevant style. The xyz-system is fixed in the lab frame with gravity along the −z direction. The origin is at the top's pivot. The first rotation (left panel) is through angle about the z-axis to form the -axes (these axes are not used in the analysis that follows). The next step (middle panel) consists of a rotation through angle θ about the -axis and the set of axes thus formed is denoted by abc. The axis of symmetry of the top lies along the c-axis and the top can be seen in profile view. The final step (right panel) is a rotation through angle ψ about the c-axis to get the dqo-axes. The gray background highlights the fact that the rotation takes place in the plane of the top's spin. The light gray color has been consistently used to indicate the top here and in Fig. 2 ; moreover the d, q, and o axes are shown in a darker shade of gray to highlight the fact that they are fixed in the top's frame. We note that abc and dqo are both principal axes for the top; however, the latter frame carries the entire angular velocity of the top while the former has only a portion of it.

Three-dimensional view of the top. The axes and angles are as shown in Fig. 1 . We should remember that the rotations through and ψ are not coplanar. As described in the text, the rotation is according to the x-convention of Goldstein (Ref. 5 ).

This figure shows the trajectory of the tip of a top satisfying Eqs. (15) and (16) with the specific set of initial conditions described in the text. The trajectory lies on the surface of a sphere, shown here in light gray, with the trajectory itself in dark gray. The lab frame axes L1, L2, and L3 are the suitable axes in which C = 0 and the motion has a simple description. The origin of the axes is at the top's pivot. Axes L1 and L2 lie in the equatorial plane and are shown as dashed lines, while L3 comes directly out of the page through the line of viewing. The angles θ and are defined with respect to L1 L2 L3. A second choice of lab-frame axes has been denoted by primes. Of these, is shown as an axis that makes a certain inclination with L3. The trajectory is very difficult to describe in terms of Euler angles defined with respect to the primed axes.

This figure shows the trajectory of the tip of a top satisfying Eqs. (15) and (16) with the specific set of initial conditions described in the text. The trajectory lies on the surface of a sphere, shown here in light gray, with the trajectory itself in dark gray. The lab frame axes L1, L2, and L3 are the suitable axes in which C = 0 and the motion has a simple description. The origin of the axes is at the top's pivot. Axes L1 and L2 lie in the equatorial plane and are shown as dashed lines, while L3 comes directly out of the page through the line of viewing. The angles θ and are defined with respect to L1 L2 L3. A second choice of lab-frame axes has been denoted by primes. Of these, is shown as an axis that makes a certain inclination with L3. The trajectory is very difficult to describe in terms of Euler angles defined with respect to the primed axes.

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