Ellipsoid of inertia

Fig. 1: Ellipsoid of inertia (blue network) and main axes of inertia (dashed blue) of a body (not shown) and an axis of rotation in the global direction (dashed black) ${\displaystyle z}$

The ellipsoid of inertia of a rigid body is a closed surface in the shape of an ellipsoid , which has a distance from the center in each direction that is a measure of the moment of inertia of the body when rotating about that direction: the moment of inertia is equal to the reciprocal of the square of the distance (see Fig. 1). As a result, the three semi-axes of the ellipsoid of inertia are parallel to the main axes of inertia of the body, and their lengths are given by the reciprocal of the square root of the corresponding main moments of inertia .

The ellipsoid of inertia is useful in considering the inertia properties of the body when it is rotated about any axis. In a body-fixed coordinate system it remains constant, i.e. i.e. it always rotates with the body.

calculation

Like every symmetrical tensor of the 2nd order in three dimensions, the inertia tensor Θ can be assigned a surface. It is formed by the endpoints of the vectors that satisfy the following equation: ${\displaystyle {\vec {x}}}$

${\displaystyle {\begin{aligned}1={\vec {x}}\cdot \mathbf {\Theta } \cdot {\vec {x}}=&{\begin{pmatrix}x\\y\\z\end{pmatrix}}\cdot {\begin{pmatrix}\Theta _{xx}&\Theta _{xy}&\Theta _{xz}\\\Theta _{xy}&\Theta _{yy}&\Theta _{yz}\\\Theta _{xz}&\Theta _{yz}&\Theta _{zz}\end{pmatrix}}\cdot {\begin{pmatrix}x\\y\\z\end{pmatrix}}\\=&\Theta _{xx}x^{2}+\Theta _{yy}y^{2}+\Theta _{zz}z^{2}+2\Theta _{xy}xy+2\Theta _{xz}xz+2\Theta _{yz}yz\end{aligned}}}$

Here x, y and z denote the components of the vector and Θ _{xx, xy, ...} the components of the inertia tensor with respect to an arbitrarily oriented orthonormal basis . In the main axis of inertia system, short main axis system, the inertia tensor becomes diagonal and the result is: ${\displaystyle {\vec {x}}}$

${\displaystyle {\begin{aligned}1={\vec {x}}\cdot \mathbf {\Theta } \cdot {\vec {x}}=&{\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}\cdot {\begin{pmatrix}\Theta _{1}&0&0\\0&\Theta _{2}&0\\0&0&\Theta _{3}\end{pmatrix}}\cdot {\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}\\=&\Theta _{1}x_{1}^{2}+\Theta _{2}x_{2}^{2}+\Theta _{3}x_{3}^{2}={\frac {x_{1}^{2}}{{\sqrt {\tfrac {1}{\Theta _{1}}}}^{2}}}+{\frac {x_{2}^{2}}{{\sqrt {\tfrac {1}{\Theta _{2}}}}^{2}}}+{\frac {x_{3}^{2}}{{\sqrt {\tfrac {1}{\Theta _{3}}}}^{2}}}\end{aligned}}}$

The components x _1,2,3 and the main moments of inertia Θ _1,2,3 relate to the main axis system fixed to the body . The inertia tensor is positive definite because the rotational energy is forever positive. Hence the surface is a three-axis ellipsoid . ${\displaystyle {\tfrac {1}{2}}{\vec {\omega }}\cdot \mathbf {\Theta } \cdot {\vec {\omega }}}$ ${\displaystyle {\vec {\omega }}\neq {\vec {0}}}$

In a frame of reference in which the body rotates, the components of the inertia tensor Θ _{xx, xy, ...} depend on time. The ellipsoid of inertia remains in alignment with the body. The six independent components of the inertia tensor correspond to the three main moments of inertia and the orientation of the main axes of inertia, i.e. the shape and orientation of the ellipsoid.

With the inertia tensor, the moments of inertia J are calculated with respect to any axis of rotation through the ellipsoid center in the direction of the unit vector (of length one and therefore written with a hat) according to . For a vector that points from the center of the ellipsoid to the point of intersection of the axis of rotation with the ellipsoid of inertia and has the magnitude x , this results ${\displaystyle {\hat {e}}}$ ${\displaystyle J={\hat {e}}\cdot \mathbf {\Theta } \cdot {\hat {e}}}$ ${\displaystyle {\vec {x}}=x{\hat {e}}}$

${\displaystyle 1={\vec {x}}\cdot \mathbf {\Theta } \cdot {\vec {x}}=x^{2}{\hat {e}}\cdot \mathbf {\Theta } \cdot {\hat {e}}=x^{2}J\quad \rightarrow \quad x={\sqrt {\frac {1}{J}}}\,.}$

The axis of rotation intersects the ellipsoid at a distance from the center of the ellipsoid. ${\displaystyle {\sqrt {\tfrac {1}{J}}}}$

The main moments of inertia Θ _1,2,3 satisfy the triangle inequalities . So that an ellipsoid with the axes a, b and c can be an ellipsoid of inertia, it must be possible to form a triangle from lines of lengths 1 / a ² , 1 / b ² and 1 / c ² .

Special bodies

The length of the semiaxes of the ellipsoid of inertia are inversely proportional to the square root of the main moments of inertia. Clearly, an ellipsoid of inertia stretched in one direction corresponds to a body that is compressed in this direction, and vice versa. With homogeneous density distribution and rotation around the center of mass, the following applies:

• Asymmetrical tops have a "real" ellipsoid as an ellipsoid of inertia, since Θ ₁ ≠ Θ ₂ ≠ Θ ₃ ≠ Θ ₁ . Examples are the cuboid with three unequal sides or angled molecules such as the water molecule H ₂ O. The ellipsoid of inertia of a brick has the shape of a strongly rounded piece of soap that lies across the brick (shortest central axis of the ellipsoid parallel to the longest axis of symmetry of the body, and vice versa ).
• Symmetrical tops have an ellipsoid of rotation as an ellipsoid of inertia, since two main moments of inertia are equal, e.g. B. Θ ₁ = Θ ₂ . In the case of rotationally symmetrical bodies , the axis of symmetry is always a main axis of inertia, the two main moments of inertia about any axes perpendicular thereto are the same. Examples: circular cylinders , linear molecules.
• Even bodies with n -fold rotational symmetry have an ellipsoid of revolution as an ellipsoid of inertia, because an ellipsoid cannot reproduce a rotational symmetry higher than . Examples: pillars or pyramids with equilateral triangular or square cross-sections, including tetrahedra etc. ${\displaystyle n=3}$ ${\displaystyle n=2}$
  • In the elongated or prolate top , Θ ₁ = Θ ₂ > Θ ₃ and therefore its ellipsoid of inertia is a cigar-shaped ellipsoid of revolution elongated in the axis of symmetry . ${\displaystyle {\hat {e}}_{3}}$
  • For a flattened or oblate top , Θ ₁ = Θ ₂ <Θ ₃ and therefore its ellipsoid of inertia is an ellipsoid of revolution compressed in the axis of symmetry . Examples: puck , approximately the flattened earth . ${\displaystyle {\hat {e}}_{3}}$
• Spherical tops or spherical tops have a sphere as an ellipsoid of inertia, since Θ ₁ = Θ ₂ = Θ ₃ . If a body has the same moments of inertia with respect to three different axes, the ellipsoid of inertia is a sphere. As a result, the moment of inertia is the same with respect to each axis. However, the shape of the body does not have to correspond to that of a sphere: if the density distribution is homogeneous, point symmetry such as that of a cube or other regular bodies is sufficient . In addition, irregularly shaped bodies can also be spherical tops.

If the density distribution is inhomogeneous, the shape of the ellipsoid of inertia cannot easily be deduced from the external shape.

Other ellipsoids associated with the rotary motion

Fig. 2: Rigid body (gray) with ellipsoid of inertia, twist and mass ellipsoid (blue, yellow or green), all of which are scaled on 2-axes of the same length here

In addition to the ellipsoid of inertia, other ellipsoids are important for the rotary motion, see Fig. 2:

• The energy ellipsoid, which is also called "Poinsotellipsoid" or "Poinsot surface" according to Louis Poinsot , contains all angular velocities that correspond to the same rotational energy for a given body . The energy ellipsoid emerges from the inertial ellipsoid through centric stretching . The movement of rigid bodies rotating without force can be visualized with Poinsot's construction using the energy ellipsoid.
• The twist ellipsoid is the geometric location of all angular velocities that correspond to the same angular momentum square. The twist ellipsoid is slimmer than the inertial ellipsoid in every respect and scales with the amount of angular momentum. With a given rotational energy, the size of the twist ellipsoid is limited upwards and downwards.
• The MacCullagh ellipsoid is the geometric location of all angular momentum that corresponds to the same rotational energy. The MacCullagh ellipsoid is in a certain way reciprocal to the energy ellipsoid, because corresponding axes have reciprocal lengths. A flattened MacCullagh ellipsoid belongs to an elongated energy ellipsoid and vice versa.
• The mass ellipsoid is a homogeneous, ellipsoid-shaped body that has the same mass and the same ellipsoid of inertia as a given body.

The ellipsoid of inertia and mass of the body are properties of a (rigid) body that are not influenced by any movements that may occur in the body, but are otherwise not generally similar. All of these ellipsoids are aligned with the body with its main axes of inertia as axes of symmetry.

Energy ellipsoid

The energy ellipsoid for a given rotational energy has the same geometrical shape and orientation as the inertial ellipsoid, whereby the distance of the points on the energy ellipsoid from the center is now given by the amount of the angular velocity that belongs to this rotational energy. This area is formed by the endpoints of the vectors which, with fixed rotational energy E _rot, satisfy the following equation: ${\displaystyle {\vec {\omega }}}$

${\displaystyle E_{\text{rot}}={\frac {1}{2}}{\vec {\omega }}\cdot \mathbf {\Theta } \cdot {\vec {\omega }}}$

This area corresponds to an ellipsoid of inertia stretched by the factor , because the defining formulas merge into one another when inserted. ${\displaystyle {\sqrt {2E_{\text{rot}}}}}$ ${\displaystyle {\vec {\omega }}={\sqrt {2E_{\text{rot}}}}{\vec {x}}}$

In a Cartesian coordinate system with xyz axes, written out component by component, the equation is

${\displaystyle {\begin{aligned}E_{\text{rot}}=&{\frac {1}{2}}{\begin{pmatrix}\omega _{x}\\\omega _{y}\\\omega _{z}\end{pmatrix}}\cdot {\begin{pmatrix}\Theta _{xx}&\Theta _{xy}&\Theta _{xz}\\\Theta _{xy}&\Theta _{yy}&\Theta _{yz}\\\Theta _{xz}&\Theta _{yz}&\Theta _{zz}\end{pmatrix}}\cdot {\begin{pmatrix}\omega _{x}\\\omega _{y}\\\omega _{z}\end{pmatrix}}\\=&{\frac {1}{2}}(\Theta _{xx}\omega _{x}^{2}+\Theta _{yy}\omega _{y}^{2}+\Theta _{zz}\omega _{z}^{2}+2\Theta _{xy}\omega _{x}\omega _{y}+2\Theta _{xz}\omega _{x}\omega _{z}+2\Theta _{yz}\omega _{y}\omega _{z})\\=:&f({\vec {\omega }})\,.\end{aligned}}}$

In the main axis system this quadratic form is simplified (where ω _{1,2,3} are} the components of the angular velocity in the main inertia system) to ${\displaystyle {\vec {\omega }}}$

${\displaystyle E_{\text{rot}}={\frac {1}{2}}(\Theta _{1}\omega _{1}^{2}+\Theta _{2}\omega _{2}^{2}+\Theta _{3}\omega _{3}^{2})}$

or transformed to

${\displaystyle 1={\frac {\omega _{1}^{2}}{{\sqrt {\tfrac {2E_{\text{rot}}}{\Theta _{1}}}}^{2}}}+{\frac {\omega _{2}^{2}}{{\sqrt {\tfrac {2E_{\text{rot}}}{\Theta _{2}}}}^{2}}}+{\frac {\omega _{3}^{2}}{{\sqrt {\tfrac {2E_{\text{rot}}}{\Theta _{3}}}}^{2}}}\,.}$

Poinsot construction of the direction of the angular momentum

If one considers the rotational energy E _rot as a function in the three-dimensional space of the angular velocities , then the angular momentum is precisely the gradient of this function. In the main axis system with the basis vectors, the following applies ${\displaystyle {\vec {\omega }}}$ ${\displaystyle {\vec {L}}}$ ${\displaystyle {\hat {e}}_{k}}$

${\displaystyle \operatorname {grad} _{\vec {\omega }}(E_{\text{rot}})\,=\,{\begin{pmatrix}{\frac {\partial E_{\text{rot}}}{\partial \omega _{1}}}&{\frac {\partial E_{\text{rot}}}{\partial \omega _{2}}}&{\frac {\partial E_{\text{rot}}}{\partial \omega _{3}}}\end{pmatrix}}\,=\,\Theta _{1}\omega _{1}{\hat {e}}_{1}+\Theta _{2}\omega _{2}{\hat {e}}_{2}+\Theta _{3}\omega _{3}{\hat {e}}_{3}\ =\,{\vec {L}}\ .}$

Since the gradient of a function is perpendicular to the surface of constant function value at every point, the angular momentum associated with an angular velocity is parallel to the perpendicular on the energy ellipsoid at the point . ${\displaystyle {\vec {\omega }}}$ ${\displaystyle {\vec {\omega }}}$

Fig. 3: Section through an energy ellipsoid along two main axes of inertia, with the main moments of inertia Θ ₁ and Θ ₂

The angular momentum is thus parallel to the normal of the energy ellipsoid at the point where the tip of the angular velocity vector touches the ellipsoid (see Fig. 3). It can thus be seen that

• ${\displaystyle {\vec {\omega }}}$and are only parallel along the main axes of inertia , ${\displaystyle {\vec {L}}}$
• Angular momentum and angular velocity always include an acute angle (<90 °), because , and ${\displaystyle 2E_{\text{rot}}={\vec {\omega }}\cdot {\vec {L}}>0}$
• the increase in rotational energy is maximum when the angular velocity increases in the direction of the angular momentum, because . ${\displaystyle \mathrm {d} E_{\text{rot}}={\vec {L}}\cdot \mathrm {d} {\vec {\omega }}}$

In the force-free case, the angular momentum and the rotational energy are constant and because of this , the component of the angular velocity in the direction of the angular momentum is also constant. The tangential plane to the energy ellipsoid at the location of the current angular velocity is therefore fixed and the angular velocity moves on so-called Herpolhodia in this plane. In the body's main inertia system, the angular velocity traces curves called "polhodia", which are the intersection of angular momentum and energy ellipsoid. (For more on this see under Poinsot's construction ). ${\displaystyle 2E_{\text{rot}}={\vec {\omega }}\cdot {\vec {L}}}$

Swirl ellipsoid

The angular velocities, which all deliver the same angular momentum square at a certain point in time, also define an ellipsoid, the helix ellipsoid: ${\displaystyle L^{2}:={\vec {L}}\cdot {\vec {L}}}$

${\displaystyle L^{2}={\vec {\omega }}\cdot \mathbf {\Theta } \cdot \mathbf {\Theta } \cdot {\vec {\omega }}=\Theta _{1}^{2}\omega _{1}^{2}+\Theta _{2}^{2}\omega _{2}^{2}+\Theta _{3}^{2}\omega _{3}^{2}\quad \rightarrow \quad 1={\frac {\omega _{1}^{2}}{\left({\frac {L}{\Theta _{1}}}\right)^{2}}}+{\frac {\omega _{2}^{2}}{\left({\frac {L}{\Theta _{2}}}\right)^{2}}}+{\frac {\omega _{3}^{2}}{\left({\frac {L}{\Theta _{3}}}\right)^{2}}}\,.}$

The twist ellipsoid is therefore slimmer than the inertial ellipsoid, see Fig. 2: ${\displaystyle l_{L,i}={\frac {L}{\Theta _{i}}}.}$

The angular velocity at a certain point in time lies both on this ellipsoid and on the energy ellipsoid. So that both ellipsoids can have points in common, must at any point in time

${\displaystyle 2\Theta _{1}E_{\text{rot}}\leq L^{2}\leq 2\Theta _{3}E_{\text{rot}}}$ or ${\displaystyle {\frac {L^{2}}{2\Theta _{3}}}\leq E_{\text{rot}}\leq {\frac {L^{2}}{2\Theta _{1}}}}$

apply if, as usual, the main moments of inertia are arranged according to Θ ₁ <Θ ₂ <Θ ₃ . Because a point that lies on both ellipsoids must meet the conditions

${\displaystyle {\begin{aligned}1=&{\frac {1}{L^{2}}}(\Theta _{1}^{2}\omega _{1}^{2}+\Theta _{2}^{2}\omega _{2}^{2}+\Theta _{3}^{2}\omega _{3}^{2})={\frac {1}{2E_{\text{rot}}}}(\Theta _{1}\omega _{1}^{2}+\Theta _{2}\omega _{2}^{2}+\Theta _{3}\omega _{3}^{2})\\\rightarrow 0=&(2E_{\text{rot}}\Theta _{1}-L^{2})\Theta _{1}\omega _{1}^{2}+(2E_{\text{rot}}\Theta _{2}-L^{2})\Theta _{2}\omega _{2}^{2}+(2E_{\text{rot}}\Theta _{3}-L^{2})\Theta _{3}\omega _{3}^{2}\\=&2\left(E_{\text{rot}}-{\frac {L^{2}}{2\Theta _{1}}}\right)\Theta _{1}^{2}\omega _{1}^{2}+2\left(E_{\text{rot}}-{\frac {L^{2}}{2\Theta _{2}}}\right)\Theta _{2}^{2}\omega _{2}^{2}+2\left(E_{\text{rot}}-{\frac {L^{2}}{2\Theta _{3}}}\right)\Theta _{3}^{2}\omega _{3}^{2}\end{aligned}}}$

fulfill. In the last two equations, all but the parentheses are zero or positive. For a nontrivial solution to exist, the smallest expression in brackets in both equations must not be positive and the largest must not be negative. With the assumed proportions of the main moments of inertia, this ensures the above limits for the angular momentum square and the rotational energy. Then the rotational energy and the amount of angular momentum are compatible with a rotational movement of the body under consideration.

For a given rotational energy, a rotation around the main axis of inertia with the smallest main moment of inertia has the smallest and a rotation around the main axis of inertia with the largest main moment of inertia has the largest amount of angular momentum.

Conversely, for a given amount of angular momentum, a rotation about the main axis of inertia with the smallest main moment of inertia has the largest and a rotation about the main axis of inertia with the largest main moment of inertia has the smallest rotational energy. This is why the axis of rotation will move in the direction of the 3-axis during dissipative processes ( air resistance , friction ).

MacCullagh ellipsoid

Fig. 4: Section through a MacCullagh ellipsoid along two main axes of inertia with the main moments of inertia Θ ₁ and Θ ₂

The MacCullagh ellipsoid, named after James MacCullagh , is the geometric location of all end points of the angular momentum that lead to the same rotational energy. The MacCullagh ellipsoid is therefore the analogue of the energy ellipsoid in angular momentum space: ${\displaystyle {\vec {L}}}$

${\displaystyle E_{\text{rot}}={\frac {1}{2}}{\vec {L}}\cdot \mathbf {\Theta } ^{-1}\cdot {\vec {L}}={\frac {L_{1}^{2}}{2\Theta _{1}}}+{\frac {L_{2}^{2}}{2\Theta _{2}}}+{\frac {L_{3}^{2}}{2\Theta _{3}}},}$

where the main axis representation of the angular momentum was used in the equation on the right. In this system the ellipsoid thus has the equation

${\displaystyle 1={\frac {L_{1}^{2}}{{\sqrt {2E_{\text{rot}}\Theta _{1}}}^{2}}}+{\frac {L_{2}^{2}}{{\sqrt {2E_{\text{rot}}\Theta _{2}}}^{2}}}+{\frac {L_{3}^{2}}{{\sqrt {2E_{\text{rot}}\Theta _{3}}}^{2}}}.}$

Like the other ellipsoids, it is solid and aligned along the main axes. The MacCullagh ellipsoid is, so to speak, reciprocal to the energy ellipsoid, because the product of the semiaxes of the energy ellipsoid and the MacCullagh ellipsoid is the same on all main axes: ${\displaystyle l_{E,i}={\sqrt {\tfrac {2E_{\text{rot}}}{\Theta _{i}}}}}$ ${\displaystyle l_{C,i}={\sqrt {2E_{\text{rot}}\Theta _{i}}}}$

${\displaystyle l_{E,1}l_{C,1}=l_{E,2}l_{C,2}=l_{E,3}l_{C,3}=2E_{\text{rot}}.}$

If the energy ellipsoid is flattened, the MacCullagh ellipsoid is elongated and vice versa.

In the force-free movement of a rigid body, the angular momentum and the rotational energy are constant. The body is then only allowed to rotate around the origin where the fixed end point of the angular momentum touches its MacCullagh ellipsoid and the swirl sphere with the radius . Analogous to Poinsot's construction, the angular velocity results as a gradient in the angular momentum space: ${\displaystyle |{\vec {L}}|}$

${\displaystyle \operatorname {grad} _{\vec {L}}(E_{\text{rot}})\,=\,{\begin{pmatrix}{\frac {\partial E_{\text{rot}}}{\partial L_{1}}}&{\frac {\partial E_{\text{rot}}}{\partial L_{2}}}&{\frac {\partial E_{\text{rot}}}{\partial L_{2}}}\end{pmatrix}}\,=\,{\frac {L_{1}}{\Theta _{1}}}{\hat {e}}_{1}+{\frac {L_{2}}{\Theta _{2}}}{\hat {e}}_{2}+{\frac {L_{3}}{\Theta _{3}}}{\hat {e}}_{3}\ =\,{\vec {\omega }}}$

Mass ellipsoid

Fig. 5: Ellipsoid with three unequal semi-axes

For every rigid body there is an ellipsoidal body as in Fig. 5, the mass ellipsoid , which has the same inertial properties (mass and inertia tensor) as the body itself. The mass ellipsoid and the inertial ellipsoid have the same axes of symmetry, but are otherwise mostly not similar. If the medium-length semi-axes coincide after a suitable scaling, the largest semi-axis of the inertial ellipsoid will be smaller, but the smallest will be larger than the corresponding one of the mass ellipsoid, see Fig. 2.

Because with homogeneous density distribution has an ellipsoidal body mass as well as the semi-axes , and in -, - or direction the principal moments of inertia ${\displaystyle m}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle c}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle z}$

${\displaystyle {\begin{aligned}\Theta _{x}=&{\frac {m}{5}}(b^{2}+c^{2})\\\Theta _{y}=&{\frac {m}{5}}(a^{2}+c^{2})\\\Theta _{z}=&{\frac {m}{5}}(a^{2}+b^{2})\end{aligned}}}$

or the semi-axes for given main moments of inertia

${\displaystyle {\begin{aligned}a=&{\sqrt {{\frac {5}{2m}}(\Theta _{y}+\Theta _{z}-\Theta _{x})}}\\b=&{\sqrt {{\frac {5}{2m}}(\Theta _{x}+\Theta _{z}-\Theta _{y})}}\\c=&{\sqrt {{\frac {5}{2m}}(\Theta _{x}+\Theta _{y}-\Theta _{z})}}\,.\end{aligned}}}$

Because the principal moments of inertia satisfy the triangle inequalities , every body has an ellipsoid of mass. In contrast to the ellipsoid of inertia, the semi-axes of the ellipsoid of mass can have any ratio to one another, so they do not have to satisfy the triangle inequalities. The semi-axes of the ellipsoid of inertia behave like

${\displaystyle {\frac {1}{\sqrt {\Theta _{z}}}}:{\frac {1}{\sqrt {\Theta _{y}}}}:{\frac {1}{\sqrt {\Theta _{x}}}}={\frac {1}{\sqrt {a^{2}+b^{2}}}}:{\frac {1}{\sqrt {a^{2}+c^{2}}}}:{\frac {1}{\sqrt {b^{2}+c^{2}}}}=\underbrace {\sqrt {\frac {1+{\frac {a^{2}}{c^{2}}}}{1+{\frac {a^{2}}{b^{2}}}}}} _{=:p}c:b:\underbrace {\sqrt {\frac {1+{\frac {c^{2}}{a^{2}}}}{1+{\frac {c^{2}}{b^{2}}}}}} _{=:q}a\,.}$

If is, then is and and therefore . The largest semiaxis of the ellipsoid of inertia is therefore relatively smaller, the smallest but relatively larger than the corresponding one of the mass ellipsoid, see also Fig. 2. ${\displaystyle a>b>c}$ ${\displaystyle {\tfrac {a^{2}}{c^{2}}}>{\tfrac {a^{2}}{b^{2}}}}$ ${\displaystyle {\tfrac {c^{2}}{a^{2}}}<{\tfrac {c^{2}}{b^{2}}}}$ ${\displaystyle p>1>q>0}$

See also

• Euler gyro

Individual evidence

1. ↑ Othmar Marti: Roundabout. Institute for Experimental Physics at Ulm University , accessed on June 11, 2017 . 
2. ↑ Magnus (1971), p. 61 ff.
3. ↑ Grammel (1950), p. 27 f.

literature

• Wolfgang Demtröder : Experimental Physics . Mechanics and warmth. 4th revised and updated edition. tape 1 . Springer Verlag, Berlin et al. 2006, ISBN 3-540-26034-X .
• Wolfgang Nolting : Basic course in theoretical physics . Classic mechanics. 8th edition. tape 1 . Springer Verlag, Berlin et al. 2006, ISBN 3-540-34832-8 (Springer textbook).
• K. Magnus : Kreisel: Theory and Applications . Springer, 1971, ISBN 978-3-642-52163-8 , pp. 61 ff . ( Limited preview in Google Book Search [accessed February 20, 2018]).
• R. Grammel : The top . Its theory and its applications. 2. revised Edition volume 1 , The theory of the gyro. Springer, Berlin, Göttingen, Heidelberg 1950, DNB  451641280 ( archive.org ).