Goryachev Chaplygin roundabout

The Gorjatschew-Tschaplygin-Kreisel ( English Goryachev-Chaplygin top ) according to Dmitri Nikanorowitsch Gorjatschew and Sergei Alexejewitsch Tschaplygin is one of the few cases in the gyro theory in which the Euler gyro equations can be solved analytically. The three main moments of inertia A, B and C of the heavy gyroscope fulfill the condition A = B = 4C and the center of gravity lies in the plane spanned by the main axes belonging to A and B. The top is thus a modification of the Kovalevskaya top .

The time integration of the gyroscopic equations is possible because, in addition to the total energy and the angular momentum L _z in the perpendicular direction, there is a third kinetic conservation quantity f, see #integrals of the motion , which is only constant if L _z initially disappears. As with the Kovalevskaya gyro solutions are hyper- or ultra elliptic functions but the complexity of the #Bifurkationsdiagramms does not reach that of Kowalewskajas case. The Goryachev-Chaplygin gyroscope is characterized by (quasi-) periodic pendulum movements where f = 0.

Goryachev discussed exactly these movements in 1899 and Chaplygin was able to give the general solution of the equations of motion in 1901.

phenomenology

Pendulum movements

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  Fig. 1

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  Fig. 2

The Goryachev-Chaplygin gyroscope is characterized by pendulum movements in which the integral f vanishes. In any case, this is the case when the gyro periodically oscillates around its 2- or 3-axis, whereby rollover can also take place, so that the gyro starts rotating. Other movements with f = 0 are spatial pendulum movements, s. Animations.

Bifurcation diagram

Fig. 3: Bifurcation diagram of the Goryachev-Chaplygin gyro

Bifurcation (branching) is the phenomenon that a system can branch into different states at a critical point. A vertically standing rod can fall over to the left or right in the event of a small disturbance if it is somehow bound to a plane. The dividing line between the two paths is a separatrix, one of which also exists in the asymmetrical Euler top . The movements along the separating dies are often unstable. The bifurcation diagram of the Goryachev-Chaplygin gyro has only one separatrix (blue in the picture) and is therefore less complex than that of the Kovalevskaya gyro .

The critical points in the Goryachev-Chaplygin gyro appear at certain values ​​of the total energy h and the integral f, see #integrals of motion . The bifurcation diagram consists of two areas, with combinations of parameters appearing in the gray area O, which the top cannot use at L _z = 0. The white areas are accessible to the roundabout. The pendulum movements described above take place on the ordinate with f = 0.

The black edge of the inaccessible area O defines a top that swings or rotates periodically through the bottom dead center, depending on h <c or h> c, see the following section.

Movements with minimal energy

Fig. 4: Movement with a given f and minimal energy

The condition that the angular momentum L _z disappears in the perpendicular direction restricts the gyro's range of motion. Rotations with L _z  ≠ 0 can also occur, but these are not considered here. Uniform rotations around a main vertical axis, as occurs with the Lagrange top and the Kowalewskaja top , are thus excluded. If a main axis is at some point perpendicular, then the axis of rotation must be perpendicular to it at that moment. An example of such a movement with minimal energy is shown in Fig. 4.

Movements on the Separatrix

Fig. 5: Movement on the separatrix

On the separatrix, the 1-axis moves through top dead center and, as with the movements with minimal energy above, the axis of rotation must be perpendicular to the 1-axis at the time, see Fig. 5. The movement on the separatrix is ​​unstable.

Pseudoregular precession

→ Main article : Pseudoregular precession

If the angular velocity around the center of gravity axis from the support point to the center of gravity is very high, then the top moves analogously to the pseudoregular precession of the Lagrange top .

Solution of the equations of motion

Euler-Poisson equations

→ Main article : Euler-Poisson equations

For reasons of symmetry, the first main axis of inertia belonging to A can be selected in such a way that the center of gravity of the top is on it. The Euler-Poisson equations for the Goryachev-Chaplygin gyro specialize in this

${\displaystyle {\begin{aligned}4{\dot {p}}=&3qr\\4{\dot {q}}=&-3pr+cn_{3}\\{\dot {r}}=&-cn_{2}\\{\dot {n}}_{1}=&rn_{2}-qn_{3}\\{\dot {n}}_{2}=&pn_{3}-rn_{1}\\{\dot {n}}_{3}=&qn_{1}-pn_{2}\end{aligned}}}$

In it are in the main axis system

• p, q, r = ω _{1,2,3 are} the components of the angular velocity ,
• n _{1,2,3 are} the components of the unit vector pointing antiparallel upwards to the weight force ,
• ${\displaystyle c={\tfrac {mgs_{1}}{C}}}$a parameter with dimension T ^–2 , which summarizes the effect of the weight force,
  • m is the mass of the top,
  • g is the acceleration due to gravity and
  • s _{1 is} the distance of the center of gravity from the support point.

As with the Kovalevskaya gyroscope, scaling the time with √ c and the angular velocities with √ c ^–1 results in equations of motion with c  = 1, so that they no longer have a free parameter. Mathematically it is sufficient to only consider this case c  = 1.

Integrals of motion

As with every heavy top, the norm of the direction vector of the weight force , the angular momentum in the perpendicular direction and the total energy E are constant: ${\displaystyle |{\hat {n}}|}$ ${\displaystyle {\vec {L}}}$

${\displaystyle {\begin{aligned}{\hat {n}}\cdot {\hat {n}}=&n_{1}^{2}+n_{2}^{2}+n_{3}^{2}=1={\text{const.}}\\l:=&{\frac {{\vec {L}}\cdot {\hat {n}}}{C}}=4pn_{1}+4qn_{2}+rn_{3}={\text{const.}}\\h:=&{\frac {E}{C}}=2(p^{2}+q^{2})+{\frac {1}{2}}r^{2}+c\,n_{1}={\text{const.}}\end{aligned}}}$

These constants are called integrals in gyro theory , the first two also Casimir invariants . In analytical mechanics , the total energy is also referred to as the Hamilton function , which is why it is named with h . In the Goryachev-Chaplygin gyroscope, 1 vanishes by assumption and then there is a fourth rational integral

${\displaystyle f=(p^{2}+q^{2})r-cpn_{3}}$

Because according to the # Euler-Poisson equations and the product rule , the time derivative proves

${\displaystyle {\dot {f}}=(2p{\dot {p}}+2q{\dot {q}})r+(p^{2}+q^{2}){\dot {r}}-c{\dot {p}}n_{3}-cp{\dot {n}}_{3}=-{\frac {cq}{4}}l}$

as proportional to the perpendicular momentum, which is zero by assumption.

If f  = 0, then there is another integral

${\displaystyle a^{n}={\frac {p^{2}+q^{2}}{\sqrt[{3}]{p^{2}}}}}$

because the time derivative turns out to be proportional to f according to the quotient rule in combination with the # Euler-Poisson equations :

${\displaystyle na^{n-1}{\dot {a}}={\frac {(2p{\dot {p}}+2q{\dot {q}})p^{\frac {2}{3}}-(p^{2}+q^{2}){\frac {2}{3}}p^{-{\frac {1}{3}}}{\dot {p}}}{p^{\frac {4}{3}}}}=-{\frac {q}{2p^{\frac {5}{3}}}}f}$

Goryachev used n =  ⁴ ⁄ ₃ , which gives a the dimension of an angular velocity, but n = 1 is also used.

Equilibrium solutions

The condition that the angular momentum l disappears in the perpendicular direction restricts the possibilities of movement of the top. Rotations with l ≠ 0 can also occur, but these are not considered here. If a main axis is at some point perpendicular, then the axis of rotation must be perpendicular to it at that moment.

Relative equilibria are fixed points of the # Euler-Poisson equations that occur when all quantities are constant . If the condition l = 0 is assumed, there are no equilibria that deviate from standstill.

Because from the third Euler-Poisson equation follows n ₂  = 0 and from the sixth n ₁  q = 0, where two cases have to be distinguished.

1. If n ₁  = 0, then n ₃  = ± 1 and the fourth and fifth Euler-Poisson equations result in p = q = 0. However, this contradicts the second equation. So n ₁  ≠ 0 must be.
2. If q = 0, because of the fifth equation, r = n ₃  p / n ₁ and l = 0 forces p = r = 0.

Without the condition l = 0, equilibria analogous to that of the Kovalevskaya gyro are established. The rotation around the vertical 1-axis is stable when the center of gravity is below the support point and unstable when it is above it.

Movements with minimal energy

On the black border of the area accessible to the gyro in the #bifurcation diagram , the 1-axis of the gyro runs through bottom dead center. For any f ∈ ℝ is the total energy

${\displaystyle \textstyle h={\tfrac {3}{2}}{\sqrt[{3}]{(2f)^{2}}}-c}$

and at bottom dead center results

${\displaystyle {\begin{aligned}h=&2q^{2}+{\frac {1}{2}}r^{2}-c,&f=&q^{2}r\\n_{1}=&-1,&n_{2}&=n_{3}=0\\p=&0,&q=&{\sqrt[{3}]{\frac {2f}{\sqrt {8}}}},&r=&{\sqrt[{3}]{2f}}\end{aligned}}}$

On the Separatrix, the gyro runs through top dead center, so it deviates there

${\displaystyle n_{1}=1\quad {\text{und}}\quad h=2q^{2}+{\frac {1}{2}}r^{2}+c={\frac {3}{2}}{\sqrt[{3}]{(2f)^{2}}}+c}$

present.

Goryachev's drafting

Goryachev started out from the integral a and showed in his equations (7) and (8) that this exists when the #integrals of the motion f and l are zero. With the help of these constants he determined the area that is generated by the angular velocity in the body-fixed reference system. To do this, he eliminated the direction _cosines n _1,2,3 with the help of the integrals h, f and l , and the sum of their squares, which is equal to one, leads to his equation (13):

(a⁴r² + 8a⁴ρ² - 2hρ⁴) ² = λρ⁴ (a⁴ - ρ⁴)

Here ρ² = p² + q² and λ = 4 (16a⁴ + 4c² - h²) are a constant of the movement. This equation gives r as a function of ρ. By the integral a is also

p² = ρ⁶ / a⁴

q² = ρ² (a⁴-ρ⁴) / a⁴

so that the angular velocity is completely represented as a function of ρ. With the elimination of the direction cosines, identities arose which also lead the n _1,2,3 back to ρ. The trajectory of the top parameterized with ρ is thus fixed in the body-fixed reference system.

Tschaplygin's solution

Tschaplygin succeeded in reducing the solution functions to a system of Abel-Jacobi equations. For this purpose, variables u and v are introduced:

4 (p² + q²) = uv and r = u - v

With the abbreviations

${\displaystyle {\begin{array}{lcllcl}U&=&u^{3}-2hu-4f,&V&=&v^{3}-2hv+4f\\U_{1}^{2}&=&U-2cu,&V_{1}^{2}&=&V-2cv\\-U_{2}^{2}&=&U+2cu,&-V_{2}^{2}&=&V+2cv\end{array}}}$

can all state variables using

${\displaystyle {\begin{array}{lcllcl}8cp&=&U_{1}V_{2}-U_{2}V_{1},&2cn_{1}&=&-{\frac {U+V}{u+v}}\\8cq&=&U_{1}V_{1}+U_{2}V_{2},&2cn_{2}&=&-{\frac {U_{1}U_{2}-V_{1}V_{2}}{u+v}}\\r&=&u-v,&2cn_{3}&=&-{\frac {U_{1}V_{2}+U_{2}V_{1}}{u+v}}\end{array}}}$

and the time derivatives

${\displaystyle {\frac {\mathrm {d} u}{\mathrm {d} t}}={\frac {U_{1}U_{2}}{2(u+v)}}\quad {\text{und}}\quad {\frac {\mathrm {d} v}{\mathrm {d} t}}={\frac {V_{1}V_{2}}{2(u+v)}}}$

being represented. thats why

${\displaystyle {\frac {\mathrm {d} u}{U_{1}U_{2}}}-{\frac {\mathrm {d} v}{V_{1}V_{2}}}=0\quad {\text{und}}\quad {\frac {2u\,\mathrm {d} u}{U_{1}U_{2}}}+{\frac {2v\,\mathrm {d} v}{V_{1}V_{2}}}=\mathrm {d} t}$

and the problem is due to hyperelliptic integrals.

Andoyer – Deprit variables L and G are connected to u and v via L = u - v and G = u + v.

Footnotes

1. ↑ Goryachev (1899), Magnus (1971), Leimanis (1965), Borisov a. Mamaev (2001), see literature.
2. ↑ Borisov, Mamaev (2001), p. 271.
3. ↑ ^{a b} Goryachev (1899), see literature.
4. ↑ ^{a b} Leimanis (1965), p. 92.
5. ↑ Magnus (1971), p. 130, and Leimanis (1965), p. 92.
6. ↑ ^{a b} Leimanis (1965), p. 93.
7. ↑ Magnus (1971), p. 130, Leimanis (1965), p. 93, Borisov u. Mamaev (2001), p. 269. There is γ _1,2,3 = -n _1,2,3 .
8. ↑ Borisov et al. Mamaev, p. 271, Leimanis (1965), p. 94, Goryachev (1899), p. 433.
9. ↑ Leimanis (1965), p. 95.
10. ↑ ^{a b} Borisov u. Mamaev (2001), p. 269.

literature

• DN Goryachev : About the movement of a heavy rigid body around a fixed point in the case of A = B = 4C . 1899 (Russian, mathnet.ru - original title: 0 движеніи тяжелаго твердаго тѣла вокругъ неподвижной точки в ъ случаѣ А  =  В  = 4C .).
• K. Magnus : Kreisel: Theory and Applications . Springer, 1971, ISBN 978-3-642-52163-8 , pp. 129 ff . ( Limited preview in Google Book Search [accessed February 20, 2018]).
• AV Borisov, IS Mamaev: Euler-Poisson Equations and Integrable Cases . 2001, p. 269 ff ., doi : 10.1070 / RD2001v006n03ABEH000176 , arxiv : nlin / 0502030 (English, contains solutions of the gyroscopic equations, their detailed description and further literature references.).
• Eugene Leimanis: The General Problem of the Motion of Coupled Rigid Bodies about a Fixed Point . Springer Verlag, Berlin, Heidelberg 1965, ISBN 978-3-642-88414-6 , p. 92 ff ., doi : 10.1007 / 978-3-642-88412-2 (English, limited preview in Google Book Search [accessed on March 21, 2018]).