Euler-Poisson equations

The Euler-Poisson equations according to Leonhard Euler and Siméon Denis Poisson are the equations of motion used in gyro theory for the heavy gyro with a base. For this gyro, they represent the components of the rate of swirl and the time derivative of the perpendicular direction or weight in the main axis system.

The classic gyro theory is almost exclusively dedicated to the heavy gyro with support point and a lot of effort has been and is put into finding exact solutions. In the age of powerful computing machines, these solutions no longer have the previously justified, central importance. Today it is not difficult to calculate the Euler-Poisson equations with any desired initial conditions by numerical simulation with any desired accuracy.

Wilhelm Hess was able to eliminate the direction cosines in 1890 with the help of the integral of the motion . With the help of this formulation, research investigates the topology of the energy surfaces of the heavy top with a base at the beginning of the 21st century .

history

Leonhard Euler set up the gyro equations in 1750 and was able to provide a solution - the Euler gyro - in 1758 . Joseph-Louis Lagrange made an important contribution in 1788 by solving the equations for the symmetrical heavy top with a fixed point . In 1829 Carl Gustav Jacob Jacobi published the theory of Jacobi's elliptic functions and theta functions , with which the Euler-Poisson equations can be solved. In 1888 Sofia Kovalevskaya discovered the last case that could be solved by theta functions, the heavy, symmetrical , inhomogeneous Kovalevskaya top , and reformulated the problem with analytical functions of a complex number .

AM Lyapunov proved in 1894 that the three cases of Euler, Lagrange and Kovalevskaya are the only ones in which the general solution is a unique function of time under any initial conditions. In other cases the solutions to certain initial conditions are ambiguous functions of time. Edouard Husson showed in 1905 that the Euler, Lagrange and Kowalewskaja gyroscopes are the only cases of the Euler-Poisson equations that can be solved with algebraic first integrals . In addition, only special movements can be integrable. A fourth algebraic integral only exists in cases where the solution is a unique function of time.

formulation

Vector equations

The moment of gravity results from the cross product × of the lever arm from the support point to the center of mass with the weight of the top: ${\displaystyle {\vec {M}}}$ ${\displaystyle {\vec {s}}}$ ${\displaystyle {\vec {G}}}$

${\displaystyle {\vec {M}}={\vec {s}}\times {\vec {G}}=-mg{\vec {s}}\times {\hat {e}}_{z}}$

Here m is the mass, g is the acceleration due to gravity and ê _{z is} the unit vector pointing vertically upwards . The moment of gravity is used in the principle of swirl : ${\displaystyle {\vec {M}}={\dot {\vec {L}}}}$

${\displaystyle -mg{\vec {s}}\times {\hat {e}}_{z}={\dot {\vec {L}}}={\frac {\mathrm {d} _{r}}{\mathrm {d} t}}{\vec {L}}+\underbrace {{\vec {L}}\cdot \mathbf {\Theta } ^{-1}} _{\vec {\omega }}\times {\vec {L}}=\mathbf {\Theta } \cdot {\frac {\mathrm {d} _{r}}{\mathrm {d} t}}{\vec {\omega }}+{\vec {\omega }}\times \underbrace {\mathbf {\Theta } \cdot {\vec {\omega }}} _{\vec {L}}}$

Here Θ is the inertia tensor , Θ ^–1 its inverse , the angular velocity and the angular momentum of the gyro with respect to the support point and forms the relative time derivative in the principal axis system. The Euler gyro equations are the components of the vector equation in this same system in case of any external torque. ${\displaystyle {\vec {\omega }}}$ ${\displaystyle {\vec {L}}}$ ${\displaystyle {\tfrac {\mathrm {d} _{r}}{\mathrm {d} t}}}$

The kinematic equations discovered by Poisson mathematically formulate the constancy of the perpendicular direction in the main axis system and are written in vector form

${\displaystyle {\dot {\hat {e}}}_{z}={\frac {\mathrm {d} _{r}}{\mathrm {d} t}}{\hat {e}}_{z}+{\vec {\omega }}\times {\hat {e}}_{z}={\frac {\mathrm {d} _{r}}{\mathrm {d} t}}{\hat {e}}_{z}+{\vec {L}}\cdot \mathbf {\Theta } ^{-1}\times {\hat {e}}_{z}={\vec {0}}}$

where again forms the relative time derivative in the main axis system. The components of this equation are called Poisson's equations in honor of their discoverer . ${\displaystyle {\tfrac {\mathrm {d} _{r}}{\mathrm {d} t}}}$

Euler-Poisson equations

The autonomous ordinary differential equation system of Euler and Poisson equations Euler-Poisson equations called and enrolled in the principal axes system

${\displaystyle {\begin{aligned}\Theta _{1}{\dot {\omega }}_{1}+(\Theta _{3}-\Theta _{2})\omega _{2}\omega _{3}={\dot {L}}_{1}+\left({\frac {1}{\Theta _{2}}}-{\frac {1}{\Theta _{3}}}\right)L_{2}L_{3}=&mg(n_{2}s_{3}-n_{3}s_{2})\\\Theta _{2}{\dot {\omega }}_{2}+(\Theta _{1}-\Theta _{3})\omega _{3}\omega _{1}={\dot {L}}_{2}+\left({\frac {1}{\Theta _{3}}}-{\frac {1}{\Theta _{1}}}\right)L_{3}L_{1}=&mg(n_{3}s_{1}-n_{1}s_{3})\\\Theta _{3}{\dot {\omega }}_{3}+(\Theta _{2}-\Theta _{1})\omega _{1}\omega _{2}={\dot {L}}_{3}+\left({\frac {1}{\Theta _{1}}}-{\frac {1}{\Theta _{2}}}\right)L_{1}L_{2}=&mg(n_{1}s_{2}-n_{2}s_{1})\end{aligned}}}$

${\displaystyle {\begin{aligned}{\dot {n}}_{1}=&\omega _{3}n_{2}-\omega _{2}n_{3}={\frac {L_{3}}{\Theta _{3}}}n_{2}-{\frac {L_{2}}{\Theta _{2}}}n_{3}\\{\dot {n}}_{2}=&\omega _{1}n_{3}-\omega _{3}n_{1}={\frac {L_{1}}{\Theta _{1}}}n_{3}-{\frac {L_{3}}{\Theta _{3}}}n_{1}\\{\dot {n}}_{3}=&\omega _{2}n_{1}-\omega _{1}n_{2}={\frac {L_{2}}{\Theta _{2}}}n_{1}-{\frac {L_{1}}{\Theta _{1}}}n_{2}\end{aligned}}}$

The superpoint forms the time derivative , mg is the weight force and for k = 1,2,3 is in each case

• ω _{k is} the component of the angular velocity ,
• L _{k is} the component of the angular momentum ,
• s _{k is} the component of the center of mass,
• n _{k is} the component of the unit vector ê _z and
• Θ _{k is} a principal moment of inertia of the top

in the main axis system. Become common

• the main moments of inertia Θ _1,2,3 with A, B or C,
• the angular velocities ω _1,2,3 with p, q or r and
• the direction _cosines n _1,2,3 with γ _1,2,3 or γ, γ ', γ ", occasionally also with the opposite sign

designated. Sometimes the gyroscopic equations are divided by the weight force or the factor mg is added to the direction cosine so that this factor no longer appears in the above formulas.

Calculation of the precession angle

Because the plumb line from the precession angle ψ is the rotation about the perpendicular direction independently, can not be calculated from the Euler-Poisson equations of precession angle. However, he can use the differential equation

${\displaystyle {\dot {\psi }}={\frac {n_{1}\omega _{1}+n_{2}\omega _{2}}{n_{1}^{2}+n_{2}^{2}}}}$

can be determined at the same time or subsequently.

Integrals of motion

For every heavy top with a base, the norm of the direction vector of the plumb line, the angular momentum in the plumb direction and the total energy E are constant:

${\displaystyle {\begin{array}{rll}{\hat {e}}_{z}\cdot {\hat {e}}_{z}=&n_{1}^{2}+n_{2}^{2}+n_{3}^{2}=1&={\text{const.}}\\L_{z}:=&{\vec {L}}\cdot {\hat {e}}_{z}=\Theta _{1}\omega _{1}n_{1}+\Theta _{2}\omega _{2}n_{2}+\Theta _{3}\omega _{3}n_{3}&={\text{const.}}\\E:=&{\frac {1}{2}}(\Theta _{1}\omega _{1}^{2}+\Theta _{2}\omega _{2}^{2}+\Theta _{3}\omega _{3}^{2})+mg(s_{1}n_{1}+s_{2}n_{2}+s_{3}n_{3})&={\text{const.}}\end{array}}}$

The constants are integrals of motion and are called integrals for short in gyro theory , the first two also Casimir invariants . The second integral L _z is, after the swirl or Flächensatz also swirl or surface integral . It is constant because the moment of gravity has no component in the vertical direction. The total energy E is also referred to as the Hamilton function in analytical mechanics . It is constant because the earth's gravitational field is conservative and thus the gyroscopic motion obeys the law of conservation of energy . The immutability of the integrals can also be proven by time derivative and inserting the Euler-Poisson equations.

Hess equations

Wilhelm Hess published alternative formulations in his 1890 essay in which he introduced the loxodromic pendulum . He succeeded in expressing the direction _cosines n _1,2,3 with the integrals and the angular momentum:

${\displaystyle {\begin{aligned}{\hat {e}}_{z}=&\zeta _{1}{\vec {L}}+\zeta _{2}{\vec {s}}+\zeta _{3}{\vec {s}}\times {\vec {L}}\\\zeta _{1}=&{\frac {mg|{\vec {s}}|^{2}L_{z}+L_{s}(T-E)}{mg|{\vec {s}}\times {\vec {L}}|^{2}}}\\\zeta _{2}=&{\frac {|{\vec {L}}|^{2}(E-T)-mgL_{s}L_{z}}{mg|{\vec {s}}\times {\vec {L}}|^{2}}}\\\zeta _{3}=&{\frac {\sqrt {f}}{|{\vec {s}}\times {\vec {L}}|^{2}}}\\f=&|{\vec {s}}\times {\vec {L}}|^{2}-\left|{\frac {E-T}{mg}}{\vec {L}}-L_{z}{\vec {s}}\right|^{2}\end{aligned}}}$

Therein T is the rotational energy , which is equal to the first summand in the above integral E , and . This gives Hess' equation of motion (6): ${\displaystyle L_{s}:={\vec {L}}\cdot {\vec {s}}}$

${\displaystyle {\dot {\vec {L}}}=-mg{\vec {s}}\times {\hat {e}}_{z}=-mg[\zeta _{1}{\vec {s}}\times {\vec {L}}+\zeta _{3}{\vec {s}}\times ({\vec {s}}\times {\vec {L}})]}$

which only depends on the angular momentum or the angular velocity. It has been proven that √ f is an integrating factor for this equation, which is why one more integral is sufficient to solve the equations of motion. The function f is important for the topology of the energy surfaces.

Hess could already

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}|{\vec {L}}|^{2}=2mg{\sqrt {f}}}$

specify. In his equations (8), he replaced the angular velocities with the square of the angular momentum , the projection of the angular momentum onto the axis of gravity and the rotational energy T : ${\displaystyle \nu :=|{\vec {L}}|^{2}}$ ${\displaystyle \rho :={\vec {L}}\cdot {\vec {s}}}$

${\displaystyle {\begin{aligned}{\dot {\nu }}^{2}=&4(mg)^{2}(|{\vec {s}}|^{2}\nu -\rho ^{2})+4mgL_{z}[\rho (E-T)-mgL_{z}|{\vec {s}}|^{2}]\\&+4(E-T)[mgL_{z}\rho +\nu (T-E)]\\{\dot {\rho }}^{2}=&|{\vec {s}}|^{2}(\nu |{\vec {\omega }}|^{2}-4T^{2})+\rho (2T\omega _{s}-\rho |{\vec {\omega }}|^{2})+\omega _{s}(2T\rho -\nu \omega _{s})\\{\dot {T}}=&{\frac {{\frac {1}{2}}(2|{\vec {s}}|^{2}T-\rho \omega _{s}){\dot {\nu }}+[(E-T)\rho -mg|{\vec {s}}|^{2}L_{z}]{\dot {\rho }}}{|{\vec {s}}|^{2}\nu -\rho ^{2}}}\end{aligned}}}$

In it is . PA Schiff found similar equations in 1904. The equations are equivalent to the Euler-Poisson equations, provided that no constraints are placed on the variables ν, ρ and T. ${\displaystyle \omega _{s}={\vec {\omega }}\cdot {\vec {s}}}$

Connection with the Riccatian differential equation

If angular velocities ω _{1,2,3 are} found or given , the question arises as to the corresponding direction _cosines n _1,2,3 . According to Gaston Darboux , the solution using the Riccat differential equation is recommended . For this purpose, the imaginary unit i and unknown functions x and y

${\displaystyle n_{1}={\frac {1-xy}{x-y}},\;n_{2}=\mathrm {i} {\frac {1+xy}{x-y}},\;n_{3}={\frac {x+y}{x-y}}}$

set. As a result of this, the functions x and y are integrals of the Riccatian differential equation

${\displaystyle {\dot {\sigma }}={\frac {1}{2}}(\omega _{2}-\mathrm {i} \omega _{1})-\mathrm {i} \omega _{3}\sigma +{\frac {1}{2}}(\omega _{2}+\mathrm {i} \omega _{1})\sigma ^{2}}$

This can be integrated by squaring as soon as a particular integral of it is known.

Solutions of the Euler-Poisson equations

For technical application there are significant special cases in which the Euler-Poisson equations are simplified to such an extent that they are integrable . In these cases, the trajectories of the gyroscope have an at least quasi-periodic course, the various movement modes can be classified and the time functions of the variables and their geometric meaning can be specified. In particular with the Kovalevskaya gyro and the Goryachev-Chaplygin gyro , the analytical solutions are so complicated that working out the aforementioned typical properties of the movement is extremely time-consuming. Topological analysis ( bifurcation diagram ), stability analysis , phase space diagrams and computer animations help to gain insights into the processes in the gyro and to work out their typical properties. The results obtained in this way can motivate practical applications.

The following table contains a selection of spatial motions of gyroscopes in which exact solutions of the Euler-Poisson equations were achieved up to the beginning of the 21st century.

Explorer	Main moments of inertia	Location of the center of gravity	Initial conditions ( t  = 0)
Leonhard Euler see Euler-Kreisel	any	s 1 = s 2 = s 3 = 0	any
For Joseph-Louis Lagrange see Lagrange top	A = B	s 1 = s 2 = 0, s 3 ≠ 0	any
Sofia Kowalewskaja see Kowalewskaja-Kreisel	A = B = 2C	${\displaystyle s_{1}^{2}+s_{2}^{2}\neq 0,\,s_{3}=0}$	any
Wilhelm Hess see Hess's pendulum	any	${\displaystyle {\frac {s_{1}}{s_{3}}}={\sqrt {\tfrac {C(A-B)}{A(B-C)}}}}$ s 2 = 0	${\displaystyle {\vec {L}}\cdot {\vec {s}}=0}$
Goryachev and Chaplygin see Goryachev-Chaplygin-gyro	A = B = 4C	${\displaystyle s_{1}^{2}+s_{2}^{2}\neq 0,\,s_{3}=0}$	L z = 0
Merzalow	A = B = 4C	${\displaystyle s_{1}^{2}+s_{2}^{2}\neq 0,\,s_{3}=0}$	r = 0
Bobylev and Steklow see the Bobylew-Steklow solution	2A = C	s 1 = s 2 = 0, s 3 ≠ 0	${\displaystyle q={\dot {q}}=0}$
Otto Staude see Staude rotation	any	any	${\displaystyle {\vec {\omega }}=\Omega {\hat {e}}_{z}}$ ${\displaystyle ({\vec {\omega }}\times {\vec {L}})\cdot {\vec {s}}=0}$
For Giuseppe Grioli, see Griolische Precession	any	${\displaystyle {\frac {s_{1}}{s_{3}}}={\sqrt {\tfrac {A-B}{B-C}}}}$ s 2 = 0	Uniquely defined except for one degree of freedom

In the table, A, B, C = θ _{1,2,3 are} the main moments of inertia, p, q, r = ω _{1,2,3 are} the angular velocities and s _{1,2,3 are} the (constant) coordinates of the center of mass in the main axis system.

Individual evidence

1. ↑ SV Ershakov: New exact solution of Euler's equations (rigid body dynamics) in the case of rotation over the fixed point . In: Archive of Applied Mechanics . tape 84 , no. 3 . Springer-Verlag, 2014, ISSN  0939-1533 , p. 385-389 , doi : 10.1007 / s00419-013-0806-x .
2. ↑ Magnus (1971), p. 109.
3. ↑ ^{a b} Wilhelm Hess : About Euler's equations of motion and about a new particular solution to the problem of the motion of a rigid body around a fixed point. In: Mathematical Annals . Vol. 37, 1890, pp.  153-181 ( digizeitschriften.de [accessed on May 2, 2018]).
4. ↑ ^{a b} IG Gashenenko, PH Richter : Enveloping Surfaces And Admissible Velocities Of Heavy Rigid Bodies . In: World Scientific Publishing Company (Ed.): International Journal of Bifurcation and Chaos . tape 14 , no. 8 , 2004, ISSN  0218-1274 , p. 2525–2553 , doi : 10.1142 / S021812740401103X ( iamm.su [PDF; accessed June 2, 2019] see p. 2537).
5. ^ Édouard Husson: Research of the intégrales algébriques dans le mouvement d'un solide pesant autour d'un point fixe . In: Annales de la faculté des sciences de Toulouse 2 ^e série . 1906, p. 73–152 , doi : 10.5802 / afst.232 (French, numdam.org [PDF; accessed on March 7, 2018] On page 74, a first attempt at evidence by Roger Liouville in 1897 is revealed to be incorrect.).
6. ↑ Leimanis (1965), p. 53 ff.
7. ↑ In the tensor algebra can be dispensed Parentheses:
   

   ${\displaystyle ({\vec {L}}\cdot \mathbf {\Theta } ^{-1})\times {\vec {L}}={\vec {L}}\cdot (\mathbf {\Theta } ^{-1}\times {\vec {L}})={\vec {L}}\cdot \mathbf {\Theta } ^{-1}\times {\vec {L}}}$

8. ↑ Magnus (1970), p. 106.
9. ^ Siméon Denis Poisson : Traité de Méchanique . 3. Edition. 1 to 6. JG Garnier, Brussels 1838 (French, archive.org [accessed on November 3, 2019]).
10. ↑ ^{a b} Leimanis (1965), p. 7.
11. ^ ^{A b} Peter H. Richter, Holger R. Dullin, Andreas Wittek: Kovalevskaya Top . Ed .: Institute for Scientific Film (IWF) . 1997, ISSN  0073-8433 , p. 41 (English, researchgate.net [accessed on March 28, 2018] There φ has the meaning of ψ here, see also gyroscopic theory # reference systems and Euler angles ).
12. ↑ PA Schiff (П. А. Шиффъ): About the equations of motion of a heavy rigid body with a fixed point . In: Матем. сб. tape 24 , no. 2 , 1904, pp. 169–177 (Russian, mathnet.ru [accessed June 10, 2019] Original title: Объ уравнен i яхъ движен i я тяжелаго твердаго тѣла, имѣющаго неподвижнуго неподвижнучу .).
13. ↑ Leimanis (1965), p. 104.
14. ↑ Felix Klein , Conr. Müller: Encyclopedia of the Mathematical Sciences with inclusion of its applications . Mechanics. Ed .: Academies of Sciences in Göttingen , Leipzig, Munich and Vienna. 4th volume, 1st part volume. BG Teubner , Leipzig 1908, ISBN 3-663-16021-1 , p. 565 , doi : 10.1007 / 978-3-663-16021-2 ( limited preview in Google book search [accessed on March 7, 2020] see also wikisource }).
15. ↑ AV Borisov, IS Mamaev: Euler-Poisson Equations and Integrable Cases . 2001, doi : 10.1070 / RD2001v006n03ABEH000176 , arxiv : nlin / 0502030 (English, contains solutions to the Euler-Poisson equations, their detailed description and further references.).
16. ↑ Magnus (1971), p. 108.

literature

• K. Magnus : Kreisel: Theory and Applications . Springer, 1971, ISBN 3-642-52163-0 , pp. 109 . ( limited preview in Google Book Search [accessed February 7, 2019]).
• Eugene Leimanis: The General Problem of the Motion of Coupled Rigid Bodies about a Fixed Point . Springer Verlag, Berlin, Heidelberg 1965, ISBN 3-642-88414-8 , p. 53 f ., doi : 10.1007 / 978-3-642-88412-2 (English, limited preview in Google Book Search [accessed on March 21, 2018]).