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Integrable and Non-Integrable Rigid Body Dynamics

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Integrable and Non-Integrable Rigid Body Dynamics Actions and Poincare Sections Classical Problems of Rigid Body Dynamics International Conference – PowerPoint PPT presentation

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Title: Integrable and Non-Integrable Rigid Body Dynamics


1
Integrable and Non-Integrable Rigid Body Dynamics
Actions and Poincare Sections
  • Classical Problems of
  • Rigid Body Dynamics
  • International Conference
  • Dedicated to the 300th anniversary of
  • Leonhard Euler
  • Donetsk 9-13 June 2007

2
Parameter space of standard rigid bodies
No translation 3 rotational degrees of freedom
of SO(3) type
4 parameters principal moments of inertia
A1, A2, A3
s1, s2, s3
center of gravity
Remark a Cardan suspension implies T3 rather
than SO(3) as configurations space it requires
at least two more parameters for the moment of
inertia of the suspension device, and the
direction of its axis relative to gravity.
3
Integrable and non-integrable rigid body dynamics
  • Phase spaces and basic equations
  • Full and reduced phase spaces
  • Equations of motion
  • Integrable cases Actions
  • Euler
  • Lagrange
  • Kovalevskaya
  • Non-integrable dynamics Poincaré sections
  • General principles
  • The PP-torus representation
  • An example

4
Phase space and conserved quantities
3 angles 3 velocities 6D phase
space
4 conserved quantities energy h, angular
momenta lx, ly, lz ? 2D invariant sets,
super-integrable dynamics
In general only 2 conserved quantities energy h
and momentum lz ? 4D invariant sets, (mildly)
chaotic dynamics
5
Reduced phase space
The 6D phase space of variables g(g1,g2,g3) and
l (l1,l2,l3) has Casimir constants g 21
(Poisson sphere) and lg lz, hence is
effectively 4D
3 conserved quantities energy h, angular
momenta lx, ly ? 1D invariant sets,
super-integrable dynamics
2 conserved quantities energy h, momentum l3 ?
2D invariant sets, integrable dynamics
2 conserved quantities energy h, Kovalevskaya K
? 2D invariant sets, integrable
In general only 1 conserved quantity energy h
? 3D invariant sets, (mildly) chaotic dynamics
6
Euler-Poisson equations
7
Canonical equations with Euler angles
Hamiltonians
8
Integrable and non-integrable rigid body dynamics
  • Phase spaces and basic equations
  • Full and reduced phase spaces
  • Equations of motion
  • Integrable cases Actions
  • Euler
  • Lagrange
  • Kovalevskaya
  • Non-integrable dynamics Poincaré sections
  • General principles
  • The PP-torus representation
  • An example

9
Actions in the Euler case
J,y-actions complete elliptic integrals of the
third kind
Energy surfaces h h(I1,I2,I3) carry information
on frequencies, winding numbers, bifurcations
(separatrices), and quantum mechanics.
10
Actions in the Euler case
Energy surfaces in action representation 11
correspondence of points and Liouville tori
11
Actions in the Lagrange case
actions
j-action
Frequencies are integrals of the first kind,
winding numbers are of the third kind
12
Actions in the Lagrange case
13
Actions in the Kovalevskaya case
ds
hyperelliptic integral with paths to be
determined on a hyperelliptic curve
14
Actions in the Kovalevskaya case
Energy surfaces in action representation agreemen
t of numerical computation with Abelian integrals
h -0.5
h 0.9
h 9.0
h 1.2
h 1.45
h 2.3
h 9.0
Ij towards right front, I2 towards left front, I3
pointing up
15
Integrable and non-integrable rigid body dynamics
  • Phase spaces and basic equations
  • Full and reduced phase spaces
  • Equations of motion
  • Integrable cases Actions
  • Euler
  • Lagrange
  • Kovalevskaya
  • Non-integrable dynamics Poincaré sections
  • General principles
  • The PP-torus representation
  • An example

16
Poincaré surfaces of section general
considerations
Find a surface P 2h,l in E3h,l such that every
trajectory meets it repeatedly. Recipe choose
bounded W E3h,l ? R and take S dW/dt 0 as
section condition. Then look for a convenient
surface to which P 2h,l can be mapped 11.
Finally, identify points on S( g) and S-(g)
whose preimages are identical.
17
Implementation of the general principle
First idea divide P 2h,l according to incoming
and outgoing intersections dS/dt lt 0 and dS/dt gt
0, respectively. Problem Projections in general
not 11
18
The situation in rigid body dynamics
Let the section condition be defined by W g
s, hence S A-1l (s x g) 0. The surface P
2h,l so defined projects to the Poisson sphere as
follows Its image is the entire accessible
region Uh,l Ul(g) h in S 2(g). Points on
the boundary ?Uh,l have one preimage. The
preimage of the two points g s /s is a
circle. All other points of Uh,l have exactly two
preimages.
This suggests the construction of the PP-torus
(from Poisson and Poincaré).
19
The PP-torus
in
in
P 2h,l M 23
n.acc.
n.acc.
out
out
A (2,1.5,1) s (1,0,0) h 80.5
l 12.8
20
An example
A (2,1.1,1) s (0.94868, 0, 0.61623)
S 2
l 3.25
h 1.8
T 2
21
T 2 S 2
2T 2
22
M 23
M 22
h 5.3
23
2S 2
M 22
T 2
h 6.5, l 3.25
h 3.1486, l 2.72
24
Integrable and non-integrable rigid body dynamics
  • Phase spaces and basic equations
  • Full and reduced phase spaces
  • Equations of motion
  • Integrable cases Actions
  • Euler
  • Lagrange
  • Kovalevskaya
  • Non-integrable dynamics Poincaré sections
  • General principles
  • The PP-torus representation
  • An example

25
Outlook
  • Semiclassical quantization of the integrable
    cases via actions
  • Collect knowledge about non-integrable dynamics
    very little seems to be known
  • Consider rigid bodies with Cardan suspension the
    configuration space is then not SO(3) but T3

26
Acknowlegdements
  • Mikhail Kharlamov
  • Igor Gashenenko
  • Alexey Bolsinov
  • Alexander Veselov
  • Holger Dullin
  • Andreas Wittek
  • Dennis Lorek
  • Sven Schmidt

27
(No Transcript)
28
An example
A (2,1.1,1) s (0.94868, 0, 0.61623)
29
(No Transcript)
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