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The study of non-integrable rigid body problems

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The study of non-integrable rigid body problems S3,S1xS2 2T2 S3 3S3 K3 RP3 (1.912,1.763) VII – PowerPoint PPT presentation

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Title: The study of non-integrable rigid body problems


1
The study of non-integrable rigid body problems
2
Types of rigid bodies parameter space
Translation R2 or R3 and Rotation SO(3) or T3
one point fixed only 3 rotational degrees of
freedom
4 parameters principal moments of inertia
2
2
center of gravity
in addition moments of inertia and direction of
axis of Cardan suspension
3
Rigid body dynamics
  • Phase spaces and basic equations
  • Full and reduced phase spaces
  • Euler-Poisson equations
  • Integrable cases
  • Euler
  • Lagrange
  • Kovalevskaya
  • Katoks general cases
  • Effective potentials
  • Bifurcation diagrams
  • Enveloping surfaces
  • Poincaré surfaces of section
  • Gashenenkos version
  • Dullins suggestion
  • Schmidts investigations

4
Phase space and conserved quantities
3 angles 3 velocities 6D phase
space
energy conservation hconst 5D energy
surfaces
one angular momentum lconst 4D invariant sets
mild chaos
integrable
super-integrable
5
Reduced phase space
3 components each of g and l 6D phase space
but g 21 (Poisson sphere) und lg l
(ang.momentum) are Casimirs ? effectively only
4D phase space
energy conservation hconst 3D energy
surfaces
6
Euler-Poisson equations
coordinates
Casimir constants
energy integral
effective potential
7
The basic scheme
8
Eulers case
l-motion decouples from g-motion
9
Lagranges case
10
Enveloping surfaces
11
Katoks cases
7
s2 s3 0
5
6
1
4
2
3
2
7 colors for 7 types of bifurcation diagrams
3
7colors for 7 types of energy surfaces
7
4
5
6
S1xS2
12
Effective potentials for case 1
(A1,A2,A3) (1.7,0.9,0.86)
13
71 types of envelopes
(A1,A2,A3) (1.7,0.9,0.86)
S1xS2
M32
(2,1.8)
III
14
71 types of envelopes
(A1,A2,A3) (1.7,0.9,0.86)
15
2 variations of types II and III
A (0.8,1.1,0.9)
A (0.8,1.1,1.0)
Only in cases II and III are the envelopes free
of singularities. Case II occurs in Katoks
regions 4, 6, 7, case III only in region 7.
This seems to complete the set of all possible
types of envelopes.
16
Poincaré section S1
17
Poincaré section S1 projections to S2(g)
18
Poincaré section S1 polar circles
19
Poincaré section S1 projection artifacts
20
Poincaré section S2
21
Comparison of the two sections
with
22
Poincaré sections S1 and S2 in comparison
23
From Kovalevskaya to Lagrange
(A1,A2,A3) (2,?,1) (s1,s2,s3) (1,0,0)
? 2 Kovalevskaya
? 1.1 almost Lagrange
24
Examples
E
B
25
Example of a bifurcation scheme
26
Acknowledgement
Igor Gashenenko Sven Schmidt Holger Dullin
27
(No Transcript)
28
Integrable cases
Euler gravity-free
29
Kovalevskayas case
Tori projected to (p,q,r)-space
Tori in phase space and Poincaré surface of
section
30
Poincaré section S1 projections to S2(g)
31
Poincaré section S1 polar circles
32
Improved projection representation
Place the polar circles at upper and lower rims
of the projection planes.
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