Stability and Integration on so(3)*-

Stability and Integration on so (3) ∗ −

Ross M. Adams ∗ , Rory Biggs, Claudiu C. Remsing

Department of Mathematics (Pure and Applied) Rhodes University, Grahamstown 6140, South Africa

XV th International Conference: Geometry, Integrability and

Quantization, Varna, Bulgaria, June 7–12, 2013

Outline

1 Introduction

2 Classification

3 Stability

4 Integration

Outline

1 Introduction

2 Classification

3 Stability

4 Integration

Problem statement

Hamilton-Poisson systems on so (3) ∗ −

Classify quadratic Hamilton-Poisson systems Stability

Integration via Jacobi elliptic functions.

Poisson structure

(minus) Lie-Poisson structure

{F , G} (p) = −p ([dF (p), dG(p)]) , p ∈ g ∗ Hamiltonian vector field:

H[F ~ ] = {F , H}

Casimir function:

{C, F } = 0

Hamilton-Poisson Systems

Quadratic Hamilton-Poisson System (g ∗ − , H) H A,Q : p 7→ pA + pQp > , A ∈ g

Equations of motion:

p ˙ i = −p ([E i , dH(p)]) H ~ = Π − · ∇H

Q (PSD) quadratic form

A = 0 - homogeneous

A 6= 0 - inhomogeneous

so (3) =

A ∈ R 3×3 | A > + A = 0

Basis:

E 1 =





0 0 0

0 0 −1

0 1 0



 E 2 =





0 0 1

0 0 0

−1 0 0



 E 3 =





0 −1 0

1 0 0

0 0 0



 ·

Commutator relations:

[E 2 , E 3 ] = E 1 [E 3 , E 1 ] = E 2 [E 1 , E 2 ] = E 3 Poisson structure on so (3) ∗ −

Π − =





0 −p 3 p 2 p 3 0 −p 1

−p 2 p 1 0





C(p) = p 1 2 + p 2 2 + p 2 3

Automorphisms: Aut(so (3) ∗ − ) = Aut(so (3)) = SO (3)

Outline

1 Introduction

2 Classification

3 Stability

4 Integration

Affine equivalence

Definition

Systems G and H are affinely equivalent if

∃ affine automorphism ψ such that ψ ∗ G ~ = H ~ Proposition

The following systems are equivalent to H A,Q :

H A,Q ◦ ψ : where ψ - linear Poisson automorphism H A,rQ : where r 6= 0

H A,Q + C : where C- Casimir function

Classification on so (3) ∗ −

H(p) = pQp >

1 2 p 2 1 p 2 1 + 1 2 p 2 2

H(p) = pA + pQp >

αp 1 p 2 + 1 2 p 1 2 p 1 + 1 2 p 1 2 p 1 + αp 2 + 1 2 p 1 2

α 1 p 1 + α 2 p 2 + γ p 3 + p 1 2 + βp 2 2

Conditions α > 0

α 1 , α 2 ≥ 0

γ ∈ R

0 < β < 1.

Homogeneous systems

Proof (sketch)

Let H Q = pQp > (Q PSD 3 × 3 matrix).

∃Ψ ∈ SO(3) such that ΨQΨ > = diag(a 1 , a 2 , a 3 ), where a 1 ≥ a 2 ≥ a 3 ≥ 0

Then a 1

1

−a

₃

ΨQΨ > − a 3 C(p)

= diag(1, α, 0), 0 ≤ α ≤ 1.

ψ = diag(− √ 2 √

1 − α, 2 p

α(1 − α), − √ 2 √

α) brings H α (p) = p 1 2 + αp 2 2 into H 1 (p) = p 2 1 + 1 2 p 2 2

ψ · H ~ α = H ~ 1 ◦ ψ :



 



 

 p ˙ 1 = 2

√ 2α √

1 − α p 2 p 3 p ˙ 2 = 4 p

α(1 − α) p 1 p 3 p ˙ 3 = 2 √

2α(1 − α) p 1 p 2 .

Inhomogeneous systems

Proof (sketch)

Let H A,Q = pA + pQp > .

By Proposition, H A,Q is A-equivalent to (for some p 0 ∈ so(3) ∗ − )

H a (p) = p 0 , H b (p) = p 0 + 1

2 p 1 2 , H c (p) = p 0 + p 2 1 + βp 2 2 Consider Ψ ∈ SO(3) s.t. ΨQΨ > = Q. Let H 1,α (p) = αp 1 , then H a (p) ∼ = H 1,α , α > 0.

Consider ψ : p 7→ ψ 0 (p) + q s.t. ψ 0 · H ~ 1,α = H ~ 1,β ◦ ψ:

For ψ 0 = [ψ ij ],

−αψ 13 p 2 + αψ 12 p 3 = 0

−αψ 23 p 2 + αψ 22 p 3 − β(ψ 31 p 1 + ψ 32 p 2 + ψ 33 p 3 + q 3 ) = 0

−αψ 33 p 2 + αψ 32 p 3 − β(ψ 21 p 1 + ψ 22 p 2 + ψ 23 p 3 + q 2 ) = 0

Inhomogeneous systems

Proof (sketch)

Consider H b (p) = p 0 + 1 2 p 2 1 . Then Ψ =

±1 0

0 S

, S ∈ O(2), for ΨQΨ > = Q Thus H b (p) is equivalent to (α 1 , α 2 ≥ 0)

H 2,α (p) = α 1 p 1 + α 2 p 2 + 1 2 p 1 2 Let α 1 = 0. Then H 2,α ∼ = H 2,1 , where H 2,1 = p 2 + 1 2 p 2 1

Indeed, H ~ 2,α and H ~ 2,1 are compatible with the affine isomorphism

p 7→ p





1 0 0

0 α 2 0 0 0 α 2



 +



 0 1 − α 2 2

0





>

Outline

1 Introduction

2 Classification

3 Stability

4 Integration

Types of stability

Lyapunov stable

∀ nbd N ∃ nbd N 0 s.t. F t (N 0 ) ⊂ N spectrally stable

Re(λ i ) ≤ 0 for eigenvalues of DH(p e ) Lyapunov stable = ⇒ spectrally stable Energy methods

Energy Casimir: (λ 1 , λ 2 ∈ R ),

d(λ 1 H + λ 2 C)(p e ) = 0 and d 2 (λ 1 H + λ 2 C)(p e ) | W ×W < 0 W = ker dH(p e ) ∩ ker dC(p e )

Continuous energy Casimir: H −1 (H(p e )) ∩ C −1 (C(p e )) = {p e }

Classification on so (3) ∗ −

H(p) = pQp >

1 2 p 2 1 p 2 1 + 1 2 p 2 2

H(p) = pA + pQp >

αp 1 p 2 + 1 2 p 1 2 p 1 + 1 2 p 1 2 p 1 + αp 2 + 1 2 p 1 2

α 1 p 1 + α 2 p 2 + γ p 3 + p 1 2 + βp 2 2

Conditions α > 0

α 1 , α 2 ≥ 0

γ ∈ R

0 < β < 1.

Stability: H(p) = p 2 + 1 2 p 1 2

Equations of motion

p ˙ 1 = −p 3 p ˙ 2 = p 1 p 3 p ˙ 3 = p 1 (1 − p 2 )

-2

0

2

E

₁^*

-2

0

E

₂^* 2 -2

0 2

E

₃^*

-2

0

2

E

₁^*

-2

0

E

₂^* 2 -2

0 2

E

₃^*

Stability: H(p) = p 2 + 1 2 p 1 2

Each state e µ = (µ, 1, 0) is stable, µ ∈ R Let H λ = λ 1 H + λ 2 C.

For λ 1 = 1 and λ 2 = − 1 2

dH λ (e µ ) = 0 and d 2 H λ (e µ ) =





0 0 0

0 −1 0

0 0 −1





W = kerdH(e µ ) ∩ kerdC(e µ ) = span









 1

−µ 0



 ,



 0 0 1









 Thus (µ 6= 0)

d 2 H λ (e µ )| W ×W =

−µ 2 0

0 −1

Stability: H(p) = p 2 + 1 2 p 1 2

Each state e µ = (0, µ, 0) is stable for µ ≤ 1 For µ < 1, the Hessian is given by

d 2 H λ (µ, 0, 0) =







−( 1−µ µ ) 0 0 0 − µ 1 0 0 0 − µ 1







For µ = 0

C −1 (C(e 0 )) = (0, 0, 0) For µ = 1

H −1 (H (e 1 )) ∩ C −1 (C(e 1 )) = (0, 1, 0)

e µ is (spectrally) unstable for µ > 1.

Classification on so (3) ∗ −

H(p) = pQp >

1 2 p 2 1 p 2 1 + 1 2 p 2 2

H(p) = pA + pQp >

αp 1 p 2 + 1 2 p 1 2 p 1 + 1 2 p 1 2 p 1 + αp 2 + 1 2 p 1 2

α 1 p 1 + α 2 p 2 + γ p 3 + p 1 2 + βp 2 2

Conditions α > 0

α 1 , α 2 ≥ 0

γ ∈ R

0 < β < 1.

Stability: H(p) = p 1 + 1 2 p 1 2

Equations of motion

p ˙ 1 = 0

p ˙ 2 = (1 + p 1 )p 3 p ˙ 3 = −(1 + p 1 )p 2 .

-2 0

2

E

₁^*

-2 0

2

E

₂^*

-2 0 2

E

₃^*

Stability: H(p) = p 1 + 1 2 p 1 2

Each state e µ,ν = (−1, µ, ν) is unstable Let = p

µ 2 + ν 2

Consider open ball B centred at e µ,ν An integral curve of H, for any ~ δ > 0,

p(t) = (−1 + δ, µ cos(δt) + ν sin(δt), ν cos(δt) − µ sin(δt))

∀ V ⊂ B , e µ,ν 2 ∈ V , there ∃ δ > 0 s.t. p(0) ∈ V Furthermore

kp( π

3δ ) − e µ,ν 2 k = p

δ 2 + µ 2 + ν 2 >

Stability summary

-2 0

2 E₁^*

-2 0

E₂^* 2 -2 0 2

E₃^*

-2 0

2 E₁^*

-2 0

E₂^* 2 -2 0 2

E₃^*

-2 0

2 E₁^*

-2 0

E₂^* 2 -2 0 2

E₃^*

(d)

¹₂

p

²₁

p

²₁

+

¹₂

p

²₂

αp

1

-2 0

2 E1*

-2 0

E2* 2 -2 0 2

E3*

-2 0

2 E1*

-2 0

E2* 2 -2 0 2

E3*

-2 0

2 E1*

-2 0

E2* 2 -2 0 2

E3*

(e) p

2

+

¹₂

p

₁²

p

1

+

¹₂

p

₁²

p

1

+ αp

2

+

¹₂

p

²₁

Outline

1 Introduction

2 Classification

3 Stability

4 Integration

Classification on so (3) ∗ −

H(p) = pQp >

1 2 p 2 1 p 2 1 + 1 2 p 2 2

H(p) = pA + pQp >

αp 1 p 2 + 1 2 p 1 2 p 1 + 1 2 p 1 2 p 1 + αp 2 + 1 2 p 1 2

α 1 p 1 + α 2 p 2 + γ p 3 + p 1 2 + βp 2 2

Conditions α > 0

α 1 , α 2 ≥ 0

γ ∈ R

0 < β < 1.

Integration: H(p) = p 2 + 1 2 p 1 2

Equations of motion

p ˙ 1 = −p 3 , p ˙ 2 = p 1 p 3 , p ˙ 3 = p 1 (1 − p 2 ) Separation: Let h 0 = H(p(0)) and c 0 = C(p(0))

Level sets H −1 (h 0 ) and C −1 (c 0 ) are tangent if their gradients are parallel

For λ ∈ R \{0}

∇H(p) = λ∇C(p) ⇐⇒



 p 1

1 0



 = λ



 2p 1 2p 2 2p 3





p 1 = 2λp 1 , p 2 = 1

2λ , p 3 = 0

Integration: H(p) = p 2 + 1 2 p 1 2

Seperation

For λ 6= 1 2 : p 3 = p 1 = 0, p 2 ∈ R \{1}:

h 0 = p 2 and c 0 = p 2 2 = ⇒ c 0 = h 2 0 Therefore we consider:

c 0 < h 2 0 , c 0 = h 2 0 , c 0 > h 2 0

For λ = 1 2 : p 3 = 0, p 2 = 1, p 1 ∈ R :

h 0 = 1 + 1 2 p 2 1 and c 0 = p 1 2 + 1 = ⇒ c 0 = 2h 0 − 1 We further consider:

c 0 = 2h 0 − 1, c 0 > 2h 0 − 1

Integration: H(p) = p 2 + 1 2 p 1 2 : 0 < 2h 0 − 1 < c 0 < h 2 0

-2 0 2

E1* -2

0 2 E2* -2

0 2

E3*

-2 0 2

E1* -2

0 2 E2* -2

0 2

E3*

Theorem: H(p) = p 2 + 1 2 p 1 2

For 0 < 2h 0 − 1 < c 0 < h 2 0 , σ ∈ {−1, 1},



 

 

 

 

 

 

 

 

p ¯ 1 (t) = σ

√

2δ 1 + k sn (Ω t, k ) dn (Ω t, k ) p ¯ 2 (t) = h 0 + δ − 2δ

1 − k sn (Ω t , k) p ¯ 3 (t) = σk Ω √

2δ cn (Ω t, k ) k sn (Ω t, k ) − 1 Here Ω = √

h 0 − 1 + δ, k =

√

h

0

−1−δ

√

h

₀

−1+δ , and δ = q

h 2 0 − c 0

Proof (Sketch)

H(p) = p 2 + 1 2 p 2 1 and C(p) = p 2 1 + p 2 2 + p 3 2

d

dt ¯ p 2 = p 1 p 3 = q

2(h 0 − p ¯ 2 )(c 0 − 2(h 0 − p ¯ 2 ) − ¯ p 2 2 ) Transform it into standard form, letting s = ¯ p ¯ p

²

−r

¹

2

−r

₂

,

√

2 t = 1

(r 1 − r 2 ) √ A 1 A 2

Z

p¯2(t)−r1

¯ p2(t)−r2

0

ds r

s 2 + B A

²

2

s 2 + B A

¹

1

A 1 = 1

4δ > 0 A 2 = h 0 − 1 − δ 2δ > 0 B 1 = − 1

4δ < 0 B 2 = − h 0 − 1 + δ

2δ < 0

r 1 = h 0 + δ r 2 = h 0 − δ.

Proof (Sketch)

Applying the integral formula Z x

a

dt

p (t 2 − a 2 )(t 2 − b 2 ) = 1 a dc −1

1 a x , b a

, b < a ≤ x we obtain

¯ p 2 (t) = r 2

q

− B A

²

2

dc



(r 1 − r 2 ) √ 2A 1 A 2

q

− B A

²

2

t, r 1

−

^B_A²

2



 − r 1

q

− B A

²

2

dc



(r 1 − r 2 ) √ 2A 1 A 2

q

− B A

²

2

t, r 1

−

^B_A²

2



 − 1

·

Hence

¯ p 2 (t) = h 0 + δ (k + dc (Ω t, k))

k − dc (Ω t, k ) ·

Proof (Sketch)

Using sn (x+K 1 ,k) = dc (x , k ) and a translation in t we get p ¯ 2 (t) = h 0 + δ + 2δ

k sn (Ω t, k) − 1 · Solve for p ¯ 1 (t) and p ¯ 3 (t) using constants of motion.

Verify dt d p(t) = ¯ H(¯ p(t)), for example d

dt p ¯ 2 (t) − ¯ p 1 (t)¯ p 3 (t)

= 2k δ(−1 + σ 2 )Ω cn (Ω t, k ) dn (Ω t, k )

(−1 + k sn (Ω t, k)) 2 ·

which is zero for σ ∈ {−1, 1}.

Proof (Sketch)

Verify p(t ) = ¯ p(t + t 0 )

We have p 2 (0) + 1 2 p 1 (0) 2 = h 0 and p 1 (0) 2 + p 2 (0) 2 + p 3 (0) 2 = c 0 . Thus

1 − p

1 + c 0 − 2h 0 ≤ p 2 (0) ≤ 1 + p

1 + c 0 − 2h 0 . Now ¯ p 2 ( K Ω ) = 1 − √

1 + c 0 − 2h 0 and ¯ p 2 ( 3K Ω ) = 1 + √

1 + c 0 − 2h 0 . Thus ∃t 1 ∈ [ K Ω , 3K Ω ] such that ¯ p 2 (t 1 ) = p 2 (0).

Then

min p

2

(h 0 − p 2 (0)) = h 0 − 1 − p

1 + c 0 − 2h 0 > 0.

Thus p 1 (0) 6= 0. Let σ = sgn (p 1 (0)).

Proof (Sketch)

Then, we have

p 1 (0) 2 = 2h 0 − 2p 2 (0) = 2h 0 − 2¯ p 2 (t 1 ) = ¯ p 1 (t 1 ) 2 . As sgn (p 1 (0)) = σ = sgn (¯ p 1 (t 1 )), p 1 (0) = ¯ p 1 (t 1 ).

Also,

p 3 (0) 2 = c 0 − p 1 (0) 2 − p 2 (0) 2

= c 0 − p ¯ 1 (t 1 ) 2 − ¯ p 2 (t 1 ) 2 = ¯ p 3 (t 0 ) 2 . Thus p 3 (0) = ±¯ p 3 (t 1 ).

Now

p ¯ 1 (−t+ 2K Ω ) = ¯ p 1 (t), ¯ p 2 (−t + 2K Ω ) = ¯ p 2 (t), ¯ p 3 (−t+ 2K Ω ) = −¯ p 3 (t).

Thus there exists t 0 ∈ R such that p(0) = ¯ p(t 0 ).

Theorem: H(p) = p 2 + 1 2 p 1 2

For h 2 0 < c 0 , σ ∈ {−1, 1}



 



 



¯ p 1 (t) = σ

√ 2 p

h 0 + δ − 1 cn (Ω t, k)

¯ p 2 (t) = h 0 − (h 0 + δ − 1) cn (Ω t, k ) 2

¯ p 3 (t) = σ √ 2 p

h 0 + δ − 1 Ω dn (Ω t, k ) sn (Ω t, k ) . Here Ω = √

δ, k =

q h

₀

+δ−1

2δ , and δ = √

1 + c 0 − 2h 0

-2 0 2 E1* -2

0 2 E2* -2 0 2

E3*

-2 0 2 E1* -2

0 2 E2* -2 0 2

E3*

Theorem: H(p) = p 2 + 1 2 p 1 2

Limitng c 0 → h 0 2 , h 0 > 1, σ ∈ {−1, 1}



 

 



 

 



p ¯ 1 (t) = 2σ p

h 0 − 1 sech p

h 0 − 1 t

p ¯ 2 (t) = h 0 − 2(h 0 − 1) sech p

h 0 − 1 t 2

p ¯ 3 (t) = 2σ(h 0 − 1) sech p

h 0 − 1 t

tanh p

h 0 − 1 t .

-2 0 2 E1* -2

0 2 E2* -2 0 2

E3*

-2 0 2 E1* -2

0 2 E2* -2 0 2

E3*

Classification on so (3) ∗ −

H(p) = pQp >

1 2 p 2 1 p 2 1 + 1 2 p 2 2

H(p) = pA + pQp >

αp 1 p 2 + 1 2 p 1 2 p 1 + 1 2 p 1 2 p 1 + αp 2 + 1 2 p 1 2

α 1 p 1 + α 2 p 2 + γ p 3 + p 1 2 + βp 2 2

Conditions α > 0

α 1 , α 2 ≥ 0

γ ∈ R

0 < β < 1.

Integration: H(p) = p 1 + αp 2 + 1 2 p 2 1

Equations of motion

p ˙ 1 = −αp 3 p ˙ 2 = (1 + p 1 )p 3

p ˙ 3 = αp 1 − (1 + p 1 )p 2 .

Consider H(p) = p 1 + p 2 + 1 2 p 1 2 Transform equation

d ¯ p 1 dt = − 1

2 q

4c 0 − 4h 0 2 + 8h 0 ¯ p 1 − 8¯ p 1 2 + 4h 0 p ¯ 2 1 − 4¯ p 3 1 − ¯ p 4 1 Decompose into

d ¯ p 1 dt

2

= (γ 1 p 2 1 + γ 2 p 1 + γ 3 )(η 1 p 2 1 + η 2 p 1 + η 3 )

Integration: H(p) = p 1 + αp 2 + 1 2 p 2 1

-2 0

2 E₁^*

-2 0

E₂^* 2 -2 0 2

E₃^*

-2 0

2 E₁^*

-2 0

E₂^* 2 -2 0 2

E₃^*

-2 0

2 E₁^*

-2 0

E₂^* 2 -2 0 2

E₃^*

-2 0

2 E₁^*

-2 0

E₂^* 2 -2 0 2

E3*

-2 0

2 E₁^*

-2 0

E₂^* 2 -2 0 2

E3*

-2 0

2 E₁^*

-2 0

E₂^* 2 -2 0 2

E3*

Conclusion

Summary

Classification of Hamilton-Poisson systems on so (3) ∗ − Stability nature of equilibria

Integration of several systems Outlook

Integration of remaining systems

Associated optimal control problems on SO (3)

Systems on so (4) ∗ −