Straight line rolling of an ellipsoid on a plane and the Chasles - - PowerPoint PPT Presentation

Straight line rolling of an ellipsoid on a plane and the Chasles theorem

Yuri F¨ edorov, UPC, Barcelona (still Spain) Q = X 2

1

A1 + X 2

1

A2 + X 2

3

A3 = 1

• ⊂ R3 = (X1, X2, X3)
• n
• γ

No slip no twist

The Jacobi geodesic problem on the ellipsoid Q

Linealization of the geodesic flow on Q (Jacobi and Weierstrass): Let λ1, λ2 be the ellipsoidal coordinates on Q, X 2

i = Ai

(Ai − λ1)(Ai − λ2) (Ai − Aj)(Ai − Ak) , i = 1, 2, 3. After time re-parametrization ds = λ1λ2 ds1, reduction to quadratures (Jacobi, 1881) dλ1 2

• R(λ1)

+ dλ2 2

• R(λ2)

= ds1, λ1dλ1 2

• R(λ1)

+ λ2dλ2 2

• R(λ2)

= 0, R(λ) = −λ(λ − A1)(λ − A2)(λ − A3)(λ − c) where c is a constant of motion such that the geodesic is tangent to the caustic Q ∩ Qc.

Geodesic flow on n − 1-dimensional quadric

Family of confocal quadrics in Rn(X1, . . . , Xn) Q(c) =

• X 2

1

A1 − c + · · · + X 2

n

An − c = 1

• ,

c ∈ R. Theorem (Chasles) Let X(s) be a geodesic on Q = Q(0) with a natural parameter s and γ(s) = dX/ds be the tangent vector. Then 1) the tangent line ℓ = {X + t γ|t ∈ R} is also tangent to n − 1 fixed confocal quadrics Q(c1) = Q(0), Q(c2), . . . , Q(cn−1). 2) Let qj be a unit normal vector of Q(cj) at the contact point pj = ℓ ∩ Q(cj). The vectors q1, . . . , qn−1, γ form an orthogonal frame in Rn.

Theorem (following J.Moser) When X(s) traces a geodesic on Q, the evolution of q1, . . . , qn−1, γ is described by d ds qj = −Ω qj, j = 1, . . . , n − 1, d ds γ = −Ωγ, Ω = X, A−2X A−1X ∧ A−1γ = q ∧ A−1γ .

• Corollary. In the reference frame R = {

q1, . . . , qn−1, γ} the ellipsoid Q rolls on the hyperplane H = span( q12, . . . , qn−1, γ) without slipping and twisting. On H the contact point Q ∩ H moves alongs the line L =span( γ). On the ellipsoid Q the contact point traces the geodesic X(s). In the frame R the angular velocity of Q has the form ¯ Ω =      Ω12 · · · Ω1,n Ω12 · · · . . . . . . ... . . . Ω1,n · · ·      For n = 3 the angular velocity vector of Q satisfies ¯ ω, q1 = 0

Relation with the Neumann system (H. Kn¨

• rrer)

Let X(s) be a geodesic on Q, and q(s) = q1(s) be the normal unit vector of Q at the point X. Then q(s1) is a solution of the Neumann problem on Sn−1 = {q, q} = 1 with H = 1

2(˙

q, ˙ q − q, Iq), I = A−1: d2 ds2

1

q = Iq + νq, provided that ds = X, A−2Xds1.

Separation of variables for the Neumann system

Let λ1, . . . , λn−1 be spheroconical coordinates on Sn−1 = {q, q = 1}: q2

i = (Ii − λ1) · · · (Ii − λn−1)

• j=i(Ii − Ij)

, then                dλ1 2

• R(λ1)

+ · · · + dλn−1 2

• R(λn−1)

= 0, · · · · · · λn−2

1

dλ1 2

• R(λ1)

+ · · · + λn−2

n−1dλn−2

2

• R(λn−1)

= ds1, R(λ) = −(λ − I1) · · · (λ − In) · (λ − C1) · · · (λ − Cn−1), C1 = 0, Cj = 1/cj j = 2, . . . , n − 1

The rotation matrix of Q in the fixed frame R = { q1, . . . , qn−1, γ}: R = ( q1 · · · qn−1 γ)T ∈ SO(n) in terms of points P1 = (λ1, µ1), . . . , Pn−1 = (λn−1, µn−1) on the genus g = n − 1 hyperelliptic curve Γ : µ2 = −(λ − I1) · · · (λ − In) · (λ − C1) · · · (λ − Cn−1) , C1 = 0 q1,i =

• U(Ii)
• Ψ′(Ii)

, i = 1, . . . , n, qs,i =

• U(Ii)
• Ψ′(Ii)
• U(Cs)
• ψ′(Cs)

n−1

• k=1

µk (Ii−λk)(Cs−λk) ,

s = 2, . . . , n − 1, γi =

• U(Ii)
• Ψ′(Ii)
• U(0)
• ψ′(0)

n−1

• k=1

µk (Ii−λk)λk ,

where U(r) = (r − λ1) · · · (r − λn−1), Ψ(λ) = (λ − I1) · · · (λ − In), ψ(λ) = (λ − C1) · · · (λ − Cn−1). ( Following Yu. F., B. Jovanovi´

• c. J. Nonl. Sci. 2004)

The angular velocity of Q in the frame R = { q1, . . . , qn−1, γ} ¯ Ω =      Ω12 · · · Ω1,n Ω12 · · · . . . . . . ... . . . Ω1,n · · ·      with Ω1,s =

• U(0)
• ψ′(0)
• U(Cs)
• ψ′(Cs)

n−1

• k=1

µk λk(Cs − λk) , s = 2, . . . , n − 1, Ω1,n =

• U(0)
• ψ′(0)

. The coordinates of the center of Q in R: (X, q1, · · · , X, qn−1, s + X, γ), X =

• λ1 · · · λn−1 Aq1,

s =

• λ1 · · · λn−1 ds1.

The angular velocity of the frame R = { q1, . . . , qn−1, γ} with respect to the axes {X1, . . . , Xn} of the ellipsoid Q: ωij =

• U(Ii)
• Ψ′(Ii)
• U(Ij)
• Ψ′(Ij)

n−1

• k=1

µk (Ii − λk)(Ij − λk) , 1 ≤ i < j ≤ n.

Theta-function solution

Let B be the g × g period matrix of the genus g = n − 1 curve Γ : µ2 = −(λ − I1) · · · (λ − In) · (λ − C1) · · · (λ − Cn−1), C1 = 0. Introduce the corresponding theta-function θ(z|B) =

• M∈Zg

exp(BM, M/2 + M, z), M, z =

g

• i=1

Mizi, BM, M =

g

• i,j=1

BijMiMj, z ∈ Cg, as well as theta-functions with characteristics α = (α1, . . . , αg), β = (β1, . . . , βg), αj, βj ∈ R, which are obtained from θ(z|B) by shifting the argument z and multiplying by an exponent: θ α β

• (z) = exp{Bα, α/2 + z + 2πβ, α} θ(z + 2πβ + Bα).

Theta-function solution (II)

• Half-integer theta-characteristics ηi = [η′′

i , η′ i] such that

2π η′′

i + Bη′ i =

(Ii,0)

∞

¯ ω

• r

= (Ci,0)

∞

¯ ω The rotation matrix of Q in the fixed frame R is R =         σ1 θ[ηI1](z) θ(z) ε1 θ[ηI1 + ηC2](z) θ(z) κ1 θ[ηI1 + ηC1](z) θ(z) σ2 θ[ηI2](z) θ(z) ε2 θ[ηI2 + ηC2](z) θ(z) κ2 θ[ηI2 + ηC1](z) θ(z) σ3 θ[ηI3](z) θ(z) ε3 θ[ηI3 + ηC2](z) θ(z) κ3 θ[ηI3 + ηC1](z) θ(z)         , z = vs1 + z0 , v ∈ Cn, z0 = const ∈ Cn.