Main Idea Idea To study the discretizations of dynamical systems - - PowerPoint PPT Presentation

On the Discretization of Nonholonomic Dynamics in Rn

Fernando Jim´ enez (Joint work with J¨ urgen Scheurle) Technische Universit¨ at M¨ unchen Nonholonomic Mechanics and Optimal Control Workshop Institut Henri Poincar´ e, Paris November, 2014

Institut Henri Poincar´ e (Paris) Nov 2014 1 / 48

Main Idea

Idea

To study the discretizations of dynamical systems viewed as perturbations: Qualitative and quantitative aspects.

Question

Can we understand a discretization of NH systems as a non-autonomous perturbation of the continuous dynamics? “On the discretization of nonholonomic dynamics on Rn”, Jim´ enez and Scheurle; Preprint arXiv:1407.2116 (Submitted).

Remark

We do not consider autonomous perturbations of the continuous dynamics (BEA).

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Autonomous Perturbation: Backward Error Analysis

ODE: ˙ x = h(x). Numerical flow xk+1 = φ(ǫ, xk), where xk ≃ x(tk), s.t. tk = t0 + ǫ k. ǫ time step. φ(ǫ, x) = x + ǫ h(x) + ǫ2 d2(x) + ǫ3 d3(x) + ... Order of consistency |x(tk) − xk| = O(ǫp+1). BEA: ˙ ˜ x = hǫ(˜ x), s.t. ˜ x(tk) = xk. ˙ ˜ x = h(˜ x) + ǫ h2(˜ x) + ǫ2 h3(˜ x) + .... Comparing series expansions h2(y) = d2(y) − 1 2!h′ h(x), h3(y) = d3(y) − 1 3!(h′′(x) + h′ h′ h(y)) − 1 2!(h′ h2(y) + h′

2h(x)).

Drawback: Backward Error Analysis is an Asymptotic Theory

Institut Henri Poincar´ e (Paris) Nov 2014 3 / 48

Outline

Introduction The Lagrangian nonholonomic problem Discretization of nonholonomic dynamics as perturbation Variational integrators Nonholonomic integrators Examples and plots

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Introduction

Fielder B and Scheurle J: “Discretization of homoclinic orbits, rapid forcing and invisible chaos.” Memoirs of the American Mathematical Society, 119(570), (1996).

Theorem

Suposse that h ∈ Cl(Rn, Rn), l ≥ 1, and consider the autonomous ODE ˙ x = h(x). (1) Let F(t, x) be the fundamental solution of (1) satisfying F(0, x) = 0, and assume that there are an integer ρ ≥ 1, a continuous function C : [0, ∞) → [0, ∞) and a one-step difference approximation

• f step size ǫ

xk+1 = φ(ǫ, xk), (0 < ǫ ≤ ǫ0; k ∈ Z) which is consistent of order p, i.e. |φ(ǫ, x) − F(ǫ, x)| ≤ C(|x|) ǫp+1. Then, there exists a function g(ǫ, τ, x), as smooth as h and periodic in τ of period 1, such that if G(t, s; ǫ, x), G(s, s; ǫ, x) = x, is the fundamental solution of the non-autonomous, ǫ−periodic ODE ˙ x = h(x) + ǫpg(ǫ, t/ǫ, x), (2) then G(ǫ, 0; ǫ, x) = φ(ǫ, x), where G(ǫ, 0; ǫ, ·) : Rn → Rn is the Poincar´ e map (period map) for (2), corresponding to initial time s = 0.

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Introduction

Theorem

Any p−th order discretization xk+1 = φ(ǫ, xk)

• f the autonomous Ordinary Differential Equations

˙ x = h(x), |x(tk) − xk| ∼ O(ǫp+1), can equivalently be viewed as the time ǫ−period map of a suitable ǫ−periodic non-autonomous perturbation of the original ODE ˙ x = h(x) + ǫpg(ǫ, t/ǫ, x), where ǫ is the fixed discretization lenght. This is, if ˜ x(t) is the solution of the perturbed equation, then ˜ x(tk+1) = xk+1 = φ(ǫ, xk).

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Outline

1

The Lagrangian nonholonomic problem

2

Discretization of nonholonomic dynamics as perturbation

3

Variational integrators

4

Nonholonomic integrators

5

Examples and plots

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The Lagrangian nonholonomic problem

Q is a n−dimensional manifold with local coordinates qi, i = 1, ..., n. TQ tangent bundle with local coordinates (qi, ˙ qi). D non-integrable constant rank (n − m) distribution on Q (D ⊂ TQ). Linear constraints µα

i (q) ˙

qi = 0, 1 ≤ α ≤ m. D◦ =

• µα = µα

i (q) dqi, 1 ≤ α ≤ m

• , µα independent.

Lagrangian function L : TQ → R.

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Lagrange-d’Alembert principle

Triple (Q, D, L) defines a nonholonomic system. Lagrange-d’Alembert principle: q : I ⊂ R → Q δ t1

t0

L (q (t) , ˙ q (t)) dt = 0 for all variations δq(t) ∈ Dq(t), q(t0), q(t1) fixed, t ∈ [t0, t1].

Nonholonomic equations (DAE)

d dt ∂L ∂ ˙ qi

• − ∂L

∂qi = λα µα

i (q),

µα

i (q) ˙

qi = 0, λα, α = 1, ..., m, set of Lagrange multipliers.

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Examples: The vertical rolling coin

Q = R2 × S1 × S1, q = (x, y, θ, ϕ). L = 1

2m(˙

x2 + ˙ y2) + 1

2I ˙

θ2 + 1

2J ˙

ϕ2, ˙ x − R cos ϕ ˙ θ = 0, ˙ y − R sin ϕ ˙ θ = 0. Lagrange-d’Alembert principle: m ¨ x = λ1, m ¨ y = λ2, J ¨ ϕ = 0, I ¨ θ = −λ1 R cos ϕ − λ2 R sin ϕ.

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Examples: The snakeboard

Q = SE(2) × S1 × S1, q = (x, y, θ, ψ, φ). L = 1

2m(˙

x2 + ˙ y2) + 1

2m r2 ˙

θ2 + 1

2J0 ˙

ψ2 + J0 ˙ ψ ˙ θ + 1

2J1 ˙

φ2, − sin (θ + φ) ˙ x + cos (θ + φ) ˙ y − r cos φ ˙ θ = 0, − sin (θ − φ) ˙ x + cos (θ − φ) ˙ y + r cos φ ˙ θ = 0. Lagrange-d’Alembert principle: m ¨ x = −λ1 sin (θ + φ) − λ2 sin (θ − φ), m ¨ y = λ1 cos (θ + φ) − λ2 cos (θ − φ), ¨ θ = (−λ1 r cos φ + λ2 r cos φ)/(m r2 − J0), ¨ ψ = (λ1 r cos φ − λ2 r cos φ)/(m r2 − J0), J1 ¨ φ = 0.

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Outline

1

The Lagrangian nonholonomic problem

2

Discretization of nonholonomic dynamics as perturbation

3

Variational integrators

4

Nonholonomic integrators

5

Examples and plots

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Question

Question

Can we understand a discretization of NH systems as a non-autonomous perturbation of the continuous dynamics?

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Nonholonomic equations

Q = Rn. Second order condition ˙ qi = vi. Regularity condition: det

• ∂2L

∂vj∂vi

• = 0.

Nonholonomic DAE

˙ qi = vi, ˙ vi = ∂2L ∂vj∂vi −1 − ∂2L ∂qk∂vj vk + ∂L ∂qj + λα µα

j (q)

• ,

µα

i (q) vi = 0,

(q(t0) = q0, v(t0) = v0), system of 2n + m Differential Algebraic Equations (together with initial conditions for q(t) and v(t)). Local Nonholonomic flow (q(t), v(t), λ(t)) = FNH

t

(q0, v0), s.t. (q(t), v(t)) ∈ D.

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Nonholonomic equations

x = (qi, vi) ∈ R2n and λα ∈ Rm, (mij) =

• ∂2L

∂vj∂vi

• ,

ϕ : R2n → Rm, ϕα(x) = µα

i (q) vi.

Nonholonomic DAE

˙ x = f(x) + λα

• m−1(x)∇vϕα(x)
• ,

ϕα(x) = 0. f(x) =

• f i

q(x), f i v(x)

• =
• vi , −mik

∂2L ∂qj∂vk vj + mij ∂L ∂qj

• .

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Question

Question

Instead of a DAE, can we obtain an ODE defining the flow within D?

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Constraint submanifold

Nonholonomic constraints ϕ(x) = µα

i (q) vi = 0 defines a regular submanifold

D ⊂ R2n. Perpendicularity condition: ∇ϕβ(x) , f(x) + λα

• m−1(x)∇vϕα(x)
• = 0,

This determines a unique λα(x) enforcing the vector field to be tangent to D. ˙ x = h(x) s.t. x(t) ∈ D ⊂ R2n and h(x) ∈ TxD ⊂ TxR2n.

Nonholonomic ODE on D (ϕ(x) = 0)

˙ x = h(x) = f(x) + λα(x)

• m−1(x)∇vϕα(x)
• ,

(x(t0) = x0 ∈ D).

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Result

Approach

Apply the result by Fielder and Scheurle on this system to obtain a corresponding perturbed ODE on D. Extending that to all of TQ × Rm = R2n × Rm, we obtain

Result

Any p−th order discretization of the nonholonomic ODE (evolving on D) may be embedded into a non-autonomous perturbation of the original nonholonomic DAE of the form ˙ x = f(x) + λα

• m−1(x)∇vϕα(x)
• + ǫp˜

g(ǫ, t/ǫ, x), ϕα(x) = 0.

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Ehresmann connection

Bloch, A: “Nonholonomic Mechanics and Control”, Springer-Verlag New-York, (2003).

Ehresmann connection: global definition

An Ehresmann connection A is a vertical vector-valued one-form on Q, which satisfies Aq : TqQ → Vq is a linear map at each point q ∈ Q, A is a projection, i.e., A(vq) = vq, for all vq ∈ Vq. The connection allows TqQ = Hq ⊕ Vq, s.t. Hq = ker Aq. Why do we need the connection? It helps to choose convenient coordinates for our

• system. Moreover, it guarantees the independence of the coordinate choice in TQ.

Hq = Dq := D ∩ TqQ, and thus the constraints are globally expressed as A · vq = 0 for any vq ∈ TqQ.

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Ehresmann connection

Original coordinates x = (qi, vi) ∈ R2n, Ehresmann connection: (qi, va, vα = −Aα

a (q) va) ∈ D ⊂ R2n,

i = 1, ..., n, a = 1, ..., n − m and α = 1, ..., m. ψ : (qi, va) ∈ R2n−m → (qi, va, −Aα

a (q) va) ∈ D ⊂ R2n. ξ = (qi, va) ∈ R2n−m.

Nonholonomic ODE

˙ x = h(x) s.t. ϕ(x) = 0 may be transformed into ˙ ξ = (∇ξψ)−1 (ξ) h (ψ(ξ)) .

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Interpolation within D

Perturbation

˙ ξ(t) = (∇ξψ)−1 (ξ(t)) h (ψ(ξ(t))) + ǫpg(ǫ, t/ǫ, ξ(t)). Before, we need to be sure that we can connect up any two points in D by a curve inside D. Interpolate between z1 = (qi

1, va 1, vα 1 = −Aα a (q) va 1) ∈ D ⊂ R2n,

z2 = (qi

2, va 2, vα 2 = −Aα a (q) va 2) ∈ D ⊂ R2n.

Cut-off functions χ0 : R → [0, 1]

χ0(τ) ≡ 1 for τ ≤ 0, χ0(τ) ≡ 0 for τ ≥ 1, s.t. χ0 is real analytic for τ = 0, 1. χ1 := 1 − χ0(τ). For instance χ0(τ) = (1 + tanh (cot(πτ))) /2, 0 < τ < 1.

Define the C∞ curves qi : [0, ǫ] → R, va : [0, ǫ] → R by qi(t) = χ0(t/ǫ) qi

1 + χ1(t/ǫ) qi 2,

va(t) = χ0(t/ǫ) va

1 + χ1(t/ǫ) va 2.

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Important

Result

Define the C∞ curve c : [0, ǫ] → R2n by c(t) = (qi(t), va(t), vα(t) = −Aα

a (q(t)) va(t)) ⊂ D.

D → TRn → TRn × Rm

Result

Any p−th order discretization of the nonholonomic ODE (evolving on D) may be embedded into a non-autonomous perturbation of the original nonholonomic DAE of the form ˙ x = f(x) + λα

• m−1(x)∇vϕα(x)
• + ǫp˜

g(ǫ, t/ǫ, x), ϕα(x) = 0. IMPORTANT: this result applies if

the integrator we are studying evolves in D, i.e. it is D−preserving, the integrator is p−th order consistent with the continuous dynamics on D.

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Questions

Questions

Is there a way of generating, systematically, D−preserving integrators? May we say anything about their consistency order?

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Outline

1

The Lagrangian nonholonomic problem

2

Discretization of nonholonomic dynamics as perturbation

3

Variational integrators

4

Nonholonomic integrators

5

Examples and plots

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Variational integrators

Idea

To discretize variational principles (Moser and Veselov; Marsden and West; et al) TQ ⇒ Q × Q. Moreover Ld : Q × Q → R: Ld(q0, q1) ≈ t0+ǫ

t0

L(q(t), ˙ q(t)) dt. q(t0) = q0, q(t0 + ǫ) = q1. Define the sequences (q0, ..., qN) ∈ QN+1, qk ≃ q(tk), and the Action Sum Sd =

N

• k=1

Ld(qk−1, qk). Extremizing Sd (fixed (q0, qN)) you obtain the discrete Euler-Lagrange equations: D1Ld(qk, qk+1) + D2Ld(qk−1, qk) = 0, 1 ≤ k ≤ N − 1. If D1D2Ld(qk, qk+1) is a regular matrix, then Discrete Flow: FLd : Q × Q → Q × Q, (qk−1, qk) → (qk, qk+1).

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Outline

1

The Lagrangian nonholonomic problem

2

Discretization of nonholonomic dynamics as perturbation

3

Variational integrators

4

Nonholonomic integrators

5

Examples and plots

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Nonholonomic integrators

Idea

To discretize the Lagrange-d’Alembert principle (DLdA). (Cort´ es and Mart´ ınez, McLachlan and Perlmutter) Discrete nonholonomic system (Q × Q, Ld, Dd):

• 1. Dd ⊂ Q × Q, dim Dd = 2n − m, s.t.

Id = {(q, q) | q ∈ Q} ⊂ Dd.

• 2. φα

d m appropriate functions, φα d : Q × Q → R, α = 1, ..., m, s.t. φα d (qk, qk+1) = 0 ⇒ Dd.

(Discretization of the constraints)

• 3. Ld : Q × Q → R is a discrete Lagrangian as defined above.

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Discrete Lagrange-d’Alembert principle

Extremize Sd (fixed (q0, qN)) with δqk ∈ Dqk and (qk, qk+1) ∈ Dd:

DLdA

D1Ld(qk, qk+1) + D2Ld(qk−1, qk) = (λα)k+1 µα(qk), (qk, qk+1) ∈ Dd, i.e. φα

d (qk, qk+1) = 0,

1 ≤ k ≤ N − 1. Non-degeneracy condition: det   D1D2Ld(qk, qk+1) µα(qk) D2φα

d (qk, qk+1)

  = 0 Discrete nonholonomic flow FNH

Ld : Dd → Dd, (qk−1, qk) → (qk, qk+1) s.t.

(qk−1, qk) ∈ Dd then (qk, qk+1) ∈ Dd, provided λk+1 is chosen appropriately.

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Discrete nonholonomic flow

Drawback

IN GENERAL, the discrete nonholonomic flow FNH

Ld does NOT preserve the original

nonholonomic distribution D. (Indeed FNH

Ld is defined in Q × Q instead of TQ)

Finite Difference Maps (McLachlan and Perlmutter)

We employ finite difference maps ρ : Q × Q → TQ (local diffeomorphisms). To define the discrete nonholonomic system (Q × Q, Ld, Dd):

Ld = ǫ L ◦ ρ, Dd defined through φα

d = µα ◦ ρ (Particular Choice of φα d ).

We define a so-called velocity nonholonomic integrator ˜ FLd : TQ → TQ through ˜ FLd := ρ ◦ FNH

Ld ◦ ρ−1.

Still, ˜ FLd does NOT necessarily preserve D.

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D preservation

D preservation

A sufficient condition to guarantee D−preservation for the velocity nonholonomic integrators, requires in general, besides φα

d = µα ◦ ρ, a redefinition of the nodes as we

show in the case of mechanical systems along with consistency results.

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Mechanical systems

Q = Rn, Lagrangian function defined as kinetic minus potential energy.

Mechanical systems

L(q, ˙ q) = 1 2 ˙ qTM˙ q − V(q)

Nonholonomic DAE

˙ q = v, ˙ v = −M−1∇qV(q) + λαM−1µα(q), 0 = µα(q) v.

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Consistency results

Finite difference map

ρ(qk, qk+1) = qk + qk+1 2 , qk+1 − qk ǫ

• {qk}0:N = {q0, q1, ..., qN−1, qN} .

Using these nodes, the velocity nonholonomic integrator is given by

Algorithm Al

qk+1 = qk + ǫ vk vk+1 = vk − 1 2ǫ M−1

• ∇qV
• qk + 1

2ǫvk

• + ∇qV
• qk − 1

2ǫvk

• + (λk+1)αM−1µα(qk),

µα qk+1 + ǫ 2vk+1

• vk+1 = 0,

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Consistency results

A1

A1: (qk, vk) → (qk+1, vk+1) First order consistent with respect to (q, v) variables. Does NOT preserve D. µα (qk+1) vk+1 = 0 µα qk+1 + ǫ

2vk+1

• vk+1 = 0,

µα (qk+1) vk+1 + ǫ

2vk+1∇µα(qk+1)vk+1 + O(ǫ2) = 0

Deformed constraints

A1 is preserving a deformed constraint manifold which is not even linear in the velocities anymore.

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Consistency results

Redefine the nodes: ˜ qk+1 =

qk+1+qk 2

, {˜ qk}1:N−1

Algorithm A2

˜ qk+1 = ˜ qk + 1 2ǫ˜ vk + 1 2ǫ˜ vk+1 ˜ vk+1 = ˜ vk − 1 2ǫ M−1 (∇qV(˜ qk+1) + ∇qV(˜ qk)) + (λk+1)αM−1µα(˜ qk + 1 2ǫ˜ vk), µα(˜ qk+1)˜ vk+1 = 0.

A2

A2: (˜ qk,˜ vk) → (˜ qk+1,˜ vk+1) Second order consistent with respect to (q, v) variables. DOES preserve D.

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Outline

1

The Lagrangian nonholonomic problem

2

Discretization of nonholonomic dynamics as perturbation

3

Variational integrators

4

Nonholonomic integrators

5

Examples and plots

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Nonholonomic particle

Nonholonomic particle

L = 1 2(˙ x2 + ˙ y2 + ˙ z2), ˙ z − y ˙ x = 0. Applying the Lagrange-d’Alembert principle and setting (˙ x, ˙ y, ˙ z)T = (vx, vy, vz)T

Nonholonomic DAE

˙ vx = −yλ, ˙ x = vx, ˙ vy = 0, ˙ y = vy, ˙ vz = λ, ˙ z = vz, vz − y vx = 0. Eliminating the Lagrange multiplier, λ =

1 1+y2 vx vy:

Nonholonomic ODE

˙ vx = −

y 1+y2 vx vy,

˙ x = vx, ˙ vy = 0, ˙ y = vy, ˙ vz =

1 1+y2 vx vy,

˙ z = vz.

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Plots

A2, redefining the nodes: preserves the constraint vz − y vx = 0.

• 2

4 6 8 10

• 1.µ10-13
• 5.µ10-14

5.µ10-14 1.µ10-13 1.5µ10-13 2.µ10-13 Time Constraint

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Plots

A1, without redefining the nodes: does not preserve the constraint vz − y vx = 0.

• 2

4 6 8 10

• 0.0010
• 0.0008
• 0.0006
• 0.0004
• 0.0002

Time Constraint

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Plots

A2 (D-preserving) vs A1 (non-preserving)

• 2

4 6 8 10

• 0.0010
• 0.0008
• 0.0006
• 0.0004
• 0.0002

0.0000 Time Constraint

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Plots

A1, without redefining the nodes: preserves the deformed constraint vz − y vx + ǫ

2vxvy = 0.

• 2

4 6 8 10

• 1.µ10-13
• 5.µ10-14

5.µ10-14 1.µ10-13 1.5µ10-13 2.µ10-13 Time Constraint

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Plots

Performance of the A2 method in the vz component.

• e = 10-3

2 4 6 8 10 1.0 1.1 1.2 1.3 1.4 Time vz

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Conclusions

Any p−th order discretization of the nonholonomic ODE (evolving on D) may be embedded into a non-autonomous perturbation of the original nonholonomic DAE. The discretization of the Lagrange-d’Alembert principle provides D−preserving integrators of the nonholonomic systems under some circumstances, namely a particular choice of the discrete constraints and the nodes. This nodes choice can be easily performed through finite difference maps.

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Finale

Thanks a lot!!

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Nonholonomic rigid body (Suslov problem)

ω ∈ so(3) ∼ = R3, l : so(3) → R, I : so(3) → so∗(3), a = (0, 0, 1).

Nonholonomic rigid body

l = Iω, ω, a, ω = ω3 = 0.

Nonholonomic DAE

d dt ∂l ∂ω

• =

∂l ∂ω

• × ω + λ a;

a, ω = ω3 = 0. Eliminating the Lagrange multiplier λ =

a,I−1(ω× ∂l

∂ω )

a,I−1a

:

Nonholonomic ODE for ω

d dt ∂l ∂ω

• =

1 a, I−1aa, ∂l ∂ω

• I−1a × ω
• .

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Nonholonomic rigid body (Suslov problem)

I = (Iij), I−1 = (Iij).

Nonholonomic ODE

I11 ˙ ω1 + I12 ˙ ω2 = ω2 I33

• I31(I11ω1 + I12ω2) + I32(I21ω1 + I22ω2)
• ,

I21 ˙ ω1 + I22 ˙ ω2 = − ω1 I33

• I31(I11ω1 + I12ω2) + I32(I21ω1 + I22ω2)
• .

I =   1 0.1 0.2 0.1 1 0.1 0.2 0.1 1  

Finite difference map

ρ : R3 × R3 → TR3 ρ(ωk, ωk+1) =

• ωk,

ωk+1−ωk ǫ

• Institut Henri Poincar´

e (Paris) Nov 2014 45 / 48

Plots

Performance of the method in the ω1 component.

• e = 10-3

2 4 6 8 10

• 0.1

0.0 0.1 0.2 0.3 0.4 Time w1

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Plots

Performance of the method in the ω2 component.

• e = 10-3

2 4 6 8 10 0.50 0.55 0.60 0.65 Time w2

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Plots

Preservation of the constraint ω3 = 0.

• 2

4 6 8 10

• 2.µ10-17
• 1.µ10-17

1.µ10-17 2.µ10-17 Time w3

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