What makes nonholonomic integrators work?

Klas Modin

Collaborator: Olivier Verdier

Constrained Lagrangian systems

Lagrange-d'Alembert

\displaystyle\delta \int_a^b L(\mathbf{q}(t),\dot{\mathbf{q}}(t))dt = 0

for virtual displacements \( \delta\mathbf{q}\)

A(\mathbf{q}(t))\delta\mathbf{q}(t) = 0

Constraint defined by distribution \(A(\mathbf{q})\dot{\mathbf{q}}=0\)

Constrained Lagrangian systems

Lagrange-d'Alembert

\displaystyle\delta \int_a^b L(\mathbf{q}(t),\dot{\mathbf{q}}(t))dt = 0

for virtual displacements \( \delta\mathbf{q}\)

A(\mathbf{q}(t))\delta\mathbf{q}(t) = 0

Constraint defined by distribution \(A(\mathbf{q})\dot{\mathbf{q}}=0\)

Holonomic systems

\displaystyle A(\mathbf q)\dot{\mathbf q} = \frac{d}{dt}\mathbf F(\mathbf q)
\mathcal C = \{ \mathbf{q}\in \mathcal Q \mid \mathbf F(\mathbf q) = 0 \}

Distribution is integrable
\(\Rightarrow\) Lagrangian dynamics on \(T\mathcal C\)

Nonholonomic systems

\displaystyle A(\mathbf q)\dot{\mathbf q} \neq \frac{d}{dt}\mathbf F(\mathbf q)
\mathcal M = \text{state space}

Distribution nonintegrable

\mathbf q
\dot{\mathbf q}
\mathbf q\cdot\dot{\mathbf q} = 0

Holonomic systems

\displaystyle A(\mathbf q)\dot{\mathbf q} = \frac{d}{dt}\mathbf F(\mathbf q)
\mathcal C = \{ \mathbf{q}\in \mathcal Q \mid \mathbf F(\mathbf q) = 0 \}

Distribution is integrable
\(\Rightarrow\) Lagrangian dynamics on \(T\mathcal C\)

Nonholonomic systems

\displaystyle A(\mathbf q)\dot{\mathbf q} \neq \frac{d}{dt}\mathbf F(\mathbf q)
\mathcal M = \text{state space}

Distribution nonintegrable

\mathbf q
\dot{\mathbf q}
\mathbf q\cdot\dot{\mathbf q} = 0

Nonholonomic examples

Rolling disk

Nonholonomic examples

Knife edge

Nonholonomic examples

Continuously variable transmission (CVT)

What makes holonomic integrators work?

Definition: Numerical integrator

Map \( \Phi_h: (\mathbf{q}_{k},\dot{\mathbf{q}}_{k})\mapsto (\mathbf{q}_{k+1},\dot{\mathbf{q}}_{k+1}) \) that preserves constraints and approximates exact flow \(\varphi^h\)

Exact flow \(\varphi^h\) \(\Rightarrow\) Hamiltonian system on \(T^*\mathcal C\)

Geometric numerical integration and backward error analysis:

  • Integrator is exact flow of modified system on \(T^*\mathcal C\)
  • Modified system Hamiltonian?
    \(\Rightarrow\) modified Hamiltonian preserved
  • Original system Arnold-Liouville integrable?
    \(\Rightarrow\) KAM stability of tori

What about nonholonomic integrators?

Exact flow \(\varphi^h\) \(\Rightarrow\) energy system on \(\mathcal M\)

  • Is that all?
  • Not clear what "structure preserving" means

Idea: discrete analog of Lagrange d'Alembert (DLA)

Energy

Time

Why so good?

Notion: additional structure behind long-time behaviour

Strategy

  • Identify structure of standard nonholonomic test problems

Clues

  • Reversibility known to be important
    [McLachlan and Perlmutter, 2006]
     
  • Most test problems integrable
    (First integrals common evaluation criterion )

Notion: additional structure behind long-time behaviour

Strategy

  • Identify structure of standard nonholonomic test problems

Clues

  • Reversibility known to be important
    [McLachlan and Perlmutter, 2006]
     
  • Most test problems integrable
    (First integrals common evaluation criterion )

Energy

Time

DLA but nonreversible

DLA and reversible

Nonholonomically coupled systems

Definition: Nonholonomically Coupled System (NCS)

 

 

 

L(\mathbf x, \xi, \dot{\mathbf x},\dot\xi) = L_1(\mathbf x,\dot{\mathbf x}) + L_2(\xi,\dot\xi)
A(\xi)\dot{\mathbf x} = 0
\displaystyle\frac{d}{dt}\frac{\partial L_2}{\partial\dot\xi} =\frac{\partial L_2}{\partial \xi}

independent subsystem (driver)

\dot{\mathbf x} = \mathbf{k}(\xi) v \qquad \dot v= \mathbf{k}(\xi)\cdot \mathbf F(\mathbf x)

Conserved energies: \(E_1(\mathbf x,v)\) and \(E_2(\xi,\dot\xi)\)

Is system on \(\mathcal M\) integrable?

Integrable NCS

\dot{\mathbf u} = B(\xi)\mathbf u

\(B(\xi)\in \mathfrak{g}\subset \mathfrak{gl}(n)\)
(\(\mathfrak{g}\)-system)

Definition: ODE system integrable \(\iff\) Action-Angle variables

Theorem (M. and Verdier)

NCS (with additional assumptions) are fibrated over integrable system

Reduced dynamics

Next step: KAM theory

Requirement for KAM:

  • dynamics either symplectic or reversible

Exact flow:

(\mathbf q_0,\dot{\mathbf q}_0)\mapsto (\mathbf q(h),\dot{\mathbf q}(h))
\mathcal M \to \mathcal M
\mathcal N \to \mathcal N

Action-Angle variables

\mathcal N \to \mathcal N

KAM stable tori

reversible perturbation

Discrete flow:

(\mathbf q_k,\dot{\mathbf q}_k)\mapsto (\mathbf q_{k+1},\dot{\mathbf q}_{k+1})
\mathcal M \to \mathcal M
\approx

Next step: KAM theory

Requirement for KAM:

  • dynamics either symplectic or reversible

Exact flow:

(\mathbf q_0,\dot{\mathbf q}_0)\mapsto (\mathbf q(h),\dot{\mathbf q}(h))
\mathcal M \to \mathcal M
\mathcal N \to \mathcal N

Action-Angle variables

\mathcal N \to \mathcal N

KAM stable tori

reversible perturbation

Discrete flow:

(\mathbf q_k,\dot{\mathbf q}_k)\mapsto (\mathbf q_{k+1},\dot{\mathbf q}_{k+1})
\mathcal M \to \mathcal M
\approx

possible failure

Next step: KAM theory

Requirement for KAM:

  • dynamics either symplectic or reversible

Exact flow:

(\mathbf q_0,\dot{\mathbf q}_0)\mapsto (\mathbf q(h),\dot{\mathbf q}(h))
\mathcal M \to \mathcal M
\mathcal N \to \mathcal N

Action-Angle variables

\mathcal N \to \mathcal N

KAM stable tori

reversible perturbation

Discrete flow:

(\mathbf q_k,\dot{\mathbf q}_k)\mapsto (\mathbf q_{k+1},\dot{\mathbf q}_{k+1})
\mathcal M \to \mathcal M
\approx

possible failure

Theorem (M. and Verdier)

If discrete flow

  1. Preserve fibration
  2. Preserves reversibility (from action-angle variables)

then all integrals are nearly conserved

Next step: KAM theory

Requirement for KAM:

  • dynamics either symplectic or reversible

Exact flow:

(\mathbf q_0,\dot{\mathbf q}_0)\mapsto (\mathbf q(h),\dot{\mathbf q}(h))
\mathcal M \to \mathcal M
\mathcal N \to \mathcal N

Action-Angle variables

\mathcal N \to \mathcal N

KAM stable tori

reversible perturbation

Discrete flow:

(\mathbf q_k,\dot{\mathbf q}_k)\mapsto (\mathbf q_{k+1},\dot{\mathbf q}_{k+1})
\mathcal M \to \mathcal M
\approx

possible failure

Theorem (M. and Verdier)

If discrete flow

  1. Preserve fibration
  2. Preserves reversibility (from action-angle variables)

then all integrals are nearly conserved

Numerical verification

CVT problem

Sub-energy \(E_1\)

Time

Is the result sharp?

Strategy: construct NCS by perturbing

  1. fibration projection
  2. reversibility map

IMPORTANT: resulting systems are still integrable NCS

Knife edge, perturbed fibration

Total energy

Is the result sharp?

Strategy: construct NCS by perturbing

  1. fibration projection
  2. reversibility map

IMPORTANT: resulting systems are still integrable NCS

CVT, perturbed reversibility

Sub-energy \(E_1\)

Conclusion and outlook

  • DLA does not imply structure preservation
     
  • Reason: class of nonholonomic systems too large for generic DLA approach
     
  • Strategy: focus on subclasses with enough structure for e.g. KAM (Hamiltonian or reversible)

    Already started: recent work by [Martín de Diego et. al.]

THANKS!

Slides available at: slides.com/kmodin

What makes nonholonomic integrators work?

By Klas Modin

What makes nonholonomic integrators work?

Online-presentation given 2021-05 in the Geometry, Dynamics, and Mechanics Seminar.

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