## 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

Rolling disk

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

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

## 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

By Klas Modin

# What makes nonholonomic integrators work?

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

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