## Joint work with

Robert McLachlan

Massey University

Olivier Verdier

Western Norway University

## Outline

• Symplectic integrators on $$\mathbb{R}^{2n}$$
• Slightly dissipative systems
• Spherical midpoint method

## Illustration

• Non-linear pendulum $$H(q,p) = p^2/2 -\cos(q)$$
\dot q = p, \quad \dot p = - \sin(q)
q
p

## Illustration

• Non-linear pendulum $$H(q,p) = p^2/2 -\cos(q)$$
\dot q = p, \quad \dot p = - \sin(q)

## Illustration

• Non-linear pendulum $$H(q,p) = p^2/2 -\cos(q)$$
\dot q = p, \quad \dot p = - \sin(q)

Why?

## Hamiltonian dynamics on co-tangent bundles

Phase space $$T^*Q$$

Simplest case $$T^*\mathbb{R}^n\simeq \mathbb{R}^{2n}$$

\displaystyle\dot q^i = \frac{\partial H}{\partial p_i}(q^1,\ldots,q^n,p_1,\ldots,p_n)
\displaystyle\dot p_i = -\frac{\partial H}{\partial q^i}(q^1,\ldots,q^n,p_1,\ldots,p_n)

Typical examples: celestial mechanics, molecular dynamics

\mathbf{z}=(q,p) \;\Rightarrow\; \dot \mathbf{z} = X_H(\mathbf{z})

## Symplectic property of the flow

Symplectic 2-form:

\Omega = dq^i\wedge dp_i

On $$\mathbb{R}^{2n}$$:

\displaystyle\Omega_z(U,V) = U^\top \begin{pmatrix} 0 & -I_{n\times n} \\ I_{n\times n} & 0 \end{pmatrix} V

Hamiltonian vector fields fulfill (defining property)

\Omega_z(X_H,\cdot) = d H
\Omega_z(X_H,X_G) = \{H,G \}(z)

Flow $$\varphi_t$$ preserves the symplectic form:

\Omega_{\varphi_t(z)}(D\varphi_t(z)\cdot U,D\varphi_t(z)\cdot V) = \Omega_z(U,V)

Geometric interpretation: the "symplectic area" of 2D surfaces in phase space are preserved

## Consequences of symplecticness

This explains why symplectic integrators are superior!

• "Almost" conservation of tori for Hamiltonian perturbations of integrable systems (KAM theory)

• Convergence towards correct macroscopic equilibrium (e.g. temperature in MD)

## Common symplectic integrators on $$\mathbb{R}^{2n}$$

• Verlet scheme (many names: Leap-frog, Störmer-Verlet, symmetric splitting, Strang splitting, etc) for $$H(q,p) = T(p) + V(q)$$
\displaystyle q^{(k+1)} = q^{(k)} + h \nabla T(p^{(k+1/2)})
\displaystyle p^{(k+1/2)} = p^{(k)} - \frac{h}{2} \nabla V(q^{(k)})
\displaystyle p^{(k+1)} = p^{(k+1/2)} - \frac{h}{2} \nabla V(q^{(k+1)})
• Midpoint method
\displaystyle z^{(k+1)} = z^{(k)} + h X_H\left(\frac{z^{(k+1)}+z^{(k)}}{2} \right)

## Slightly dissipative systems

• Elastic 3D pendulum
\displaystyle\dot q = \frac{p}{m}, \quad \dot p = -k\left(1-\frac{\ell}{|q|} \right)q + mg
\displaystyle -\;\varepsilon \frac{qq^\top p}{|q|^2}

## Slightly dissipative systems

• Elastic 3D pendulum
\displaystyle\dot q = \frac{p}{m}, \quad \dot p = -k\left(1-\frac{\ell}{|q|} \right)q + mg
\displaystyle -\;\varepsilon \frac{qq^\top p}{|q|^2}

non-symplectic

symplectic

## Slightly dissipative systems

• Elastic 3D pendulum
\displaystyle\dot q = \frac{p}{m}, \quad \dot p = -k\left(1-\frac{\ell}{|q|} \right)q + mg
\displaystyle -\;\varepsilon \frac{qq^\top p}{|q|^2}

Energy behaviour

## Classical spin systems

\displaystyle \dot w_i = w_i\times \frac{\partial H}{\partial w_i}

Phase space $$(S^2)^n$$

Symplectic structure on $$S^2$$ $\Omega_w(u,v)=\mathrm{det}(w,u,v)$

+\cdots +

## Some example spin systems

Free rigid body $\dot w = w\times I^{-1}w$

Heisenberg spin chain
(discrete Landau-Lifshitz) $\dot w_i = w_i\times (w_{i-1}+w_{i+1})$

Fluid particle tracking on sphere $\dot w = \xi(t,w), \quad \xi(t,\cdot)\in \mathfrak{X}_\mu(S^2)$

Point vortex dynamics on spheres
(Jupiter's great red spot)

## Known symplectic integrators

• Splitting methods

• $$S^2$$ coadjoint orbit of reduced system on $$T^*SU(2)$$
RATTLE $$\Rightarrow$$ 9 variables
Variational Lie $$\Rightarrow$$ 8 variables

• Collective symplectic integrator
$$\Rightarrow$$ 4 variables
T^*SU(2)
\mathfrak{su}(2)^*

Drawbacks: many auxiliary variables, complicated, large error constants

## and symplectic?

• Classical midpoint method on $$\mathbb{R}^3$$

• Riemannian midpoint method on $$S^2$$

$$\Rightarrow$$ not symplectic

$$\Rightarrow$$ not symplectic

Candidates:

## Spherical midpoint method

\displaystyle\frac{W_i - w_i}{h} = \frac{W_i+w_i}{|W_i+w_i|}\times \frac{\partial H}{\partial w_i}\left(\frac{W_1+w_1}{|W_1+w_1|},\ldots,\frac{W_n+w_n}{|W_n+w_n|} \right)

Main result

1. Second order
2. Equivariant w.r.t. $$SO(3)^n$$
3. Symplectic
S^3 \simeq

S^2
\pi

### Extended Hopf fibration

\mathbb{R}^3_*
\pi
T^*\mathbb{R}^2_*
\pi

Classical
midpoint

Riemannian
midpoint

## Simple interpretation

\displaystyle\frac{W - w}{h} = \frac{W+w}{|W+w|}\times \frac{\partial H}{\partial w}\left(\frac{W+w}{|W+w|} \right)
\displaystyle X_H(w) = w\times \frac{\partial H}{\partial w}(w)
\displaystyle \Rightarrow W-w = h X_H\left(\frac{W+w}{|W+w|}\right)
\displaystyle \bar X_H(w) = X_H\left(\frac{w}{|w|} \right)
\displaystyle \Rightarrow W-w = h \bar X_H\left(\frac{W+w}{2}\right)

Classical midpoint method applied to $$\bar X_H$$

## Example 1: free rigid body

\dot w = w \times I^{-1}w

## Example 2: irreversible rigid body

H(w)= \frac{1}{2} w\cdot I(w)^{-1}w

## Example 3: development of chaos

H(w)= \frac{1}{2} w\cdot I^{-1}w + \varepsilon \sin(t)w^1

## Example 4: Landau-Lifshitz PDE

\dot w = w\times w'', \qquad w\colon \mathbf{R}\times S^1 \to S^2

# THANKS!

References:

Symplectic integrators for spin systems, Phys. Rev. E, 89:061301, 2014

arXiv:1402.4114

A minimal-variable symplectic integrator on spheres, Math. Comp., 86, 2325-2344, 2016

arXiv:1402.3334

Geometry of discrete-time spin systems, J. Nonlin. Sci., 26(5):1507-1523, 2016

arXiv:1505.04035

*slides at slides.com/kmodin

By Klas Modin

# Symplectic integrators for classical spin systems

Presentation given 2017-12 to physicists in Uppsala.

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