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)

\dot q = p, \quad \dot p = - \sin(q)

q

q

p

p

Illustration

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

\dot q = p, \quad \dot p = - \sin(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)

\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 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)

\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

\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

\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,\cdot) = d H

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

\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)

\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 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+1/2)} = 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)})

\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)

\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\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}

\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\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}

\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\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}

\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}

\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 +

+\cdots +

Motivation for spin systems

Abundance in physics
(e.g. atomistic spin dynamics)
Simplest Kähler manifold
Simplest coadjoint orbit
Simplest inexact symplectic form

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)

T^*SU(2)

\mathfrak{su}(2)^*

\mathfrak{su}(2)^*

Drawbacks: many auxiliary variables, complicated, large error constants

Minimum variables (=3)

and symplectic?

Classical midpoint method on $\mathbb{R}^3$
Riemannian midpoint method on $S^2$

$\Rightarrow$ 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)

\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