Variational integrators for stochastic Hamiltonian systems on Lie groups: properties and convergence

François Gay-Balmaz; Meng Wu

doi:10.3934/jcd.2026004

Article Contents

2026, Volume 15: 1-62. Doi: 10.3934/jcd.2026004

This volume Previous Article A new perspective on symplectic integration of constrained mechanical systems via discretization maps Next Article Rough differential equations and planarly branched universal limit theorem

Variational integrators for stochastic Hamiltonian systems on Lie groups: properties and convergence

François Gay-Balmaz^1, , and
Meng Wu^{1, 2,}

1.
Division of Mathematical Sciences, Nanyang Technological University, Singapore
2.
LMD, École Normale Supérieure, France

^*Corresponding author: François Gay-Balmaz
^*Corresponding author: François Gay-Balmaz

Received: January 29, 2025

Revised: January 14, 2026

Published online: February 12, 2026

The first author is partially supported by a startup grant from Nanyang Technological University and by the Ministry of Education, Singapore, under Academic Research Fund (AcRF) Tier 1 Grant RG99/24.

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Abstract

We derive variational integrators for stochastic Hamiltonian systems on Lie groups using a discrete version of the stochastic Hamiltonian phase space principle. The structure-preserving properties of the resulting scheme, such as symplecticity, preservation of the Lie-Poisson structure, preservation of the coadjoint orbits, and preservation of Casimir functions, are discussed, along with a discrete Noether theorem for subgroup symmetries. We also consider in detail the case of stochastic Hamiltonian systems with advected quantities, studying the associated structure-preserving properties in relation to semidirect product Lie groups. A full convergence proof for the scheme is provided for the case of the Lie group of rotations. Several numerical examples are presented, including simulations of the free rigid body and the heavy top.

Keywords:

Stochastic variational integrators,
stochastic Hamiltonian systems on Lie groups,
convergence analysis,
Poisson integrators,
coadjoint orbits,
Casimir functions,
discrete Noether theorem,
semidirect products.

Mathematics Subject Classification: Primary: 65P10, 37M15; Secondary: 65C30, 70L10.

Citation:

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Figure 1. Two stochastic paths on the angular momentum sphere, with the initial condition $ \Pi_0 = (-0.5878, 0, 0.8090) $, marked by the red dot. The deterministic path of the same initial condition is highlighted in red

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Figure 2. Energy evolution of the two stochastic paths presented in Figure 1. The blue line corresponds to the stochastic path on the left, and the orange line corresponds to the stochastic path on the right

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Figure 3. With the initial condition $ \Pi_0 = (-0.5878, 0, 0.8090) $, an ensemble of 20 stochastic paths are generated. Each image shows the positions of $ \Pi_k $ of the ensemble at the given time horizon. The deterministic $ \pi_k $ is marked in red for reference. The possibility of bifurcation significantly influences the scattering pattern of $ \Pi_k $

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Figure 4. Plots of the mean square errors of $ \Pi $ for different step size $ \Delta t $ on $ \mathrm{log}_2 $ scale. The plot on the left corresponds to the numerical solutions with $ N = 1 $; the best fit line has slope $ k = 0.9782 $. The plot on the right corresponds to the numerical solutions with $ N = 3 $; the best fit line has slope $ k = 0.5397 $

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Figure 5. $ V(\theta) $ as function of $ \theta $, for $ p_{\psi} = 0.891 $ and $ p_{\phi} = 1 $. The dashed line marks the level of $ E' = 0.8732 $. This is the setting of the deterministic experiment to be presented below in Section 7.3.4

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Figure 6. Deterministic experiment: the colour pattern indicates the time

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Figure 7. Deterministic experiment: evolution of various parameters. $ \theta_0 $, $ p_{\psi} $ and $ E_{p_{\psi}} $ are exactly conserved and $ E $ is nearly conserved

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Figure 8. One sample path from the stochastic experiment with stochastic Hamiltonian $ h_1(\Pi, \Gamma) = 0.1 \Pi_z $; the colour pattern indicates the time

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Figure 9. Stochastic experiment with stochastic Hamiltonian $ h_1(\Pi, \Gamma) = 0.1 \Pi_z $: evolution of various parameters of the same sample path as presented in Figure 8. $ \theta_0 $, $ p_{\psi} $ and $ E_{p_{\psi}} $ are exactly conserved and $ E $ is nearly conserved

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Figure 10. $ V(\theta) $ as function of $ \theta $, for $ p_{\psi} = 0.891 $ and $ p_{\phi} = 1 $. The partial energy jumped from $ E'(t = 0) = 0.8732 $ to $ E'(t = 50) = 0.8886 $, causing the expansion of the nutation angle range. The nutation angle $ \theta_0 $ of the lowest "effective potential" stays unchanged

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Figure 11. One sample path from stochastic experiment with stochastic Hamiltonian $ h_1(\Pi, \Gamma) = Mg \Gamma_z $; the colour pattern indicates the time

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Figure 12. Stochastic experiment with stochastic Hamiltonian $ h_1(\Pi, \Gamma) = Mg \Gamma_z $: evolution of various parameters of the same sample path as presented in Figure \ref{Gyros_ex_Ga_z}. $ \theta_0 $, $ p_{\psi} $ and $ E_{p_{\psi}} $ are exactly conserved while $ E $ is perturbed

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Figure 13. $ V(\theta)_t $ as functions of $ \theta $, for $ p_{\psi} = 0.891 $ and $ p_{\psi} = 0.891 $, $ p_{\phi}(t = 50) = 0.8048 $. The partial energy jumps from $ E'(t = 0) = 0.8732 $ to $ E'(t = 50) = 0.8886 $, causing the shift of both the nutation angle range $ [\theta_1, \theta_2] $ and the nutation angle $ \theta_0 $ corresponding to the lowest "effective potential"

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Figure 14. One sample path from stochastic experiment with stochastic Hamiltonians $ h_1(\Pi, \Gamma) = Mg\Gamma_x $, $ h_2(\Pi, \Gamma) = Mg\Gamma_y $; the colour pattern indicates the time

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Figure 15. Stochastic experiment with stochastic Hamiltonians $ h_1(\Pi, \Gamma) = mg \Gamma_x $, $ h_2(\Pi, \Gamma) = mg\Gamma_y $: evolution of various parameters of the same sample path as presented in Figure 14. $ \theta_0 $, $ p_{\psi} $ and $ E_{p_{\psi}} $ and $ E $ are all perturbed

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Figure 16. $ ||(A, B)|| $ as function of $ \Pi $ on one coadjoint orbit — the unit sphere. Here we consider multiple stochastic Hamiltonians, with $ \Delta t = 0.01 $, $ N = 3 $, $ \Delta t = 0.01 $, $ N = 3 $, $ \chi_1 = (1, 0, 0) $, $ \chi_2 = (0, 1, 0) $, $ \chi_3 = (0, 0, 1) $, the value $ \overline{\Delta W} $ is indicated above each figure

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	$ T^*_g G $	$ \mathfrak{g}^* $
	...	...
$ t_{k-\frac{1}{2}} $	$ (\tilde g_k, \tilde p^1_k) $	$ \tilde g_k^{-1} \tilde p^1_k = \tilde \mu^1_k $
$ t_{k-\frac{1}{2}} $	$ (\tilde g_k , \tilde p^2_k) $	$ \tilde g_k^{-1} \tilde p^2_k = \tilde \mu^2_k $
$ t_k $	$ (g_k, p_k) $	$ g_k^{-1} p_k = \mu_k = \begin{cases} \operatorname{Ad}^_{\tilde g_k ^{-1} g _k } [ {\rm d} _{ \tau ^{-1} (\tilde g_k ^{-1} g _k )} \tau ^{-1} ] ^ \tilde\mu^2 _k\\ [ {\rm d} _{ \tau ^{-1} ( g_{k} ^{-1} \tilde g _{k+1} )} \tau ^{-1} ] ^* \tilde\mu^1 _{k+1} \end{cases} $
$ t_{k+\frac{1}{2}} $	$ (\tilde g_{k+1}, \tilde p^1_{k+1}) $	$ \tilde g_{k+1}^{-1} \tilde p^1_{k+1} = \tilde \mu^1_{k+1} $
$ t_{k+\frac{1}{2}} $	$ (\tilde g_{k+1} , \tilde p^2_{k+1}) $	$ \tilde g_{k+1}^{-1} \tilde p^2_{k+1} = \tilde \mu^2_{k+1} $
	...	...

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