辛积分器对可积分哈密顿系统数值不变分量的研究

On the accuracy of symplectic integrators for secularly evolving planetary systems

Symplectic integrators have made it possible to study the long-term evolution of planetary systems with direct N-body simulations. In this paper we reassess the accuracy of such simulations by running a convergence test on 20Myr integrations of the Solar System using various symplectic integrators. We find that the specific choice of metric for determining a simulation's accuracy is important. Only looking at metrics related to integrals of motions such as the energy error can overestimate the accuracy of a method. As one specific example, we show that symplectic correctors do not improve the …

Hybrid Symplectic Integrators for Planetary Dynamics

Hybrid symplectic integrators such as MERCURY are widely used to simulate complex dynamical phenomena in planetary dynamics that could otherwise not be investigated. A hybrid integrator achieves high accuracy during close encounters by using a high order integration scheme for the duration of the encounter while otherwise using a standard 2nd order Wisdom-Holman scheme, thereby optimizing both speed and accuracy. In this paper we reassess the criteria for choosing the switching function that determines which parts of the Hamiltonian are integrated with the high order integrator. We show that t…

Non-resonant tori in symplectic twist maps without conjugate points

We study the dynamics of a symplectic twist map without conjugate points. We show that in a neighborhood of a totally periodic Lagrangian manifold, there exists a large family of invariant Lagrangian tori on which the twist map is conjugated to a translation of non-resonant vector.

Totally non-symplectic Anosov actions on tori and nilmanifolds

We show that sufficiently irreducible totally non-symplectic Anosov actions of higher rank abelian groups on tori and nilmanifolds are smoothly conjugate to affine actions.

通过辛算法研究近可积哈密顿系统的椭圆不变环面

The elliptical invariant tori of nearly integrable Hamiltonian system through symplectic algorithms

本文利用辛算法证明近可积哈密顿系统在 Rüssmann 非退化条件下可保持大量椭圆低维不变环面，并给出相应测度估计。

Quasi-periodic motions on symplectic tori

The KAM (Kolmogorov-Arnold-Moser) theorem guarantees the stability of quasi-periodic invariant tori by perturbation in some Hamiltonian systems. Michel Herman proved a similar result for quasi-periodic motions, with $k$-dimensional involutive manifolds in Hamiltonian systems with $n$ degrees of freedom $n \leq k < 2n $. In this paper, we extend this result to the case of a quasi-periodic motion on symplectic tori $k = 2n$.

Symplectic non-convexity of toric domains

We investigate the convexity up to symplectomorphism (called symplectic convexity) of star-shaped toric domains in $\mathbb R^4$. In particular, based on the criterion from Chaidez-Edtmair via Ruelle invariant and systolic ratio of the boundary of star-shaped toric domains, we provide elementary operations on domains that can kill the symplectic convexity. These operations only result in $C^0$-small perturbations in terms of domains' volume. Moreover, one of the operations is a systematic way to produce examples of dynamically convex but not symplectically convex toric domains. Finally, we are…