依赖构型动能的耗散系统的投影半显式积分器：接触-赫尔格洛茨公式与基准测试
A Projected Semiexplicit Integrator for Dissipative Systems with Configuration-Dependent Kinetic Energy: Contact-Herglotz Formulation and Benchmarks
AI总结：
本文针对含动量交叉项的稠密度量耗散系统，提出投影Pihajoki-接触积分器，其二阶精度可控制能量漂移，在双摆等场景优于现有方法。
AI中文摘要：
接触哈密顿动力学为耗散力学提供了内在的作用变量，但显式接触分裂积分器仅能处理项可精确积分的动能：包括冻结坐标对角度量（如单摆、环面粒子），而含动量交叉项的稠密度量（以双摆为代表）则无法处理。针对这种非可分场景，我们提出一种投影Pihajoki-接触积分器，结合相空间复制、向物理对角的对称投影以及恒定摩擦阻尼半步，作用因子由精确的赫尔格洛茨更新携带。如其所基于的投影扩展相空间框架，该构造无需绑定参数，每一步都将副本归位到对角，且将非线性求解限制在2n个投影变量中。对于恒定摩擦，当投影被精确求解时，步长会将ω=dη乘以精确因子e^(-γτ)（这是经典的保形辛恒等式，此处针对该类系统实现）；同时，时间对称性、一致性与光滑性产生O(τ^3)阶的单步接触形式残差，该界并非接触形式所特有。在阻尼双摆、单摆和环面粒子上，该方法具有二阶精度，能复现接触衰减律，并在Tao基线和未投影平均法失效的粗网格或刚性工况下控制长期能量与接触漂移。与精确接触同胚分裂的直接对比界定了其适用场景：当存在冻结坐标分裂时，它能精确保持接触形式且在匹配成本下更优；对于稠密双摆度量，可行的替代方法是一阶的，且具有难以接受的常数项，而该投影方法占优。该接触形式估计是局部的、单步的且针对恒定摩擦。
英文摘要：
Contact Hamiltonian dynamics gives dissipative mechanics an intrinsic action variable, but explicit contact splittings reach only kinetic energies whose terms are exactly integrable: frozen-coordinate diagonal metrics (the spherical pendulum, a torus particle) are included, while dense metrics with momentum cross terms, with the double pendulum as flagship, are not. We introduce a projected Pihajoki-contact integrator for this non-separable setting, combining phase-space duplication, symmetric projection onto the physical diagonal, and constant-friction damping half-steps, with the action factor carried by an exact Herglotz update. As in the projected extended-phase-space framework it builds on, the construction needs no binding parameter, returns the copies to the diagonal at every step, and confines the nonlinear solve to the $2n$ projection variables. For constant friction the step rescales $ω=dη$ by the exact factor $e^{-γτ}$ when the projection is solved exactly (a classical conformally symplectic identity, realized here for this class), while time-symmetry, consistency, and smoothness yield an $O(τ^3)$ one-step contact-form residual, a bound not specific to the contact form. On the damped double pendulum, spherical pendulum, and torus particle the method is second-order accurate, reproduces the contact decay law, and controls long-time energy and contact drift in coarse or stiff regimes where the Tao baseline and the unprojected average lose the solution. A head-to-head with exact-contactomorphism splittings delimits the niche: where a frozen-coordinate splitting exists it preserves the contact form exactly and wins at matched cost; for the dense double-pendulum metric the realizable alternative is first-order with a prohibitive constant and the projected method prevails. The contact-form estimate is local, one-step, and constant-friction.
