守恒牛顿系统的交换平面多项式向量场

Joel Nagloo, A. Ovchinnikov, Peter Thompson
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引用次数: 3

摘要

研究了平面上与给定多项式向量场交换的多项式向量场的刻画问题。这是一个经典的结果,我们可以写出ODE的解公式,它对应于一个具有线性无关(横向)交换向量场的平面向量场(见定理2.1)。在接下来的内容中,我们将使用(多项式)向量场与(多项式)环上的导数之间的标准对应关系。设[MATH HERE]是一个导数,其中f是一个多项式,其系数在域K中为零特征。这个推导对应于微分方程= f(x),它被称为保守牛顿系统,因为它是在保守力的影响下被限制在一条直线上的粒子的牛顿第二定律的表达式。设H为常数项为零的d的哈密顿多项式,即[MATH HERE]。则与d交换的所有多项式导数的集合形成K[H]-模Md[6,推论7.1.5]。我们表明,对于每个这样的d,当且仅当度数f大于或等于2时,模块Md为1级。例如,经典椭圆方程= 6x2 + a,其中a∈C。关于本摘要中所述结果的证明,请参阅[5]。
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Commuting planar polynomial vector fields for conservative newton systems
We study the problem of characterizing polynomial vector fields that commute with a given polynomial vector field on a plane. It is a classical result that one can write down solution formulas for an ODE that corresponds to a planar vector field that possesses a linearly independent (transversal) commuting vector field (see Theorem 2.1). In what follows, we will use the standard correspondence between (polynomial) vector fields and derivations on (polynomial) rings. Let [MATH HERE] be a derivation, where f is a polynomial with coefficients in a field K of zero characteristic. This derivation corresponds to the differential equation ẍ = f(x), which is called a conservative Newton system as it is the expression of Newton's second law for a particle confined to a line under the influence of a conservative force. Let H be the Hamiltonian polynomial for d with zero constant term, that is [MATH HERE]. Then the set of all polynomial derivations that commute with d forms a K[H]-module Md [6, Corollary 7.1.5]. We show that, for every such d, the module Md is of rank 1 if and only if deg f ⩾ 2. For example, the classical elliptic equation ẍ = 6x2 + a, where a ∈ C, falls into this category. For proofs of the results stated in this abstract, see [5].
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