We study variational systems for space curves, for which the Lagrangian or action principle has a Euclidean symmetry, using the Rotation Minimising frame, also known as the Normal, Parallel or Bishop frame. Such systems have previously been studied using the Frenet-Serret frame. The Rotation Minimising frame has many advantages, however, and can be used to study a wider class of examples. We achieve our results by extending the powerful symbolic invariant calculus for Lie group based moving frames, to the Rotation Minimising frame case. To date, the invariant calculus has been developed for frames defined by algebraic equations. By contrast, the Rotation Minimising frame is defined by a differential equation. In this paper, we derive the recurrence formulae for the symbolic invariant differentiation of the symbolic invariants. We then derive the syzygy operator needed to obtain Noether's conservation laws as well as the Euler-Lagrange equations directly in terms of the invariants, for variational problems with a Euclidean symmetry. We show how to use the six Noether laws to ease the integration problem for the minimising curve, once the Euler-Lagrange equations have been solved for the generating differential invariants. Our applications include variational problems used in the study of strands of proteins, nucleid acids and polymers.

中文翻译：

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欧氏对称变分系统旋转最小化框架的应用

我们使用旋转最小化框架（也称为法线、平行或 Bishop 框架）研究空间曲线的变分系统，其中拉格朗日或作用原理具有欧几里得对称性。以前曾使用 Frenet-Serret 框架研究过此类系统。然而，旋转最小化框架具有许多优点，可用于研究更广泛的示例。我们通过将基于李群的移动坐标系的强大符号不变演算扩展到旋转最小化坐标系的情况来实现我们的结果。迄今为止，已经为代数方程定义的框架开发了不变微积分。相比之下，旋转最小化框架由微分方程定义。在本文中，我们推导出符号不变量的符号不变量微分的递推公式。然后，对于欧几里得对称性的变分问题，我们推导出直接根据不变量获得 Noether 守恒定律和 Euler-Lagrange 方程所需的 syzygy 算子。我们展示了如何使用六个诺特定律来缓解最小化曲线的积分问题，一旦欧拉-拉格朗日方程已经被求解以生成微分不变量。我们的应用包括用于研究蛋白质链、核酸和聚合物的变分问题。一旦欧拉-拉格朗日方程已经被求解以生成微分不变量。我们的应用包括用于研究蛋白质链、核酸和聚合物的变分问题。一旦欧拉-拉格朗日方程已经被求解以生成微分不变量。我们的应用包括用于研究蛋白质链、核酸和聚合物的变分问题。