AI 中文总结
研究弹性杆平衡方程，借鉴数学物理观点，通过特定方法构建框架和理论得出动量，使其有泊松结构，应用于肌腱驱动杆时能给出明确平衡方程并简化反演。
AI 中文摘要
弹性杆的平衡方程可通过平衡力和力矩或使势能平稳得到。对于复杂细丝，能量途径对机械直觉要求较低。在经典哈密顿图景中，广义动量的分量是先验假设的，其物理意义随坐标图随意变化。本文借鉴数学物理的两个观点：一是用左不变向量场构建构型李群的扩展切丛框架；二是采用变分形式的一维卡当 - 勒帕热 - 克鲁普卡理论，使动量是被强制得出而非假设。勒让德变换成为框架的线性变化，扩展切丛上有泊松结构。对于孤立的科塞尔杆，在旋转群的每种编码中，动量与杆理论中熟悉的材料力和力矩一致。存在相互作用时，仅依赖构型的相互作用（如重力）使这种识别不变，依赖应变的相互作用则破坏它，共轭动量和内应力分离。由于能量相加，动量分解为内应力和相互作用贡献。在不同框架中表示泊松二向量的两个因子，能直接在内部变量中揭示哈密顿流。应用于肌腱驱动杆时，该构造给出明确的平衡方程，所需的反演简化为一阶修正。
英文摘要
The equilibrium equations of elastic rods can be obtained by balancing forces and moments, or by rendering a potential energy stationary. For complex filaments the energy route asks less of one's mechanical intuition. However, in the classical Hamiltonian picture, the components of the generalized momenta are postulated a priori and their physical meaning changes at the whim of the coordinate chart. Here, we draw on two ideas from mathematical physics. First, the extended tangent bundle of the configuration Lie group is framed by left-invariant vector fields. Second, we follow a one-dimensional reading of the Cartan--Lepage--Krupka theory of variational forms: in this setting the momenta are not postulated but forced. The Legendre transform becomes a linear change of frame, and a Poisson structure on the extended tangent bundle follows. For an isolated Cosserat rod, the momenta coincide, in every encoding of the rotation group, with the material force and moment familiar from rod theories. In the presence of interactions, two cases arise: interactions that depend only on the configuration, such as gravity, leave this identification intact; interactions that depend on the strains destroy it --- the conjugate momenta and the internal stresses part company. Because energies add, the momenta decompose into the internal stresses and an interaction contribution. Expressing the two factors of the Poisson bivector in different frames --- one adapted to the internal stresses, the other to the conjugate momenta --- then exposes the Hamiltonian flow directly in the internal variables. Applied to a tendon-actuated rod, where the standard passage to the Hamiltonian picture demands a nonlinear inversion, the construction delivers explicit equilibrium equations, the required inversion collapsing to a rank-one correction.
