论文标题
减少修改后的牛顿重力理论相似性组的拉格朗日运动方程
Reduction of Lagrangian Equations of Motion of Modified Newtonian Theory of Gravity with respect to the Similarity Group
论文作者
论文摘要
在相似性转换组下，系统的绝对配置的等效类别$ SIM（3）$称为系统的形状。修改后的牛顿理论的$ SIM（3）$不变的拉格朗日可确保其在形状空间上的运动定律的存在。 To deduce the equations of motion for a system's shape degrees of freedom from its evolution equations for the $3N$ absolute configuration degrees of freedom, the Boltzman-Hamel equations of motion in an non-holonomic frame on the tangent space $T(Q)$ to the system's absolute configuration space $Q$ is adapted to the $Sim(3)$ fiber bundle structure of the configuration space.形状空间上的运动方程式使我们能够预测由该理论控制的经典系统形状的演变，而无需提及其绝对位置，方向或空间中的大小。本文将解释说，通过将测量仪器作为理论中的一部分，在配置空间上的质量度量$ \ textbf {m} $如何唯一地定义了在减少的切线$ \ frac {t（q）} {sim（3）} $上的分别$ s $ shyp y shape $ s $ s $ s $ s $ s $ s $ s $ s上的指标。在为一般$ n $体系统的形状空间上衍生出还原的运动方程之后，给出了适当的坐标中三体系统的形状运动方程式。
The equivalence class of absolute configurations of a system under the group of similarity transformations $Sim(3)$ is called the shape of the system. The $Sim(3)$ invariant Lagrangian of the modified Newtonian theory ensures the existence of the its law of motion on shape space. To deduce the equations of motion for a system's shape degrees of freedom from its evolution equations for the $3N$ absolute configuration degrees of freedom, the Boltzman-Hamel equations of motion in an non-holonomic frame on the tangent space $T(Q)$ to the system's absolute configuration space $Q$ is adapted to the $Sim(3)$ fiber bundle structure of the configuration space. The derived equations of motion on shape space enable us, among other things, to predict the evolution of the shape of a classical system governed by this theory without any reference to its absolute position, orientation, or size in space. The paper will explain, that by treating the measuring instruments as part of the matter in the theory, how the mass metric $\textbf{M}$ on the configuration space $Q$ uniquely defines a metric on the reduced tangent bundle $\frac{T(Q)}{Sim(3)}$, and how the unique metric structure on shape space $S$ can be derived. After deriving the reduced equations of motion on shape space for a general $N$-body system, the shape equations of motion for a three-body system in suitable coordinates is given as an illustration.
