Celestial Mechanics and Dynamical Astronomy Equations for the orbital elements : Hidden symmetry
Celestial Mechanics and Dynamical Astronomy Equations for the orbital elements : Hidden symmetry
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轨道元素的天体力学和动力天文学方程：隐藏对称性
DOI：
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发表时间：
2003
期刊：
影响因子：
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通讯作者：
M. Efroimsky
中科院分区：
文献类型：
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作者：
M. Efroimsky
我们重新讨论了轨道根数的拉格朗日和德劳内方程组，并指出了这些方程以前被忽视的一个方面：在这两种情况下，轨道都位于由轨道根数及其时间导数所构成的12维空间的某个9维子流形上。我们证明，有一个巨大的自由选择这个子流形。这种选择的自由（=规范固定的自由）揭示了隐藏在拉格朗日和德劳内系统背后的对称性，这在数学上类似于电动力学中的规范不变性。就像规范的方便选择简化了电动力学中的计算一样，子流形的自由选择也可能被用来创建更简单的轨道积分方案。另一方面，这一特征的存在可能是以前未认识到的数值不稳定性的来源。我们提供了一个实际的例子，如果不考虑所述量规类型的自由度，就不能正确处理这种情况。
We revisit the Lagrange and Delaunay systems of equations for the orbital elements, and point out a previously neglected aspect of these equations: in both cases the orbit resides on a certain 9-dimensional submanifold of the 12-dimensional space spanned by the orbital elements and their time derivatives. We demonstrate that there exists a vast freedom in choosing this submanifold. This freedom of choice (=freedom of gauge fixing) reveals a symmetry hiding behind Lagrange’s and Delaunay’s systems, which is, mathematically, analogous to the gauge invariance in electrodynamics. Just like a convenient choice of gauge simplifies calculations in electrodynamics, so the freedom of choice of the submanifold may, potentially, be used to create simpler schemes of orbit integration. On the other hand, the presence of this feature may be a previously unrecognised source of numerical instability. We provide a practical example of a situation that cannot be correctly handled without the said gauge-type freedom taken into account.