论文标题
没有变化的变化：重新聚集，差异性，一般协方差，谎言衍生物以及所有这些
Varying without varying: Reparameterisations, Diffeomorphisms, General Covariance, Lie derivatives, and all that
论文作者
论文摘要
得出给定点粒子动作的Euler-Lagrange（EL）方程的标准方法是改变轨迹并将动作的第一个变化设置为零。但是，如果该动作是（i）重新聚集不变的，并且（ii）通常协变，我表明一个人可以通过明智的坐标转换来适当地“无效”变异来得出EL方程。其最终结果是曲线保持固定，而动作中的所有其他几何对象都会发生变化，这是由沿变异矢量场的谎言衍生物确切给出的。因此，这是阐明一般协方差，差异不变性和谎言衍生物之间联系的最直接和透明的方法，而无需指协变量衍生物。 我通过将其应用于最简单的现场理论，使讨论的水平易于访问高级本科生，从而强调了上述想法的几何基础和一般性。作为这些思想的非平凡应用，i（i）使用一阶差异形态得出了地球偏差方程，并且（ii）证明了它们如何突出现场理论中规范和度量应力量张量之间的连接。
The standard way of deriving Euler-Lagrange (EL) equations given a point particle action is to vary the trajectory and set the first variation of the action to zero. However, if the action is (i) reparameterisation invariant, and (ii) generally covariant, I show that one may derive the EL equations by suitably "nullifying" the variation through a judicious coordinate transformation. The net result of this is that the curve remains fixed, while all other geometrical objects in the action undergo a change, given precisely by the Lie derivatives along the variation vector field. This, then, is the most direct and transparent way to elucidate the connection between general covariance, diffeomorphism invariance, and Lie derivatives, without referring to covariant derivative. I highlight the geometric underpinnings and generality of above ideas by applying them to simplest of field theories, keeping the discussion at a level easily accessible to advanced undergraduates. As non-trivial applications of these ideas, I (i) derive the Geodesic Deviation Equation using first order diffeomorphisms, and (ii) demonstrate how they can highlight the connection between canonical and metric stress-energy tensors in field theories.
