不可逆性、最小作用量原理与因果关系
Irreversibility, least action principle and causality
摘要：
最小作用量原理，通过其变分表述，具有一个finalist方面。它明确地出现在分数阶微积分的框架中，其中迄今为止获得的欧拉-拉格朗日方程违反因果关系原理。为了澄清这两个原理之间的关系，我们首先指出，用来描述因果物理现象的导数实际上是左导数。这导致了一个正式的不可逆动力学的方法，向前和向后的时间演化解耦。这种形式主义，然后集成到拉格朗日系统，通过一个特定的嵌入过程。在该装置中，应用最小作用量原理，可区分轨迹和变化的动力学状态。更确切地说，当轨迹和变化的时间箭头相反，我们证明了最小作用量原理提供因果欧拉-拉格朗日方程，即使在分数的情况下。此外，所开发的嵌入是相干的。
英文摘要：
The least action principle, through its variational formulation, possesses a finalist aspect. It explicitly appears in the fractional calculus framework, where Euler-Lagrange equations obtained so far violate the causality principle. In order to clarify the relation between those two principles, we firstly remark that the derivatives used to described causal physical phenomena are in fact left ones. This leads to a formal approach of irreversible dynamics, where forward and backward temporal evolutions are decoupled. This formalism is then integrated to the Lagrangian systems, through a particular embedding procedure. In this set-up, the application of the least action principle leads to distinguishing trajectories and variations dynamical status. More precisely, when trajectories and variations time arrows are opposed, we prove that the least action principle provides causal Euler-Lagrange equations, even in the fractional case. Furthermore, the embedding developped is coherent.
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