Ghost from constraints: a generalization of Ostrogradsky theorem
Ghost from constraints: a generalization of Ostrogradsky theorem
复制标题
约束中的幽灵：奥斯特罗格勒茨基定理的推广
DOI：
10.1088/1475-7516/2020/08/026
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发表时间：
2020
影响因子：
6.4
通讯作者：
Hayato Motohashi
中科院分区：
文献类型：
--
作者：
Katsuki Aoki;Hayato Motohashi
Ostrogradsky定理指出，在非退化假设下，当Euler-Lagrange方程高于二阶微分方程时，Hamilton是无界的。由于高阶非退化拉格朗日量总是可以通过引入辅助变量和约束来重铸为至多具有一阶导数的等效系统，因此可以想象，鬼与高阶导数之间的联系可以被重新解释为鬼与约束和/或辅助变量之间的联系。我们发现，后者的观点实际上提供了更一般的角度比前者，通过探索的无界性的哈密顿量的一般理论包含辅助变量，欧拉-拉格朗日方程可以基本上是二阶或低于。对于拉格朗日包括辅助变量非线性，我们推导出退化条件，以避免Ostrogradsky鬼，可以适用，即使辅助变量只能在本地解决。对于具有拉格朗日乘子约束的理论，我们建立了包含非完整（速度相关）约束的标准，导致局部极小的哈密顿量的情况下。我们的标准包括Ostrogradsky定理作为一个特殊的情况下，不仅可以检测鬼与高阶导数，但也鬼来自系统的低阶导数的约束。我们讨论如何躲避这样的鬼魂。我们还提供了各种具体的例子来强调我们的一般论点的应用和局限性。
Ostrogradsky theorem states that Hamiltonian is unbounded when Euler-Lagrange equations are higher than second-order differential equations under the nondegeneracy assumption. Since higher-order nondegenerate Lagrangian can be always recast into an equivalent system with at most first-order derivatives by introducing auxiliary variables and constraints, it is conceivable that the link between ghost and higher derivatives may be reinterpreted as a link between ghost and constraints and/or auxiliary variables. We find that the latter point of view actually provides more general perspective than the former, by exploring the un/boundedness of the Hamiltonian for general theories containing auxiliary variables, for which Euler-Lagrange equations can be essentially second order or lower than that. For Lagrangians including auxiliary variables nonlinearly, we derive the degeneracy condition to evade the Ostrogradsky ghost that can apply even if auxiliary variables can be solved only locally. For theories with constraints with Lagrange multipliers, we establish criteria for inclusion of nonholonomic (velocity-dependent) constraints leading to the absence of local minimum of Hamiltonian. Our criteria include the Ostrogradsky theorem as a special case, and can detect not only ghost associated with higher-order derivatives, but also ghost coming from lower-order derivatives in system with constraints. We discuss how to evade such a ghost. We also provide various specific examples to highlight application and limitation of our general arguments.