论文标题
最小动作原理的扩展，重点是耗散方程
Extension of the principle of least action with focus on dissipative equations
论文作者
论文摘要
在本文中，我们扩展了\ emph {最少动作的原理}，并表明\ emph {lagrange密度}始终存在于通常的线性PDE或线性分数问题$ \ oa \，u = f $物理学中的u = f $，如果通常的因果关系条件$ | _ | _ | _ | _ {t <0} = 0 $ f <0 $ f |是$ f |________________________________。 （该方法实际上适用于存在伴随的唯一可解决的线性算子方程。）Lagrange密度的集合以及零向量形成一个非平凡的向量空间，以及每个不同的变量集，例如。 $ \ {u_t，f \} $，$ \ {u_ {u_ {xt}，f \} $或$ \ {u_t，u_x，u_y，u_y，u_z，f \} $，存在一个lagrange密度，暗示着lagrange方程，这与被认为是问题的问题相当。通常的拉格朗日密度使其暗示“原始方程式”。但是有PDE的标准理论并不意味着拉格朗日密度。我们表明，对于每个方程式，存在一个（协变）的拉格朗日密度，它导致等效\ emph {高阶pde}（如果在上述因果关系条件下配制）。对于每个方程式中，存在一个拉格朗日密度，这意味着等于原始方程的Lagrange方程，但是该拉格朗日密度至少包含一个\ emph {线性积分运算符}。一个新的观点是，对于给定的一组变量，这些等效的拉格朗日密度中的每一个都意味着（通常不同的）\ emph {广义的汉密尔顿密度}，如果$ \ oa $和$ f $适当，则相应的“汉密尔顿”是适当的。标准的Lagrange密度意味着（经常）对能量进行建模的哈密顿量。此外，每个保守的汉密尔顿人都意味着可计数的许多高阶哈密顿人是保守的（如果被认为的问题的解决方案足够平滑。）
In this paper, we extend the \emph{principle of least action} and show that a \emph{Lagrange density} always exists for the usual linear pde or linear fractional problems $\oA\,u=f$ in physics, if the usual causality conditions $u|_{t<0}=0$ and $f|_{t<0}=0$ are assumed. (The approach is actually applicable to uniquely solvable linear operator equations for which an adjoint exist.) The set of Lagrange densities together with the zero vector form a non-trivial vector space and for each different set of variables, e.g. $\{u_t,f\}$, $\{u_{xt},f\}$ or $\{u_t,u_x,u_y,u_z,f\}$, there exists a Lagrange density that implies a Lagrange equation, which is equivalent to the considered problem. The usual Lagrange density is such that it implies the 'original equation'. But there are pde's for which the standard theory does not imply a Lagrange density. We show that for each of these equations a (covariant) Lagrange density exists that leads to an equivalent \emph{higher order pde} (if it is formulated with the above causality conditions). For each of these equations, there exists a Lagrange density that implies a Lagrange equation that equals the original equation, but this Lagrange density contains at least one \emph{linear integral operator}. A new point of view is that each of these equivalent Lagrange densities for a given set of variables implies a (usually different) \emph{generalized Hamiltonian density}, where the respective 'Hamiltonian' is conserved if $\oA$ and $f$ are appropriate. The standard Lagrange density implies an Hamiltonian that (frequently) models the energy. Morever, each conserved Hamiltonian implies countable many higher order Hamiltonians that are conserved (if the solution of the considered problem is sufficiently smooth.)
