PhyslibAlpha.ClassicalFieldTheory.Local.Lagrangian

This definition allows a local $k$-th order Lagrangian $L$ to be treated as a function that maps a jet point $J \in \text{JetPoint}(d, m, k)$ to a real number in $\mathbb{R}$. Here, the jet point $J$ represents the local coordinates of a field and its derivatives up to order $k$ at a specific point in a $d$-dimensional space.

For a local $k$-th order Lagrangian $L$, a jet point $J$, and a vector $V$ in the fiber of the jet space, the fiber derivative is defined as the sum over all multi-indices $I$ with $|I| \le k$ and all field components $a \in \{1, \dots, m\}$ of the product of the coordinate derivative of $L$ and the components of $V$: $$\sum_{|I| \le k} \sum_{a=1}^m \frac{\partial L}{\partial u^a_I}(J) \cdot V^a_I$$ where $\frac{\partial L}{\partial u^a_I}(J)$ is the derivative of the Lagrangian with respect to the jet coordinate $u^a_I$ evaluated at the point $J$, and $V^a_I$ denotes the component of the fiber-jet direction $V$ corresponding to the multi-index $I$ and field index $a$. This value represents the directional derivative of $L$ along the fiber direction $V$ at the point $J$.

For a local $k$-th order Lagrangian $L$, a jet point $J$, and a fiber-direction vector $V$, the function $s \mapsto L(J + sV)$ has a derivative at $s = 0$ equal to the fiber derivative of $L$ at $J$ in the direction $V$: $$\left. \frac{d}{ds} L(J + sV) \right|_{s=0} = \sum_{|I| \le k} \sum_{a=1}^m \frac{\partial L}{\partial u^a_I}(J) \cdot V^a_I$$ where $J + sV$ denotes the jet point whose base coordinates are those of $J$ and whose fiber coordinates are $J.\text{fiber}_{I, a} + s \cdot V^a_I$.

A local $k$-th order Lagrangian $L$ satisfies this property if the map $(x, u) \mapsto L(J(x, u))$ is continuous, where $x \in \text{Space } d$ represents the base coordinates and $u \in \text{JetCoordinates } d m k$ represents the collection of field and derivative coordinates $\{u^a_I\}_{|I| \le k, 0 \le a < m}$. This defines the continuity of the Lagrangian with respect to the explicit local jet coordinates $(x, u^a_I)$.

A property of a local $k$-th order Lagrangian $L$ asserting that for every multi-index $I$ with total order $|I| \le k$ and every field component index $a \in \{0, \dots, m-1\}$, the coordinate derivative $\frac{\partial L}{\partial u^a_I}$ is an infinitely differentiable ($C^\infty$) function of the base space coordinates $x \in \text{Space } d$ and the jet coordinates $u \in \text{JetCoordinates } d m k$.

A local $k$-th order Lagrangian $L$ is smooth in coordinates if it satisfies the following two conditions: 1. The map $(x, u) \mapsto L(x, u)$ is continuous with respect to the base space coordinates $x \in \text{Space } d$ and the jet coordinates $u \in \text{JetCoordinates } d m k$. 2. For every field component index $a \in \{0, \dots, m-1\}$ and every multi-index $I$ with order $|I| \le k$, the coordinate derivative $\frac{\partial L}{\partial u^a_I}$ is an infinitely differentiable ($C^\infty$) function of $(x, u)$.

For a local $k$-th order Lagrangian $L$ and a field $f: \text{Space } d \to \mathbb{R}^m$, this property states that for all multi-indices $I$ with $|I| \le k$ and all field component indices $a \in \{0, \dots, m-1\}$, the function $$x \mapsto \frac{\partial L}{\partial u_I^a}(j^k f(x))$$ is infinitely differentiable ($C^\infty$) on $\text{Space } d$. Here, $j^k f(x)$ denotes the $k$-jet of the field $f$ at the point $x$, and $\frac{\partial L}{\partial u_I^a}$ represents the coordinate derivative of the Lagrangian with respect to the jet coordinate $u_I^a$.

Let $L$ be a local $k$-th order Lagrangian and $F : \mathbb{R} \to (\text{Space } d \to \mathbb{R}^m)$ be a one-parameter family of fields. This property states that for every multi-index $I$ with $|I| \le k$ and every field component index $a \in \{0, \dots, m-1\}$, the mapping $$(s, x) \mapsto \frac{\partial L}{\partial u^a_I}(j^k F_s(x))$$ is continuous as a function from $\mathbb{R} \times \text{Space } d$ to $\mathbb{R}$. Here, $j^k F_s(x)$ is the $k$-jet of the field $F_s$ at the point $x$, and $\frac{\partial L}{\partial u^a_I}$ denotes the coordinate derivative of the Lagrangian with respect to the jet coordinate $u^a_I$.

For any local $k$-th order Lagrangian $L$ that is continuous in its jet coordinates and any infinitely differentiable ($C^\infty$) field $f: \text{Space } d \to \mathbb{R}^m$, the function mapping each point $x \in \text{Space } d$ to the Lagrangian evaluated at the $k$-jet of $f$ at $x$, denoted by $x \mapsto L(j^k f(x))$, is continuous.

For a local $k$-th order Lagrangian $L$ and an infinitely differentiable ($C^\infty$) field $f: \text{Space } d \to \mathbb{R}^m$, if the coordinate derivatives of the Lagrangian $\frac{\partial L}{\partial u^a_I}$ are $C^\infty$ functions of the jet coordinates, then the functions $x \mapsto \frac{\partial L}{\partial u_I^a}(j^k f(x))$ (the coordinate derivatives evaluated along the $k$-jet of the field $f$) are also $C^\infty$ on $\text{Space } d$.

Let $L$ be a local $k$-th order Lagrangian for fields on a $d$-dimensional space with values in $\mathbb{R}^m$. Suppose that the coordinate derivatives of the Lagrangian, denoted by $\frac{\partial L}{\partial u^a_I}$ for each multi-index $I$ (with total order $|I| \le k$) and each field component $a \in \{0, \dots, m-1\}$, are infinitely differentiable ($C^\infty$) functions of the base space coordinates $x \in \text{Space } d$ and the jet coordinates $u$. Let $F: \mathbb{R} \times \text{Space } d \to \mathbb{R}^m$ be a one-parameter family of fields $(s, x) \mapsto F_s(x)$. If the mapping $(s, x) \mapsto (x, j^k F_s(x))$ is continuous, where $j^k F_s(x)$ represents the $k$-jet coordinates of the field $F_s$ at the point $x$, then for every $a$ and $I$, the coordinate derivative evaluated along the family, $(s, x) \mapsto \frac{\partial L}{\partial u^a_I}(x, j^k F_s(x))$, is continuous as a function of $(s, x)$.

Given a local $k$-th order Lagrangian $L$ (a function defined on the $k$-th order jet bundle of fields from a base space of dimension $d$ to a target space of dimension $m$) and a field $f: \mathbb{R}^d \to \mathbb{R}^m$, this definition returns a scalar function on the base space $\mathbb{R}^d \to \mathbb{R}$. For any point $x \in \mathbb{R}^d$, the value of the function is $L(j^k f(x))$, where $j^k f(x)$ denotes the $k$-jet of the field $f$ at $x$.

Let $L$ be a local $k$-th order Lagrangian, $f: \text{Space } d \to \mathbb{R}^m$ be a field, and $x \in \text{Space } d$ be a point in the base space. The evaluation of the Lagrangian $L$ along the field $f$ at the point $x$, denoted as $(L.\text{alongField } f)(x)$, is equal to the value of $L$ applied to the $k$-jet of the field $f$ at $x$: $$(L.\text{alongField } f)(x) = L(j^k f(x))$$ where $j^k f(x)$ represents the $k$-jet of the field $f$ at the point $x$, containing the base point $x$ and the derivatives of $f$ at $x$ up to order $k$.