PhyslibAlpha.ClassicalFieldTheory.Local.FirstOrder

For a base manifold of dimension $d$ and a field with $m$ components, this defines the space of coordinate data for first-order jets ($k=1$). These coordinates $\{u^a_I\}$ consist of the field values $u^a$ (where $|I|=0$) and their first-order partial derivatives $\partial_i u^a$ (where $|I|=1$), for component indices $0 \le a < m$ and spatial indices $0 \le i < d$. It is the specialization of the general $k$-th order jet coordinates $\text{JetCoordinates}(d, m, k)$ to the case $k=1$.

For a classical field theory with $d$ independent variables (base manifold dimension) and $m$ dependent variables (field components), a **first-order jet point** is an element of the jet bundle of order $k=1$. It represents the local data of a field at a point, consisting of the coordinates of the base manifold, the values of the fields, and their first-order partial derivatives.

Given a base manifold of dimension $d$ and a field space of dimension $m$, this definition represents the fiber data of a first-order jet bundle ($k=1$). It consists of the field values and their first-order partial derivatives at a point in the base manifold, serving as the configuration space for first-order classical field theories.

For a field theory with $d$ spacetime dimensions and $m$ field components, a first-order local Lagrangian is defined as a Lagrangian of order $k=1$. This represents a Lagrangian density that depends on the coordinates, the field values, and their first derivatives.

Given a dimension $d$ and a coordinate index $i \in \{0, \dots, d-1\}$, this function returns the multi-index of order 1 that represents a partial derivative in the $i$-th direction. It is defined as the multi-index $e_i$ (where the $i$-th component is 1 and all others are 0), treated as an element of the set of multi-indices $\{ I \in \mathbb{N}^d \mid |I| \le 1 \}$.

For a given dimension $d$ and a coordinate index $i \in \{0, \dots, d-1\}$, the first-derivative multi-index of order 1 in the $i$-th direction is equal to the unit multi-index $e_i$, which is obtained by incrementing the $i$-th component of the zero multi-index.

For any dimension $d \in \mathbb{N}$ and coordinate index $i \in \{0, \dots, d-1\}$, let $I$ be the multi-index representing the first partial derivative in the $i$-th direction (often denoted $e_i$). The order of this multi-index, defined as the sum of its components $|I| = \sum_{j=0}^{d-1} I_j$, is equal to $1$.

Given a first-order jet point $J$ in a theory with $d$ independent variables and $m$ dependent variables, this function returns the jet coordinate $u^a_i \in \mathbb{R}$. Here, $i \in \{0, \dots, d-1\}$ denotes the direction of the partial derivative and $a \in \{0, \dots, m-1\}$ denotes the field component index. This coordinate represents the value of the first-order partial derivative $\frac{\partial u^a}{\partial x^i}$ at the point represented by $J$.

Given a first-order jet point $J$ (representing the local data of a field with $d$ independent and $m$ dependent variables) and a coordinate direction $i \in \{0, \dots, d-1\}$, this function returns the vector in the Euclidean space $\mathbb{R}^m$ consisting of the first-order partial derivatives of all field components in that direction. Specifically, the $a$-th component of the resulting vector is the jet coordinate $u^a_i$, which represents the value of the partial derivative $\frac{\partial u^a}{\partial x^i}$ at the point $J$.

Let $J$ be a first-order jet point in a classical field theory with $d$ independent variables and $m$ field components. For any spatial direction index $i \in \{0, \dots, d-1\}$ and field component index $a \in \{0, \dots, m-1\}$, the $a$-th component of the vector of first-order partial derivatives in the direction $i$ (denoted as `firstDerivVector i`) is equal to the first derivative coordinate $u^a_i$ (denoted as `firstDerivCoord i a`).

Let $x \in \text{Space } d$ be a base point and $u$ be a set of first-order jet coordinates (representing field values $u^a$ and first derivatives $u^a_i$). Let $J$ be the jet point constructed from $x$ and $u$ via `JetPoint.ofBaseCoordinates`. For any spatial direction $i \in \{0, \dots, d-1\}$ and field component index $a \in \{0, \dots, m-1\}$, the first derivative coordinate $u^a_i$ of the jet point $J$ is equal to the entry in $u$ corresponding to the first-order multi-index $e_i$ and the field index $a$. That is, $$(x, u).u^a_i = u(e_i, a)$$ where $e_i$ is the multi-index with $1$ in the $i$-th position and $0$ elsewhere.

For a field $f: \text{Space } d \to \mathbb{R}^m$ and a base point $x \in \text{Space } d$, the **first-order jet of $f$ at $x$**, denoted $j^1 f(x)$, is the jet point of order $k=1$. It captures the local data of the field at $x$, consisting of the base point $x$, the field values $u^a = f_a(x)$, and the first-order partial derivatives $u^a_i = \frac{\partial f_a}{\partial x^i}(x)$ for $a \in \{1, \dots, m\}$ and $i \in \{1, \dots, d\}$.

For a field $f : \text{Space } d \to \mathbb{R}^m$ and a point $x \in \text{Space } d$, the base point component of the first-order jet $j^1 f(x)$ is equal to $x$.

For a field $f: \text{Space } d \to \mathbb{R}^m$ and a point $x \in \text{Space } d$, the value component of the first-order jet of $f$ at $x$, denoted $(j^1 f(x)).\text{value}$, is equal to the value of the field at that point, $f(x)$.

For a field $f: \text{Space } d \to \mathbb{R}^m$, a point $x \in \text{Space } d$, a coordinate index $i \in \{0, \dots, d-1\}$, and a field component index $a \in \{0, \dots, m-1\}$, the first derivative coordinate $u^a_i$ of the first-order jet $j^1 f(x)$ is equal to the partial derivative of the $a$-th component of the field with respect to the $i$-th coordinate evaluated at $x$: $$(j^1 f(x))^a_i = \frac{\partial f_a}{\partial x^i}(x)$$ where $\text{Space } d$ represents the $d$-dimensional base space (typically $\mathbb{R}^d$) and $j^1 f(x)$ is the first-order jet of $f$ at $x$.

Given a first-order local Lagrangian $L$ (defined for $m$ field components in $d$ spacetime dimensions) and a field $f: \mathbb{R}^d \to \mathbb{R}^m$, the first-order action density is the scalar-valued function on the base space $\mathbb{R}^d \to \mathbb{R}$ defined by evaluating $L$ at the first-order jet of the field at each point. For any point $x \in \mathbb{R}^d$, the value is $L(j^1 f(x))$, where $j^1 f(x)$ comprises the field values $f(x)$ and their first-order partial derivatives $\partial_\mu f(x)$.

Given a first-order local Lagrangian $L$ (defined for $m$ field components in $d$ dimensions) and a field $f: \text{Space } d \to \mathbb{R}^m$, the first-order action is the real number defined by the integral of the Lagrangian density evaluated at the first-order jet $j^1 f(x)$ over the base space $\text{Space } d$: $$S(L, f) = \int_{\text{Space } d} L(j^1 f(x)) \, dx$$ where $j^1 f(x)$ represents the first-order jet of the field $f$ at point $x$, comprising the field values and their first-order partial derivatives.

Given a first-order local Lagrangian $L$ (a Lagrangian of order $k=1$) and a field $f: \mathbb{R}^d \to \mathbb{R}^m$, the first-order Euler-Lagrange operator $\mathcal{E}(L)$ maps the field $f$ to a function $\mathbb{R}^d \to \mathbb{R}^m$. For each field component $a \in \{1, \dots, m\}$, the $a$-th component of the operator's value at a point $x \in \mathbb{R}^d$ is given by the standard first-order Euler-Lagrange expression: $$(\mathcal{E}(L)[f](x))_a = \frac{\partial L}{\partial u^a} - \sum_{\mu=1}^d D_\mu \left( \frac{\partial L}{\partial u^a_\mu} \right)$$ where $\frac{\partial L}{\partial u^a}$ is the partial derivative of the Lagrangian with respect to the field value, $\frac{\partial L}{\partial u^a_\mu}$ is the partial derivative with respect to the first-order derivative $\partial_\mu f^a$, and $D_\mu$ denotes the total derivative with respect to the spatial coordinate $x^\mu$.

Consider a first-order local Lagrangian $L$ for $m$ field components in $d$ dimensions and a field $f: \mathbb{R}^d \to \mathbb{R}^m$. Suppose that the pair $(L, f)$ is admissible for the action functional (meaning $f$ is smooth and the action integral $S(L, f) = \int_{\mathbb{R}^d} L(j^1 f(x)) \, dx$ is finite) and that $L$ is smooth in its jet coordinates. Then, the field $f$ is a critical point for the action functional if and only if it satisfies the first-order Euler-Lagrange equations, i.e., $$\mathcal{E}(L)[f] = 0$$ where $\mathcal{E}(L)[f]$ is the Euler-Lagrange operator whose components are given by $$(\mathcal{E}(L)[f](x))_a = \frac{\partial L}{\partial u^a} - \sum_{\mu=1}^d D_\mu \left( \frac{\partial L}{\partial u^a_\mu} \right)$$ for each field component $a \in \{1, \dots, m\}$.