PhyslibAlpha.ClassicalFieldTheory.Local.Action

Given a local Lagrangian $L$ of order $k$ and a field $f: \mathbb{R}^d \to \mathbb{R}^m$, the action density is 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 $x$.

For a local Lagrangian $L$ of order $k$ and a field $f: \mathbb{R}^d \to \mathbb{R}^m$, the property `HasFiniteAction L f` holds if the action density function $x \mapsto L(j^k f(x))$ is integrable over the space $\mathbb{R}^d$ with respect to the Lebesgue measure.

For a local Lagrangian $L$ of order $k$ and a field $f: \mathbb{R}^d \to \mathbb{R}^m$, the pair $(L, f)$ is admissible for the action functional if the field $f$ is smooth ($C^\infty$) and the action density function $x \mapsto L(j^k f(x))$ is integrable over $\mathbb{R}^d$ (i.e., the field has a finite action).

Given a local Lagrangian $L$ of order $k$ and a field $f: \text{Space } d \to \mathbb{R}^m$, the action is the real number defined as the integral of the action density over the base space $\text{Space } d$: $$S(L, f) = \int_{\text{Space } d} L(j^k f(x)) \, dx$$ where $j^k f(x)$ is the $k$-jet of the field $f$ at point $x$.

Let $L$ be a local Lagrangian of order $k$, $f: \mathbb{R}^d \to \mathbb{R}^m$ be a field, and $x \in \mathbb{R}^d$ be a point in the base space. The action density of the field $f$ for the Lagrangian $L$ evaluated at $x$ is equal to the Lagrangian evaluated at the $k$-jet of the field at $x$, denoted by $j^k f(x)$. That is, $\text{actionDensity}(L, f, x) = L(j^k f(x))$.

For a local Lagrangian $L$ of order $k$ and a field $f: \text{Space } d \to \mathbb{R}^m$, the action $\text{action}(L, f)$ is defined as the integral of the action density over the base space $\text{Space } d$: $$\text{action}(L, f) = \int_{\text{Space } d} \text{actionDensity}(L, f, x) \, dx$$ where $\text{actionDensity}(L, f, x)$ is the Lagrangian evaluated at the $k$-jet of the field $f$ at $x$.

Given a field $f \colon \text{Space } d \to \mathbb{R}^m$, an admissible variation $\eta \colon \text{Space } d \to \mathbb{R}^m$, and a scalar parameter $s \in \mathbb{R}$, the **varied field** is the function mapping each point $x$ in the domain to $f(x) + s\eta(x)$.

Let $f: \text{Space } d \to \mathbb{R}^m$ be an infinitely differentiable ($C^\infty$) field. For any admissible variation $\eta: \text{Space } d \to \mathbb{R}^m$ and any scalar $s \in \mathbb{R}$, the varied field $x \mapsto f(x) + s\eta(x)$ is also infinitely differentiable ($C^\infty$).

Given a local Lagrangian $L$ of order $k$, a field $f \colon \text{Space } d \to \mathbb{R}^m$, and an admissible variation $\eta \colon \text{Space } d \to \mathbb{R}^m$, the **action variation** is the function that maps a scalar parameter $s \in \mathbb{R}$ to the action of the varied field $f + s\eta$. It is defined as: $$s \mapsto S(L, f + s\eta) = \int_{\text{Space } d} L(j^k(f + s\eta)(x)) \, dx$$ where $j^k(f + s\eta)(x)$ denotes the $k$-jet of the varied field at point $x$.

Let $L$ be a local Lagrangian of order $k$, $f: \text{Space } d \to \mathbb{R}^m$ be a field, and $\eta$ be an admissible variation. The property `HasFiniteActionVariation` states that for every scalar parameter $s \in \mathbb{R}$, the action density associated with the varied field $f + s\eta$ is integrable over $\text{Space } d$. Mathematically, this means that for all $s \in \mathbb{R}$, the function $x \mapsto L(j^k(f + s\eta)(x))$ is integrable.

For a local Lagrangian $L$, a field $f: \mathbb{R}^d \to \mathbb{R}^m$, and an admissible variation $\eta$, the property `HasCompactlySupportedActionVariationDifference` holds if, for every scalar parameter $s \in \mathbb{R}$, the difference between the action density of the varied field $f + s\eta$ and the action density of the original field $f$, given by $$x \mapsto \mathcal{L}(j^k(f + s\eta)(x)) - \mathcal{L}(j^k f(x))$$ is a continuous function with compact support on $\mathbb{R}^d$. Here, $\mathcal{L}(j^k f(x))$ denotes the action density evaluated at the $k$-jet of the field $f$ at point $x$.

Let $L$ be a local Lagrangian of order $k$, $f \colon \mathbb{R}^d \to \mathbb{R}^m$ be a field, and $\eta$ be an admissible variation. Suppose that the field $f$ has finite action, meaning the action density $x \mapsto \mathcal{L}(j^k f(x))$ is integrable over $\mathbb{R}^d$. Furthermore, suppose that for every scalar parameter $s \in \mathbb{R}$, the difference between the action density of the varied field and the original field, $x \mapsto \mathcal{L}(j^k(f + s\eta)(x)) - \mathcal{L}(j^k f(x))$, is a continuous function with compact support on $\mathbb{R}^d$. Then the varied field $f + s\eta$ also has finite action for every $s \in \mathbb{R}$, i.e., the function $x \mapsto \mathcal{L}(j^k(f + s\eta)(x))$ is integrable over $\mathbb{R}^d$.

Given a field $f \colon \text{Space } d \to \mathbb{R}^m$, an admissible variation $\eta \colon \text{Space } d \to \mathbb{R}^m$, and a scalar parameter $s \in \mathbb{R}$, the value of the varied field evaluated at a point $x \in \text{Space } d$ is given by $f(x) + s \eta(x)$.

Given a local Lagrangian $L$ of order $k$, a field $f \colon \text{Space } d \to \mathbb{R}^m$, an admissible variation $\eta \colon \text{Space } d \to \mathbb{R}^m$, and a scalar parameter $s \in \mathbb{R}$, the value of the action variation at $s$ is equal to the action of the varied field $f + s\eta$: $$\text{actionVariation}(L, f, \eta)(s) = S(L, f + s\eta)$$

Let $k \in \mathbb{N}$ be a derivative order, $f : \text{Space } d \to \mathbb{R}^m$ be an infinitely differentiable ($C^\infty$) field, and $\eta : \text{Space } d \to \mathbb{R}^m$ be an admissible variation. For any scalar $s \in \mathbb{R}$ and any point $x \in \text{Space } d$, if $x$ is not in the topological support of $\eta$ (denoted $\text{supp}(\eta)$), then the $k$-jet of the varied field $f + s\eta$ at $x$ is equal to the $k$-jet of the field $f$ at $x$: $$j^k (f + s\eta)(x) = j^k f(x)$$

Let $L$ be a local Lagrangian of order $k$ for fields mapping from $\mathbb{R}^d$ to $\mathbb{R}^m$. Let $f: \mathbb{R}^d \to \mathbb{R}^m$ be an infinitely differentiable ($C^\infty$) field and $\eta: \mathbb{R}^d \to \mathbb{R}^m$ be an admissible variation. For any scalar $s \in \mathbb{R}$ and any point $x \in \mathbb{R}^d$, if $x$ is not in the topological support of $\eta$ (denoted $\text{supp}(\eta)$), then the action density of the varied field $f + s\eta$ at the point $x$ is equal to the action density of the original field $f$ at $x$: $$\text{actionDensity}(L, f + s\eta, x) = \text{actionDensity}(L, f, x)$$

Let $L$ be a $k$-th order local Lagrangian and $f: \mathbb{R}^d \to \mathbb{R}^m$ be an infinitely differentiable ($C^\infty$) field. If $L$ is continuous in its local jet coordinates $(x, u^a_I)$, then for any admissible variation $\eta$, the difference between the action density of the varied field $f + s\eta$ and the original field $f$ has compact support. Specifically, for every scalar $s \in \mathbb{R}$, the function $$x \mapsto \mathcal{L}(j^k(f + s\eta)(x)) - \mathcal{L}(j^k f(x))$$ is a continuous function with compact support on $\mathbb{R}^d$, where $\mathcal{L}(j^k f(x))$ denotes the action density evaluated at the $k$-jet of the field $f$ at the point $x$.

Let $L$ be a local Lagrangian of order $k$ and $f \colon \text{Space } d \to \mathbb{R}^m$ be a field. The field $f$ is **critical** for the action functional if, for every admissible variation $\eta$ such that the action of the varied field $f + s\eta$ is finite for all $s \in \mathbb{R}$ (i.e., `HasFiniteActionVariation L f η`), the derivative of the action variation with respect to the parameter $s$ at $s = 0$ is zero. This is expressed by the condition: $$\left. \frac{d}{ds} \right|_{s=0} S(L, f + s\eta) = 0$$ where $S(L, f + s\eta) = \int_{\text{Space } d} L(j^k(f + s\eta)(x)) \, dx$ is the action functional and $j^k(f + s\eta)(x)$ denotes the $k$-jet of the varied field at point $x$.