Physlib.ClassicalMechanics.HarmonicOscillator.ConfigurationSpace

In the configuration space $Q$ of a harmonic oscillator, which is modeled as a one-dimensional manifold with coordinates in $\mathbb{R}$, two configurations $x, y \in Q$ are equal if their coordinate values are equal, i.e., $x.\text{val} = y.\text{val} \implies x = y$.

The configuration space $Q$ of a harmonic oscillator is equipped with a zero element $0 \in Q$. This element represents the origin of the one-dimensional manifold, such that its coordinate value in the underlying space $\mathbb{R}$ is $0$.

The natural number $0$ is defined as an element of the configuration space $Q$ of a harmonic oscillator. This allows the numeral $0$ to be used to represent the configuration whose underlying coordinate value is $0 \in \mathbb{R}$.

In the configuration space of a harmonic oscillator, the coordinate value of the zero element $0$ is equal to the real number $0$, denoted as $(0 : \text{ConfigurationSpace}).\text{val} = 0$.

The addition operation on the configuration space of a harmonic oscillator is defined by the sum of the underlying coordinate values. For any configurations $x$ and $y$ in the configuration space, their sum $x + y$ is the configuration such that $(x + y).\text{val} = x.\text{val} + y.\text{val}$.

For any two configurations $x$ and $y$ in the configuration space of a harmonic oscillator, the coordinate value of their sum $x + y$ is equal to the sum of their individual coordinate values, i.e., $(x + y).\text{val} = x.\text{val} + y.\text{val}$.

The negation operation on the configuration space of a harmonic oscillator is defined by taking the additive inverse of the underlying coordinate value. For a configuration $x$ with coordinate value $x.\text{val}$, its negation $-x$ is the configuration such that $(-x).\text{val} = -(x.\text{val})$.

In the configuration space of a harmonic oscillator, which is modeled as a one-dimensional manifold with a chosen coordinate, for any configuration $x$, the coordinate value of its negation $-x$ is equal to the negation of its coordinate value $x.\text{val}$. That is, $(-x).\text{val} = -x.\text{val}$.

The subtraction operation on the configuration space of a harmonic oscillator is defined by taking the difference of the underlying coordinate values. For two configurations $x$ and $y$ with coordinate values $x.\text{val}$ and $y.\text{val}$ respectively, their difference $x - y$ is the configuration such that $(x - y).\text{val} = x.\text{val} - y.\text{val}$.

For any two points $x$ and $y$ in the configuration space of the harmonic oscillator, the coordinate value of their difference $x - y$ is equal to the difference of their respective coordinate values $x.\text{val}$ and $y.\text{val}$, that is, $(x - y).\text{val} = x.\text{val} - y.\text{val}$.

The definition equips the configuration space of the one-dimensional harmonic oscillator with a scalar multiplication by real numbers $\mathbb{R}$. For a scalar $r \in \mathbb{R}$ and a point $x$ in the configuration space, the scalar multiplication $r \cdot x$ is defined as the point in the configuration space whose coordinate value is the product $r \cdot x.\text{val}$.

For any real number $r \in \mathbb{R}$ and any point $x$ in the configuration space of the one-dimensional harmonic oscillator, the coordinate value of the scalar multiplication $r \cdot x$ is equal to the product of $r$ and the coordinate value of $x$, expressed as $(r \cdot x).\text{val} = r \cdot x.\text{val}$.

This definition provides a coercion of an element $x$ in the configuration space of a one-dimensional harmonic oscillator to a function from the index set $\{0\}$ (represented by $\text{Fin } 1$) to the real numbers $\mathbb{R}$. For any element $x \in \text{ConfigurationSpace}$, it is treated as a function such that $x(i) = x.\text{val}$ for the index $i = 0$.

For any element $x$ in the configuration space of a one-dimensional harmonic oscillator, the evaluation of $x$ at the index $0$ is equal to its underlying scalar value $x.\text{val}$.

For any element $x$ in the configuration space of a one-dimensional harmonic oscillator and any index $i$ in the set $\{0\}$ (represented by $\text{Fin } 1$), the coordinate evaluation $x(i)$ is equal to the underlying scalar value $x.\text{val}$.

The configuration space of a one-dimensional harmonic oscillator is equipped with the structure of an additive group. This means that for any configurations $x, y, z$ in the space, the following properties hold: 1. Addition is associative: $(x + y) + z = x + (y + z)$. 2. There exists a zero element $0$ such that $0 + x = x$ and $x + 0 = x$. 3. For every configuration $x$, there is an additive inverse $-x$ such that $-x + x = 0$. These operations are defined via the underlying real-valued coordinates of the configurations.

The configuration space of a one-dimensional harmonic oscillator, $\text{ConfigurationSpace}$, is equipped with the structure of an additive commutative group (abelian group). In addition to the properties of an additive group, for any configurations $x, y \in \text{ConfigurationSpace}$, the addition operation is commutative, satisfying $x + y = y + x$.

The configuration space of the one-dimensional harmonic oscillator, $\text{ConfigurationSpace}$, is equipped with the structure of a module over the field of real numbers $\mathbb{R}$. This implies that $\text{ConfigurationSpace}$ functions as a vector space, where the addition of configurations and scalar multiplication by real numbers satisfy the standard axioms, such as $1 \cdot x = x$, $r \cdot (x + y) = r \cdot x + r \cdot y$, and $(r + s) \cdot x = r \cdot x + s \cdot x$ for any $r, s \in \mathbb{R}$ and $x, y \in \text{ConfigurationSpace}$.

The norm $\|x\|$ for an element $x$ in the configuration space of the harmonic oscillator is defined as the norm (absolute value) of its underlying coordinate value $x.val \in \mathbb{R}$.

The distance function $d$ on the configuration space of the harmonic oscillator is defined as the norm of the difference between two configurations $x$ and $y$, such that $d(x, y) = \|x - y\|$.

For any two configurations $x$ and $y$ in the configuration space of the harmonic oscillator, the distance $d(x, y)$ between them is equal to the norm of the difference of their coordinate values, $d(x, y) = \|x_{\text{val}} - y_{\text{val}}\|$, where $x_{\text{val}}$ and $y_{\text{val}}$ are the real numbers representing the coordinates of the configurations.

The configuration space of the one-dimensional harmonic oscillator, $\text{ConfigurationSpace}$, is equipped with the structure of a seminormed additive commutative group. This structure utilizes the existing additive commutative group property and defines a distance function $d(x, y) = \|x - y\|$ that satisfies the triangle inequality $d(x, z) \leq d(x, y) + d(y, z)$ and the symmetry property $d(x, y) = d(y, x)$, consistent with the norm defined on the space.

The configuration space of the one-dimensional harmonic oscillator, $\text{ConfigurationSpace}$, is equipped with the structure of a normed additive commutative group. This structure ensures that $\text{ConfigurationSpace}$ is an additive abelian group where the distance between two configurations $x$ and $y$, defined as $d(x, y) = \|x - y\|$, satisfies the identity of indiscernibles: $d(x, y) = 0$ if and only if $x = y$. The norm $\|x\|$ corresponds to the absolute value of the underlying real coordinate of the configuration.

The configuration space of the one-dimensional harmonic oscillator, $\text{ConfigurationSpace}$, is equipped with the structure of a normed space over the real numbers $\mathbb{R}$. This means that $\text{ConfigurationSpace}$ is a real vector space endowed with a norm $\|\cdot\|$ that is compatible with scalar multiplication, satisfying $\|r \cdot x\| = |r| \|x\|$ for all scalars $r \in \mathbb{R}$ and configurations $x \in \text{ConfigurationSpace}$.

The configuration space of the harmonic oscillator is equipped with an inner product over the real numbers $\mathbb{R}$. For any two elements $x$ and $y$ in this space, their inner product is defined as $\langle x, y \rangle_{\mathbb{R}} = x \cdot y$.

For any two elements $x$ and $y$ in the configuration space of the harmonic oscillator, their real inner product $\langle x, y \rangle_{\mathbb{R}}$ is equal to the product of their coordinate values $x \cdot y$ in $\mathbb{R}$.

The configuration space of the one-dimensional harmonic oscillator, $\text{ConfigurationSpace}$, is equipped with the structure of a real inner product space over $\mathbb{R}$. This structure ensures that the space is a real vector space endowed with an inner product $\langle \cdot, \cdot \rangle_{\mathbb{R}}$ that is symmetric, linear in the first argument, and consistent with the norm such that $\|x\|^2 = \langle x, x \rangle_{\mathbb{R}}$ for all $x \in \text{ConfigurationSpace}$.

The function mapping each configuration $x$ in the configuration space of the one-dimensional harmonic oscillator to its real inner product with itself, $x \mapsto \langle x, x \rangle_{\mathbb{R}}$, is differentiable over $\mathbb{R}$.

Let $\text{ConfigurationSpace}$ be the configuration space of the one-dimensional harmonic oscillator, which is equipped with a real inner product $\langle \cdot, \cdot \rangle_{\mathbb{R}}$. For any configuration $x \in \text{ConfigurationSpace}$, the function $y \mapsto \langle y, y \rangle_{\mathbb{R}}$ is differentiable at $x$ over the real numbers $\mathbb{R}$.

In the configuration space of the one-dimensional harmonic oscillator, which is a real inner product space, the function $x \mapsto \langle x, x \rangle_{\mathbb{R}}$ mapping a configuration to its inner product with itself is $n$-times continuously differentiable (of class $C^n$) for any $n \in \mathbb{N} \cup \{\infty\}$.

The definition describes a linear map from the configuration space of the one-dimensional harmonic oscillator, denoted as $\text{ConfigurationSpace}$, to the field of real numbers $\mathbb{R}$. This map sends an element $x \in \text{ConfigurationSpace}$ to its corresponding underlying real value.

This definition establishes an $\mathbb{R}$-linear map from the real numbers $\mathbb{R}$ to the configuration space of the one-dimensional harmonic oscillator, denoted as $\text{ConfigurationSpace}$. This map embeds a real scalar $x \in \mathbb{R}$ into the configuration space by identifying it with its corresponding coordinate point.

The continuous linear map from the configuration space of the one-dimensional harmonic oscillator, denoted as $\text{ConfigurationSpace}$, to the field of real numbers $\mathbb{R}$, which maps an element of the configuration space to its underlying real coordinate value.

This definition establishes a continuous $\mathbb{R}$-linear map from the real numbers $\mathbb{R}$ to the configuration space of the one-dimensional harmonic oscillator, denoted as $\text{ConfigurationSpace}$. This map embeds a real scalar $x \in \mathbb{R}$ into the configuration space by identifying it with its corresponding coordinate point.

The configuration space of the one-dimensional harmonic oscillator, denoted as $\text{ConfigurationSpace}$, is homeomorphic to the real numbers $\mathbb{R}$. This homeomorphism is defined by the mapping $\text{val}: \text{ConfigurationSpace} \to \mathbb{R}$, which assigns each point in the configuration space to its corresponding coordinate value, and its inverse, which maps a real number $t \in \mathbb{R}$ back to the configuration space. Both the coordinate map and its inverse are continuous.

The configuration space of the harmonic oscillator, denoted as $\text{ConfigurationSpace}$, is equipped with the structure of a charted space modeled on the real numbers $\mathbb{R}$. This structure is defined by an atlas containing a single global chart, which is the homeomorphism $\text{val}: \text{ConfigurationSpace} \to \mathbb{R}$ that maps each point in the configuration space to its corresponding real coordinate.

The configuration space of the one-dimensional harmonic oscillator, denoted as $\text{ConfigurationSpace}$, is a smooth manifold of class $C^\omega$ (real analytic) modeled on the real numbers $\mathbb{R}$ with the identity model with corners $\mathcal{I}(\mathbb{R}, \mathbb{R})$.

The configuration space of the one-dimensional harmonic oscillator, denoted as $\text{ConfigurationSpace}$, is a finite-dimensional vector space over the field of real numbers $\mathbb{R}$.

The configuration space of the one-dimensional harmonic oscillator, denoted as $\text{ConfigurationSpace}$, is a complete space. This means that every Cauchy sequence in $\text{ConfigurationSpace}$ converges to a limit within the space, with respect to the metric induced by its real normed space structure.

This function maps a configuration $q$ in the configuration space of a one-dimensional harmonic oscillator to its corresponding position in the one-dimensional space $\mathbb{R}^1$. For a given configuration $q$, the resulting vector $x \in \mathbb{R}^1$ has its unique component $x_0$ equal to the coordinate value $q$.

For any configuration $q$ in the configuration space of a one-dimensional harmonic oscillator, the $i$-th component of its corresponding position vector in the one-dimensional space $\mathbb{R}^1$ is equal to the coordinate value of $q$. That is, for $i \in \{0\}$, $(q.\text{toSpace})_i = q$.