Physlib.ClassicalMechanics.HarmonicOscillator.WithDim

Given a harmonic oscillator model $S$ whose parameters carry physical dimensions (of type `HarmonicOscillatorWithDim`), this function constructs a corresponding numerical harmonic oscillator (of type `HarmonicOscillator`). It achieves this by extracting the raw numerical values of the mass $m$ and the spring constant $k$ from the dimension-tagged quantities $S.m$ and $S.k$, while preserving the requirements that $m > 0$ and $k > 0$.

Given a position $x$ in a 1-dimensional Euclidean space $\mathbb{R}^1$ that carries the physical dimension of length $L$ (represented by $L_{\mathfrak{d}}$), this function calculates the squared norm $\|x\|^2 = \langle x, x \rangle$ of the position's underlying value. The resulting real number is tagged with the physical dimension of length squared $L^2$.

For any position $x$ in a 1-dimensional Euclidean space $\mathbb{R}^1$ that carries the physical dimension of length $L$, the numerical value of its dimension-tagged squared norm $\|x\|^2$ is equal to the real inner product of the underlying numerical value of $x$ with itself, denoted by $\langle x_{\text{val}}, x_{\text{val}} \rangle_{\mathbb{R}}$.

For a harmonic oscillator $S$ with dimension-tagged parameters, the potential energy function maps a position $x$ in a 1-dimensional Euclidean space $\mathbb{R}^1$ (with physical dimension of length $L$) to the quantity $\frac{1}{2} k \|x\|^2$, where $k$ is the spring constant of the oscillator. The resulting value is a real number carrying the physical dimension of energy, $M \cdot L^2 \cdot T^{-2}$.

For a dimension-tagged harmonic oscillator $S$ and a position $x$ in a 1-dimensional Euclidean space with physical dimension of length $L$, the numerical value of the dimension-tagged potential energy $V_S(x)$ is equal to the potential energy calculated by the corresponding numerical harmonic oscillator $S_{\text{num}}$ at the numerical position $x_{\text{val}}$. That is, $$(V_S(x))_{\text{val}} = V_{S_{\text{num}}}(x_{\text{val}})$$ where $(\cdot)_{\text{val}}$ denotes the extraction of the underlying numerical value from a dimension-tagged quantity.

For a harmonic oscillator $S$ with dimension-tagged parameters, the potential energy function $V_S$ is equal to the transport of the numerical potential energy function of its corresponding numerical oscillator (constructed via `toHarmonicOscillator`). The transport operation lifts the numerical function to one that acts on an input with the dimension of length $L_{\mathfrak{d}}$ and produces an output with the dimension of energy $M_{\mathfrak{d}} \cdot L_{\mathfrak{d}}^2 \cdot T_{\mathfrak{d}}^{-2}$.

Let $S$ be a harmonic oscillator with dimension-tagged parameters and let $V_S$ be its potential energy function, which maps a position $x$ with physical dimension of length $L$ to a value with dimension of energy $M \cdot L^2 \cdot T^{-2}$. For any such position $x$, the potential energy function $V_S$ is Fréchet differentiable at $x$. Its derivative is given by the transported linear map of the Fréchet derivative of the corresponding numerical potential energy function $V_{\text{num}}$ (obtained from the numerical oscillator $S_{\text{num}}$) evaluated at the numerical value of $x$.

For a harmonic oscillator $S$ with dimension-tagged parameters, the potential energy function $V_S$ is continuous.