Physlib.ClassicalMechanics.HarmonicOscillator.Geometric.KineticEnergy

For a harmonic oscillator $S$ with mass $m$ and a point $q$ in its configuration space, `massMetricVal S q` is the continuous bilinear form on the tangent space $T_q Q$ defined by the mass-scaled Euclidean inner product. For any two tangent vectors $v, w \in T_q Q$, the metric value is given by: $$g_q(v, w) = m \cdot \langle \phi_q(v), \phi_q(w) \rangle_{\mathbb{R}}$$ where $\phi_q$ (represented by `tangentCoord q`) is the identification of the tangent space with the Euclidean model space $\mathbb{R}^1$.

For a harmonic oscillator $S$ with mass $m$ and a configuration $q$ in its configuration space $Q$, the mass-scaled Riemannian metric evaluated at two tangent vectors $v, w \in T_q Q$ is equal to the mass $m$ multiplied by the Euclidean inner product of their coordinate representatives: $$\text{massMetricVal}(S, q, v, w) = m \cdot \langle \phi_q(v), \phi_q(w) \rangle_{\mathbb{R}}$$ where $\phi_q$ (represented by `tangentCoord q`) is the continuous linear equivalence that identifies the tangent space $T_q Q$ with the Euclidean model space $\mathbb{R}^1$.

Let $Q$ be the configuration space of a one-dimensional harmonic oscillator. For any configuration $q \in Q$ and any tangent vector $v \in T_q Q$ in the tangent space at $q$, if $v \neq 0$, then its coordinate representation under the identification $\text{tangentCoord}_q : T_q Q \cong \mathbb{R}^1$ is also non-zero, i.e., $\text{tangentCoord}_q(v) \neq 0$.

For a harmonic oscillator $S$ with configuration space $Q$, let $g_q$ denote the mass-scaled metric (represented by `massMetricVal S q`) on the tangent space $T_q Q$. For any non-zero tangent vector $v \in T_q Q$, the metric evaluated at $v$ is strictly positive, i.e., $g_q(v, v) > 0$.

For a harmonic oscillator $S$ and a configuration point $q$, the unit ball in the tangent space $T_q Q$ defined by the mass-scaled metric $g_q$ (the set $\{v \in T_q Q \mid g_q(v, v) < 1\}$) is von Neumann bounded. Here, $g_q$ is the continuous bilinear form `massMetricVal S q` determined by the mass-scaled Euclidean inner product.

For a harmonic oscillator $S$ with configuration space $Q$ (an analytic manifold modeled on $\mathbb{R}^1$), the mass-scaled metric $g_q$ (defined as `massMetricVal S q`) varies analytically with respect to the configuration $q$. Specifically, the map $q \mapsto g_q$, viewed as a section of the bundle of continuous bilinear forms on the tangent spaces $T_q Q$, is of class $C^\omega$.

For a harmonic oscillator $S$ with mass $m$, the mass Riemannian metric is the analytic ($C^\omega$) Riemannian metric on the configuration space $Q$. At each point $q \in Q$, the metric provides a symmetric, positive-definite bilinear form on the tangent space $T_q Q$ defined by the mass-scaled Euclidean inner product: $$g_q(v, w) = m \cdot \langle \phi_q(v), \phi_q(w) \rangle_{\mathbb{R}}$$ where $v, w \in T_q Q$ and $\phi_q$ is the identification of the tangent space with the 1-dimensional Euclidean model space $\mathbb{R}^1$.

For a harmonic oscillator $S$ with configuration space $Q$, the geometric kinetic energy is the function that maps a configuration $q \in Q$ and a velocity vector $v$ in the tangent space $T_q Q$ to the real value: $$K(q, v) = \frac{1}{2} g_q(v, v)$$ where $g_q$ is the mass Riemannian metric of the oscillator $S$ evaluated at the point $q$.

For a harmonic oscillator $S$ with configuration space $Q$, the geometric kinetic energy $K$ at a configuration $q \in Q$ and tangent vector $v \in T_q Q$ is equal to one half of the mass Riemannian metric $g_q$ evaluated on the vector $v$: $$K(q, v) = \frac{1}{2} g_q(v, v)$$

Let $S$ be a harmonic oscillator and $Q$ be its configuration space. Let $g$ be the mass Riemannian metric on $Q$. For any configuration $q \in Q$ and any non-zero tangent vector $v \in T_q Q$ in the tangent space at $q$, the inner product defined by the metric satisfies $g_q(v, v) > 0$.

For a harmonic oscillator $S$ with mass $m$ and configuration space $Q$, the mass Riemannian metric $g$ at a point $q \in Q$ applied to tangent vectors $v, w \in T_q Q$ is equal to the mass $m$ multiplied by the standard Euclidean inner product of their coordinate representatives in $\mathbb{R}^1$: $$g_q(v, w) = m \langle \phi_q(v), \phi_q(w) \rangle_{\mathbb{R}}$$ where $\phi_q : T_q Q \to \mathbb{R}^1$ is the linear equivalence identifying the tangent space with the model space via global coordinates.

For a harmonic oscillator $S$ with mass $m$ and configuration space $Q$, the geometric kinetic energy $K$ at a configuration $q \in Q$ and tangent vector $v \in T_q Q$ is equal to the standard coordinate-based kinetic energy: $$K(q, v) = \frac{1}{2} m \langle \phi_q(v), \phi_q(v) \rangle_{\mathbb{R}}$$ where $\phi_q : T_q Q \cong \mathbb{R}^1$ is the continuous linear equivalence (`tangentCoord`) identifying the tangent space at $q$ with the Euclidean model space $\mathbb{R}^1$ via global coordinates, and $\langle \cdot, \cdot \rangle_{\mathbb{R}}$ is the standard Euclidean inner product. This identity shows that the metric-induced kinetic energy for the mass metric has the standard harmonic-oscillator coordinate form.