Physlib

Physlib.ClassicalMechanics.HarmonicOscillator.Geometric.Trajectory

Geometric trajectories of the harmonic oscillator

i. Overview

A trajectory of the harmonic oscillator is a time-parametrized curve in the configuration manifold `Q`. Since this model of `Q` has a single global coordinate valued in `EuclideanSpace ℝ (Fin 1)`, every geometric trajectory has an associated coordinate curve.

The coordinate diffeomorphism from `Q` to its model space lets smoothness of a geometric trajectory be tested as ordinary smoothness of its coordinate curve.

ii. Key results

- `Trajectory` : a curve from `Time` into the configuration manifold. - `Trajectory.coord` : the global Euclidean coordinate curve of a trajectory. - `Trajectory.toSpace` : the physical-space position along a trajectory. - `Trajectory.contMDiff_iff_contDiff_coord` : geometric smoothness of a trajectory is equivalent to ordinary smoothness of its coordinate curve. - `Trajectory.velocity` : the geometric velocity of a trajectory as a tangent vector. - `Trajectory.velocity_eq_deriv_coord` : in the global coordinate, geometric velocity is represented by the time derivative of the coordinate curve.

iii. Table of contents

  • A. The trajectory type and coordinate projection
  • B. Smoothness of trajectories
  • C. Velocity in the tangent bundle

iv. References

- Ivo Terek, Introductory Variational Calculus on Manifolds, pages 1-2 (Section 1, Basic definitions and examples).

A. The trajectory type and coordinate projection

A trajectory is a curve in the configuration manifold `Q`, parametrized by `Time`. The coordinate projection reads the same curve in the chosen global coordinate, while `toSpace` forgets the manifold structure and returns the corresponding physical-space position.

B. Smoothness of trajectories

Because the global coordinate is a diffeomorphism, composing a trajectory with it preserves and reflects manifold smoothness. Since both `Time` and the coordinate model are normed spaces, this manifold-smoothness statement then becomes ordinary `ContDiff` smoothness.

C. Velocity in the tangent bundle

The velocity of a trajectory at time `t` is the tangent vector obtained by differentiating the curve in the direction of the unit time vector. In this one-chart model, the tangent space at each configuration is represented by the same Euclidean model space, so the geometric velocity can be compared with the derivative of the coordinate curve.

8 declarations

abbrev

Trajectory of a harmonic oscillator

A trajectory γ\gamma is a curve γ:Time→Q\gamma: \text{Time} \to Q that maps a time tt to its corresponding point in the configuration manifold QQ of the harmonic oscillator.

definition

Coordinate curve of a trajectory γ\gamma

For a trajectory γ:Time→Q\gamma: \text{Time} \to Q of a harmonic oscillator, the coordinate curve coord(γ)\text{coord}(\gamma) is the function from Time\text{Time} to the Euclidean model space R1\mathbb{R}^1 that maps each time tt to the reading of the configuration γ(t)\gamma(t) in the global coordinate chart.

definition

Physical position of a trajectory γ\gamma

Given a trajectory γ\gamma of a harmonic oscillator, which is a curve in the configuration manifold QQ, this function maps each time tt to the corresponding physical position in the one-dimensional space R1\mathbb{R}^1 via the transformation (γ(t)).toSpace(\gamma(t)).\text{toSpace}.

theorem

coord(γ)(t)=γ(t).val\text{coord}(\gamma)(t) = \gamma(t).\text{val}

For a trajectory γ\gamma of a harmonic oscillator and any time t∈Timet \in \text{Time}, the value of the coordinate curve coord(γ)\text{coord}(\gamma) at time tt is equal to the coordinate representation (the value in the model space) of the configuration γ(t)\gamma(t).

theorem

CnC^n smoothness of trajectory γ\gamma is equivalent to CnC^n smoothness of its coordinate curve coord(γ)\text{coord}(\gamma)

For a trajectory γ:Time→Q\gamma: \text{Time} \to Q of a harmonic oscillator and any smoothness order n∈N∪{∞}n \in \mathbb{N} \cup \{\infty\}, the map γ\gamma is CnC^n smooth as a map between manifolds if and only if its coordinate curve coord(γ):Time→R1\text{coord}(\gamma): \text{Time} \to \mathbb{R}^1 is CnC^n smooth as a map between normed spaces. Here, the manifold structure of Time\text{Time} and the configuration space QQ are modeled on R\mathbb{R} and the Euclidean space R1\mathbb{R}^1, respectively.

definition

Geometric velocity of a trajectory γ\gamma at time tt

For a trajectory γ\gamma of a harmonic oscillator and a time t∈Timet \in \text{Time}, the geometric velocity is the tangent vector in the tangent space Tγ(t)QT_{\gamma(t)} Q defined by the manifold derivative of γ\gamma at tt applied to the unit vector 11 of the tangent space TtTimeT_t \text{Time}.

theorem

Manifold Differentiability of a Trajectory   ⟺  \iff Differentiability of its Coordinate Curve

For a trajectory γ:Time→Q\gamma : \text{Time} \to Q of a harmonic oscillator and a time t∈Timet \in \text{Time}, γ\gamma is manifold-differentiable at tt if and only if its coordinate curve coord(γ):Time→R1\text{coord}(\gamma) : \text{Time} \to \mathbb{R}^1 is differentiable at tt in the sense of standard calculus. Here, the configuration space QQ is modeled on the Euclidean space R1\mathbb{R}^1.

theorem

velocity(γ,t)=∂t(coord γ)(t)\text{velocity}(\gamma, t) = \partial_t (\text{coord } \gamma)(t)

For a trajectory γ\gamma of a harmonic oscillator and any time t∈Timet \in \text{Time}, the geometric velocity velocity(γ,t)\text{velocity}(\gamma, t)—defined as the tangent vector in Tγ(t)QT_{\gamma(t)}Q obtained by the manifold derivative of γ\gamma—is equal to the time derivative ∂t(coord γ)(t)\partial_t (\text{coord } \gamma)(t) of the trajectory's coordinate curve in the global Euclidean coordinate chart.