Physlib

Physlib.ClassicalMechanics.Pendulum.SimplePendulum.LiftInvariance

Independence of the lift for the simple pendulum

i. Overview

The dynamics of the simple gravity pendulum in `SimplePendulum.Basic` is written on a lift of the motion: the real angle `θ t 0` stands for the configuration `ConfigurationSpace.ofAngle (θ t 0)`, and two lifts differing by a whole number of turns carry the same configurations. For the lifted formulation to describe the pendulum faithfully, nothing dynamical may depend on the choice of the lift. This module proves that the dynamical quantities — the potential and kinetic energies, the torque, the energy, and the equation of motion together with its solutions — are invariant under the deck transformations `θ ↦ θ + 2π n` of the angular lift, and closes by making the starting point precise: the shifted lift describes the same configuration, by the periodicity of the angular lift of the geometric configuration space. The packaging of this invariance at the level of configuration-space trajectories comes with the geometric bridge in a later module.

ii. Key results

- `SimplePendulum.potentialEnergy_add_int_mul_two_pi` and `SimplePendulum.torque_add_int_mul_two_pi`: the potential energy and the torque are unchanged by shifting the angle by a whole number of turns. - `SimplePendulum.kineticEnergy_add_const` and `SimplePendulum.energy_add_int_mul_two_pi`: the kinetic energy is unchanged by any constant shift of the lift, and the energy by a shift by a whole number of turns. - `SimplePendulum.equationOfMotion_add_int_mul_two_pi` and `SimplePendulum.isSolution_add_int_mul_two_pi`: the equation of motion and its solutions are invariant under shifting the lift by a whole number of turns. - `SimplePendulum.ofAngle_add_int_mul_two_pi_coord`: the shifted lift describes the same configuration.

iii. Table of contents

- A. Independence of the lift - A.1. Invariance of the potential energy and the torque - A.2. Invariance of the energy - A.3. Invariance of the equation of motion and its solutions - A.4. The shifted lift describes the same configuration

iv. References

References for the simple gravity pendulum include: - Landau & Lifshitz, Mechanics, 3rd ed., §5 and §21. - Arnold, Mathematical Methods of Classical Mechanics, 2nd ed., §4.

A. Independence of the lift

The dynamics of `SimplePendulum.Basic` are written on a lift of the motion: the real angle `θ t 0` stands for the configuration `ConfigurationSpace.ofAngle (θ t 0)`, and two lifts differing by a whole number of turns carry the same configurations. This section proves that the dynamical quantities listed below — the energies, the torque, and the equation of motion and its solutions — are invariant under the deck transformations `θ ↦ θ + 2π n` of the angular lift; the packaging of this invariance at the level of configuration-space trajectories comes with the geometric bridge in a later module. The section closes by making the starting point precise: the shifted lift does describe the same configuration, by the periodicity of the angular lift of the geometric configuration space.

A.1. Invariance of the potential energy and the torque

The potential energy and the torque depend on the angle only through its cosine and its sine, and both have period `2π`: neither quantity changes when the angle is shifted by a whole number of turns.

A.2. Invariance of the energy

The shift of the lift is constant in time, so it drops out of the velocity, and the kinetic energy is unchanged by any constant shift at all; the potential energy is unchanged by the invariance of A.1. Together the two give the invariance of the energy under shifting the lift by a whole number of turns.

A.3. Invariance of the equation of motion and its solutions

Both sides of the equation of motion are invariant under the shift: the angular momentum, because the shift is constant in time, and the torque, by the invariance of A.1. Smoothness is likewise unaffected by adding a constant, so being a solution is invariant as well.

A.4. The shifted lift describes the same configuration

Finally the statement giving the previous invariances their meaning: the lift and its shift by a whole number of turns project to the same point of the configuration space, by the periodicity of the angular lift `ConfigurationSpace.ofAngle` with period `2π`.

7 declarations

theorem

Simple Pendulum Potential Energy V(x)V(x) is Invariant under 2πn2\pi n Shifts

Consider a simple pendulum where x∈R1x \in \mathbb{R}^1 represents the angular position (the angle θ\theta from the downward vertical). For any integer n∈Zn \in \mathbb{Z}, the potential energy VV is invariant under a shift of the angle by a whole number of turns (2πn2\pi n): V(x+2πn⋅e0)=V(x) V(x + 2\pi n \cdot \mathbf{e}_0) = V(x) where e0\mathbf{e}_0 is the unit vector in the 1-dimensional Euclidean space. This reflects the fact that the potential energy V(θ)=mgℓ(1−cos⁡θ)V(\theta) = m g \ell (1 - \cos \theta) depends on the angle only through the cosine function, which has a period of 2π2\pi.

theorem

Simple Pendulum Torque is Invariant under 2πn2\pi n Shifts

For a simple gravity pendulum, let x∈R1x \in \mathbb{R}^1 represent the angular position. For any integer n∈Zn \in \mathbb{Z}, the torque τ\tau is invariant under a shift of the angle by a whole number of turns (2πn2\pi n): τ(x+2πn⋅e0)=τ(x)\tau(x + 2\pi n \cdot \mathbf{e}_0) = \tau(x) where e0\mathbf{e}_0 is the unit vector in the 1-dimensional Euclidean space. This reflects the periodicity of the physical system, where the torque depends on the angle through the sine function which has a period of 2π2\pi.

theorem

Kinetic Energy of the Simple Pendulum is Invariant under Constant Angular Shifts T(θ+c)=T(θ)T(\theta + c) = T(\theta)

For a simple pendulum, let θ:Time→R1\theta: \text{Time} \to \mathbb{R}^1 be a trajectory representing the angular position. For any constant angular shift c∈R1c \in \mathbb{R}^1, the kinetic energy TT is invariant under the shift θ↦θ+c\theta \mapsto \theta + c: T(θ+c)=T(θ)T(\theta + c) = T(\theta) This invariance arises because the constant shift cc disappears when calculating the angular velocity (the time derivative of the angle), which is the only dynamical variable the kinetic energy depends on.

theorem

Simple Pendulum Energy is Invariant under 2πn2\pi n Shifts

For a simple pendulum, let θ:Time→R1\theta: \text{Time} \to \mathbb{R}^1 be a trajectory representing the angular lift of the motion. For any integer n∈Zn \in \mathbb{Z}, the total energy EE is invariant under a shift of the angular lift by a whole number of turns (2πn2\pi n): E(θ+2πn⋅e0)=E(θ) E(\theta + 2\pi n \cdot \mathbf{e}_0) = E(\theta) where e0\mathbf{e}_0 is the unit vector in the 1-dimensional Euclidean space.

theorem

The Simple Pendulum Equation of Motion is Invariant under 2πn2\pi n Shifting

Let θ:Time→R\theta : \text{Time} \to \mathbb{R} represent the angular lift of a simple pendulum over time. For any integer n∈Zn \in \mathbb{Z}, the shifted trajectory θ′(t)=θ(t)+2πn\theta'(t) = \theta(t) + 2\pi n satisfies the simple pendulum's equation of motion if and only if the original trajectory θ(t)\theta(t) does.

theorem

The shifted lift θ+2πn\theta + 2\pi n is a solution iff θ\theta is a solution

Let θ:Time→R\theta: \text{Time} \to \mathbb{R} represent the angular lift of a simple pendulum SS over time. For any integer n∈Zn \in \mathbb{Z}, the shifted trajectory defined by t↦θ(t)+2πnt \mapsto \theta(t) + 2\pi n is a solution to the simple pendulum equations if and only if the original trajectory θ\theta is a solution. Here, the angular lift is treated as a function into R\mathbb{R} (modeled as a 1-dimensional Euclidean space), and 2πn2\pi n represents a shift by a whole number of turns.

theorem

ofAngle(x0+2πn)=ofAngle(x0)\text{ofAngle}(x_0 + 2\pi n) = \text{ofAngle}(x_0)

For a simple pendulum, let x∈R1x \in \mathbb{R}^1 be a vector where x0x_0 represents the angular lift. For any integer n∈Zn \in \mathbb{Z}, shifting the angle by a whole number of turns 2πn2\pi n results in the same physical configuration. That is, ofAngle(x0+2πn)=ofAngle(x0)\text{ofAngle}(x_0 + 2\pi n) = \text{ofAngle}(x_0) where ofAngle:R→ConfigurationSpace\text{ofAngle} : \mathbb{R} \to \text{ConfigurationSpace} is the map from the angular lift to the pendulum's configuration space.