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Physlib.ClassicalMechanics.Pendulum.SimplePendulum.Equilibria

Equilibria and energy regimes of the simple pendulum

i. Overview

The equation of motion of the simple gravity pendulum, `I θ̈ = -m g ℓ sin θ` in the lifted formulation of `SimplePendulum.Basic`, has two elementary consequences that follow from the vanishing of the torque and from energy conservation alone, before any non-constant solution is constructed. The first is the equilibria: the torque of gravity vanishes exactly where `sin θ` does, so the constant lifts at the angle `0` — the bob hanging at rest below the pivot — and at the angle `π` — the bob balanced above it — solve the equation of motion, and conversely a constant lift is a solution only at the multiples of `π`. The second is the division of the smooth motions into regimes by the value of the conserved energy. The threshold is the energy `2 m g ℓ` of the inverted equilibrium, the separatrix energy: below it the potential energy cannot reach its value at the top of the swing, so the bob never gets there — classically the regime of libration, the bob swinging back and forth; above it the kinetic energy never vanishes, so the bob never halts — classically the regime of rotation, the pendulum circulating over the top. This is the phase portrait of the pendulum drawn in Arnold §4, whose level curves of the energy are closed ovals below the threshold and unbounded waves above it.

As in `SimplePendulum.Basic`, the motion is written on the Euclidean lift `Time → EuclideanSpace ℝ (Fin 1)` of the angle. Only the bounds characteristic of each regime are proved here: the librating and rotating motions themselves, and the instability of the inverted equilibrium, are statements about non-constant solutions and are not constructed in this module.

ii. Key results

- `SimplePendulum.equationOfMotion_const_zero` and `SimplePendulum.equationOfMotion_const_pi` are the hanging and the inverted equilibrium, packaged as the simplest explicit solutions of the pendulum by `SimplePendulum.isSolution_const_zero` and `SimplePendulum.isSolution_const_pi`, and `SimplePendulum.equationOfMotion_const_iff` shows that the constant solutions are exactly the equilibria. - `SimplePendulum.separatrixEnergy` is the energy `2 m g ℓ` of the inverted equilibrium (`SimplePendulum.energy_const_pi`), the threshold between libration and rotation. - `SimplePendulum.neg_one_lt_cos_of_energy_lt`: below the threshold the bob never reaches the top of the swing. `SimplePendulum.deriv_ne_zero_of_energy_gt`: above it the angular velocity never vanishes. `SimplePendulum.potentialEnergy_eq_energy_of_deriv_eq_zero`: at a turning point the potential energy equals the total energy.

iii. Table of contents

- A. Equilibria - A.1. The hanging equilibrium - A.2. The inverted equilibrium - A.3. The constant solutions are the equilibria - B. Energy regimes - B.1. The separatrix energy - B.2. Energy bounds - B.3. Libration and rotation - B.4. Turning points

iv. References

References for the equilibria and the energy regimes of the simple pendulum include: - Landau & Lifshitz, Mechanics, 3rd ed., §11 (motion in one dimension: the turning points, and finite and infinite motion according to the energy). - Arnold, Mathematical Methods of Classical Mechanics, 2nd ed., §4 (the phase portrait of the pendulum).

A. Equilibria

The two configurations at which the torque of gravity vanishes — the bob hanging at rest below the pivot and the bob balanced above it — give constant solutions of the equation of motion, the simplest explicit solutions of the pendulum. This section verifies the two, and proves the converse: a constant lift solves the equation of motion only where the torque vanishes, that is only at the angles `π n`. The constant solutions are exactly the equilibria.

A.1. The hanging equilibrium

At the angle `0` the bob hangs at rest at the bottom of its swing. The lift is constant, so the angular momentum does not change, and the torque vanishes with `sin 0`: both sides of the equation of motion are zero.

A.2. The inverted equilibrium

At the angle `π` the bob is balanced directly above the pivot, where the torque vanishes with `sin π`; the pendulum stays there. That this balance is unstable — neighbouring solutions run away from it — is a statement about non-constant solutions, and is not proved here.

A.3. The constant solutions are the equilibria

For a constant lift the angular momentum does not change, so the equation of motion reduces to the vanishing of the torque, that is to `sin θ = 0`, which holds exactly at the multiples of `π`. The constant solutions are therefore exactly the equilibria: the hanging equilibrium, the inverted equilibrium, and their copies shifted by whole turns.

B. Energy regimes

Energy conservation divides the smooth motions of the pendulum into regimes according to the value of the conserved energy, the threshold being the energy `2 m g ℓ` of the inverted equilibrium. Below the threshold the potential energy cannot reach its value at the top of the swing, so the bob never reaches the top; classically this is the regime of libration, the bob swinging back and forth. Above the threshold the kinetic energy can never vanish, so the bob never halts; classically this is the regime of rotation, the pendulum circulating over the top. This is the phase portrait of the pendulum drawn in Arnold §4, whose level curves of the energy are closed ovals below the threshold and unbounded waves above it. This section proves the below- and above-threshold bounds characteristic of each regime, from two elementary bounds relating the energies — the librating and rotating motions themselves are not constructed here — and characterizes the turning points, the instants at which the velocity vanishes and the potential energy exhausts the total energy.

B.1. The separatrix energy

The threshold between the regimes is the energy of the inverted equilibrium: no kinetic energy, and the potential energy `2 m g ℓ` of the top of the swing. It is called the separatrix energy after the curve it names in the phase portrait, the level set of the energy separating the closed orbits of libration from the unbounded orbits of rotation. Only the threshold value is used in this file: the separatrix motions themselves — the non-constant solutions asymptotic to the inverted equilibrium — are not constructed here.

B.2. Energy bounds

Two elementary bounds drive the regime theorems: the kinetic energy is non-negative, so the potential energy is at most the total energy; and the potential energy is non-negative, so `I θ̇²` is at most twice the total energy. None of the bounds of this subsection uses the equation of motion — they hold along every lift of the angle.

B.3. Libration and rotation

Along a smooth solution with energy below the separatrix energy, the potential energy — being at most the conserved total energy — stays strictly below `2 m g ℓ`, so the cosine of the angle stays strictly above `-1`: the bob never reaches the top of the swing, and the motion is a libration, swinging back and forth — though only the bound is proved here. Along a smooth solution with energy above the separatrix energy the angular velocity can never vanish, for at such an instant the whole energy would be potential, and the potential energy never exceeds `2 m g ℓ`; the velocity being continuous, it keeps a fixed sign, and the motion is a rotation over the top — though only the non-vanishing is proved here.

B.4. Turning points

At an instant where the angular velocity vanishes the kinetic energy vanishes with it, and the conserved total energy is purely potential. These are the turning points of the motion, where a librating bob halts at the extremes of its arc before swinging back; by the rotation theorem of B.3 they can occur only at energies not above the separatrix energy.

15 declarations

theorem

The constant angle θ=0\theta = 0 satisfies the equation of motion

The constant function θ:Time→R1\theta : \text{Time} \to \mathbb{R}^1 defined by θ(t)=0\theta(t) = 0 for all tt satisfies the equation of motion of the simple pendulum Iθ¨(t)=τ(θ(t))I \ddot{\theta}(t) = \tau(\theta(t)), where τ(θ)=−mgℓsin⁡θ\tau(\theta) = -m g \ell \sin \theta is the gravitational torque. This represents the hanging equilibrium where the pendulum bob remains at rest at the bottom of its swing.

theorem

The Constant Angle θ=0\theta = 0 is a Solution to the Pendulum Equation

Let SS be a simple pendulum. The constant function θ:Time→R1\theta : \text{Time} \to \mathbb{R}^1 defined by θ(t)=0\theta(t) = 0 for all t∈Timet \in \text{Time} is a solution to the equation of motion Iθ¨=−mgℓsin⁡θI \ddot{\theta} = -m g \ell \sin \theta. This solution represents the hanging equilibrium, where the pendulum bob remains at rest at the bottom of its swing.

theorem

The Constant Angle π\pi Satisfies the Pendulum Equation of Motion

The constant function θ(t)=π\theta(t) = \pi for all t∈Timet \in \text{Time} satisfies the simple pendulum's equation of motion Iθ¨=−mgℓsin⁡θI \ddot{\theta} = -mg\ell \sin \theta, where II is the moment of inertia, mm is the mass, gg is the gravitational acceleration, and ℓ\ell is the length of the pendulum.

theorem

The Inverted Equilibrium θ(t)=π\theta(t) = \pi is a Solution to the Simple Pendulum Equation

For a simple pendulum SS, the constant function θ(t)=π\theta(t) = \pi for all t∈Timet \in \text{Time} is a smooth solution to the pendulum's equation of motion Iθ¨=−mgℓsin⁡θI \ddot{\theta} = -m g \ell \sin \theta. This configuration represents the inverted equilibrium where the bob is balanced directly above the pivot.

theorem

Constant Solutions of the Simple Pendulum satisfy sin⁡θ=0\sin \theta = 0

For a constant angular position θ∈R\theta \in \mathbb{R}, the constant function θ(t)=θ\theta(t) = \theta satisfies the equation of motion of the simple pendulum if and only if sin⁡θ=0\sin \theta = 0.

definition

Separatrix energy 2mgℓ2 m g \ell

The separatrix energy of a simple pendulum is defined as 2mgℓ2 m g \ell, where mm is the mass of the bob, gg is the acceleration due to gravity, and ℓ\ell is the length of the pendulum. This value corresponds to the energy of the inverted equilibrium and serves as the threshold between two regimes of motion: libration (swinging back and forth) for energies below this value, and rotation (circulating over the top) for energies above it.

theorem

The Separatrix Energy Equals 2mgℓ2 m g \ell

For a simple pendulum with mass mm, acceleration due to gravity gg, and length ℓ\ell, the separatrix energy is equal to 2mgℓ2 m g \ell.

theorem

Separatrix Energy is Positive (Esep>0E_{\text{sep}} > 0)

The separatrix energy of the simple pendulum, defined as 2mgℓ2 m g \ell (where mm is the mass of the bob, gg is the acceleration due to gravity, and ℓ\ell is the length of the pendulum), is strictly positive.

theorem

The Energy of the Inverted Equilibrium equals the Separatrix Energy

For a simple pendulum SS, if the angular position is constant at θ(t)=π\theta(t) = \pi for all time tt (representing the inverted equilibrium where the bob is balanced at the top), then the total energy of the pendulum E(t)E(t) is equal to the separatrix energy Esep=2mgℓE_{\text{sep}} = 2 m g \ell for all tt.

theorem

Kinetic Energy of a Simple Pendulum is Non-negative (T≥0T \geq 0)

Let SS be a simple pendulum. For any angular trajectory θ:Time→R1\theta: \text{Time} \to \mathbb{R}^1 and any time tt, the kinetic energy T(t)T(t) of the pendulum is non-negative, satisfying T(t)≥0T(t) \geq 0.

theorem

I∥θ˙∥2≤2EI \|\dot{\theta}\|^2 \le 2E for a Simple Pendulum

For a simple pendulum with moment of inertia II, let θ:Time→R1\theta: \text{Time} \to \mathbb{R}^1 be its angular trajectory and E(t)E(t) be its total energy at time tt. For any time tt, the product of the moment of inertia and the square of the angular speed ∥θ˙(t)∥2\|\dot{\theta}(t)\|^2 is at most twice the total energy: I⋅∥θ˙(t)∥2≤2E(t) I \cdot \|\dot{\theta}(t)\|^2 \le 2 E(t) where θ˙(t)\dot{\theta}(t) denotes the time derivative of the trajectory θ\theta at time tt, and the square of the speed is computed using the inner product on R1\mathbb{R}^1.

theorem

Potential Energy ≤\leq Total Energy for a Simple Pendulum

For a simple pendulum SS, given any angular trajectory θ:Time→R1\theta: \text{Time} \to \mathbb{R}^1 and any time tt, the potential energy V(θ(t))V(\theta(t)) is less than or equal to the total energy E(t)E(t) of the system at that instant: V(θ(t))≤E(t) V(\theta(t)) \leq E(t)

theorem

cos⁡θ>−1\cos \theta > -1 for Pendulum Energy below Separatrix

Let θ:Time→R1\theta : \text{Time} \to \mathbb{R}^1 be an infinitely differentiable function representing the angular position of a simple pendulum. Suppose θ\theta satisfies the equation of motion Iθ¨=−mgℓsin⁡θI \ddot{\theta} = -mg\ell \sin \theta. If the total conserved energy EE of the motion is strictly less than the separatrix energy Esep=2mgℓE_{\text{sep}} = 2mg\ell, then for any time tt, the cosine of the angle satisfies cos⁡(θ(t))>−1\cos(\theta(t)) > -1. This indicates that in the regime of libration, the pendulum bob never reaches the inverted equilibrium position at the top of its swing.

theorem

E>2mgℓ  ⟹  θ˙≠0E > 2 m g \ell \implies \dot{\theta} \neq 0 for the simple pendulum

Let θ:Time→R1\theta: \text{Time} \to \mathbb{R}^1 be a smooth trajectory of a simple pendulum that satisfies the equation of motion. If the total energy EE of the system is strictly greater than the separatrix energy 2mgℓ2 m g \ell, then for any time tt, the angular velocity θ˙(t)\dot{\theta}(t) (the time derivative ∂tθ(t)\partial_t \theta(t)) is non-zero. This signifies that in the rotation regime, the pendulum bob never halts.

theorem

Potential Energy Equals Total Energy at Turning Points (θ˙=0  ⟹  V=E\dot{\theta} = 0 \implies V = E)

Let θ:Time→R1\theta : \text{Time} \to \mathbb{R}^1 be a smooth trajectory of a simple pendulum that satisfies the equation of motion. If at some time tt, the angular velocity vanishes (i.e., θ˙(t)=0\dot{\theta}(t) = 0), then the potential energy V(θ(t))V(\theta(t)) at that instant is equal to the total energy EE of the system (evaluated here at t=0t=0).