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The ideal planar pendulum: equation of motion, conserved energy, Lyapunov stability and the frequency $\sqrt{g / l}$

Example

Let $m, l, g > 0$ and let $Ω : = (- π, π) \subseteq R$ . Consider the embedding

$X : Ω \to R^{2}, X (θ) : = (l \sin θ, - l \cos θ),$

the ambient mass $M : = m I_{2}$ and the constant force $F : = (0, - m g)$ . Let $I \subseteq R$ be a nonempty open interval (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length), let $θ : I \to Ω$ be a $C^{2}$ curve and put $x (t) : = X (θ (t))$ . Then:

$X$ is a $C^{3}$ immersion with $∥ X^{'} (θ) ∥_{2} = l$ for every $θ$ ; if the ambient motion satisfies $M x^{''} = F + R$ with $R (t) \cdot X^{'} (θ (t)) δ = 0$ for all $δ \in R$ , then

$m l^{2} θ^{''} (t) + m g l \sin θ (t) = 0 (t \in I);$

the reduced energy $E (θ, θ^{'}) : = \frac{1}{2} m l^{2} θ^{' 2} - m g l \cos θ$ is constant along every solution of the equation;
$θ = 0$ is a strict local minimum of the reduced potential $U_{r e d} (θ) : = - m g l \cos θ$ on $Ω$ , and the equilibrium $(0, 0)$ of the pendulum system $θ^{'} = v$ , $v^{'} = - (g / l) \sin θ$ is Lyapunov stable;
the linearisation at $θ = 0$ is $m l^{2} ξ^{''} + m g l ξ = 0$ , a quadratic mode with angular frequency

$ω = \sqrt{g / l} .$

Facts & Assumptions

Given: The parameters $m, l, g > 0$ , the embedding $X$ , the matrix $M = m I_{2}$ , the force $F = (0, - m g)$ , the nonempty open interval $I$ , a $C^{2}$ curve $θ : I \to Ω$ with $x = X \circ θ$ , and a reaction $R$ as in clause 1.

[F1]

Derivatives of $X$ : since $\sin^{'} = \cos$ , $\cos^{'} = - \sin$ (The derivatives of sine and cosine are cosine and minus sine), $X^{'} (θ) = (l \cos θ, l \sin θ)$ and $X^{''} (θ) = (- l \sin θ, l \cos θ)$ ; hence $∥ X^{'} (θ) ∥_{2}^{2} = l^{2}$ , $X^{'} (θ) \cdot X^{''} (θ) = 0$ and $X^{'} (θ) \cdot F = - m g l \sin θ$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ). Since $∥ X^{'} (θ) ∥_{2} = l \neq 0$ , the derivative $X^{'} (θ)$ is injective and $X$ is an immersion (Submersions and immersions between Euclidean open sets).

[F2]

The reduced d'Alembert corollary (Reduced d'Alembert equation for an immersion: generalized forces and the quadratic correction): under the hypothesis on $R$ in clause 1, $X^{'} (θ)^{T} M X^{'} (θ) θ^{''} = X^{'} (θ)^{T} F - X^{'} (θ)^{T} M X^{''} (θ) θ^{' 2} .$

[F3]

The batch-3 energy-stability theorem (A strict local minimum of the potential gives a Lyapunov stable equilibrium of the mechanical system, by energy conservation, The mechanical first-order field, Lagrangian and energy of a configuration-dependent kinetic metric in a chart): for an open $Ω \subseteq R$ , a constant positive mass $M_{0}$ , a $C^{2}$ potential $U$ and a strict local minimum $θ_{0}$ of $U$ with $U^{'} (θ_{0}) = 0$ , the equilibrium $(θ_{0}, 0)$ of $θ^{'} = v$ , $v^{'} = - U^{'} (θ) / M_{0}$ is Lyapunov stable.

[F4]

The quadratic-mode theorem (Decoupling of the constant-coefficient quadratic system $M q^{''} + K q = 0$ into normal modes, with the zero-mode and energy identities): for a symmetric positive definite mass $M$ and a symmetric non-negative stiffness $K$ , the linear system $M ξ^{''} + K ξ = 0$ has modes of frequency $\sqrt{λ}$ with $λ$ the eigenvalues of $M^{- 1 / 2} K M^{- 1 / 2}$ .

[F5]

Trigonometry and calculus: $\cos^{2} θ + \sin^{2} θ = 1$ , so $∣ \cos θ ∣ \leq 1$ , with $\cos θ = 1$ only when $θ \in 2 π Z$ (Parity and the Pythagorean identity for sine and cosine, The zero sets of sine and cosine and the least positive common period 2 pi); sums and products of differentiable functions are differentiable with the usual rules (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ), and a differentiable function with vanishing derivative on an interval is constant (A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant, $C^{k}$ Euclidean maps and diffeomorphisms, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length, Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes).

Verification

technique · direct; contract the ambient equation with $X'$ and identify the resulting one-dimensional mechanical system, then invoke the stability and mode theorems

1.1F1F2algebra

Clause 1: by [F2] and [F1], $X^{' T} M X^{'} θ^{''} = l^{2} m θ^{''} = X^{' T} F - X^{' T} M X^{''} θ^{' 2} = - m g l \sin θ - 0 \cdot θ^{' 2},$ because $M = m I_{2}$ and $X^{'} \cdot X^{''} = 0$ . Dividing by $m l^{2} > 0$ gives $m l^{2} θ^{''} + m g l \sin θ = 0$ .

2.1F5step 1.1algebra

Clause 2: along a solution, by the product and chain rules [F5], $\frac{d}{d t} E (θ, θ^{'}) = m l^{2} θ^{'} θ^{''} + m g l \sin θ θ^{'} = θ^{'} (m l^{2} θ^{''} + m g l \sin θ) = 0,$ so by [F5] the energy is constant on $I$ .

2.2F5step 1.1algebra

Clause 3: the reduced potential $U_{r e d} (θ) = - m g l \cos θ$ is $C^{2}$ with $U_{r e d}^{'} (θ) = m g l \sin θ$ , so $U_{r e d}^{'} (0) = 0$ ; and for $0 < ∣ θ ∣ < π$ one has $θ \notin 2 π Z$ , hence $\cos θ < 1$ by [F5], so $U_{r e d} (θ) - U_{r e d} (0) = m g l (1 - \cos θ) > 0$ . Thus $0$ is a strict local minimum of $U_{r e d}$ on $Ω$ . With $M_{0} : = m l^{2} > 0$ , $U : = U_{r e d}$ and $θ_{0} : = 0$ , the mechanical vector field of [F3] is $(θ, v) \mapsto (v, (m l^{2})^{- 1} (- m g l \sin θ)) = (v, - (g / l) \sin θ)$ , which is the pendulum system; [F3] makes $(0, 0)$ Lyapunov stable.

2.3F4step 1.1algebra

Clause 4: the right-hand side of the pendulum equation is $- (g / l) \sin θ$ ; since $\sin 0 = 0$ and $\sin^{'} (0) = \cos 0 = 1$ (The derivatives of sine and cosine are cosine and minus sine), the linearisation at $θ = 0$ is $m l^{2} ξ^{''} + m g l ξ = 0$ . This is the constant-coefficient quadratic system of [F4] with $M = m l^{2}$ and $K = m g l$ ; the transformed stiffness is $λ = K / M = g / l > 0$ , so [F4] gives a mode of angular frequency $ω = \sqrt{λ} = \sqrt{g / l}$ .

3.1step 1.1step 2.1step 2.2step 2.3∎

Clauses 1-4 are proved.

Remarks

The example is a mathematical model of an ideal frictionless planar pendulum in a uniform downward force field; statements about a real pendulum require apparatus, friction and measurement assumptions that are not part of this item. The stability conclusion is Lyapunov stability, not asymptotic attraction, and it concerns the exact pendulum equation in the chart $Ω = (- π, π)$ .
The frequency $\sqrt{g / l}$ is the frequency of the linearised mode by [F4]; no claim is made that the nonlinear pendulum oscillates periodically with that frequency for finite amplitude.

The ideal planar pendulum: equation of motion, conserved energy, Lyapunov stability and the frequency $\sqrt{g / l}$

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources

The ideal planar pendulum: equation of motion, conserved energy, Lyapunov stability and the frequency g/lg/l​

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources

The ideal planar pendulum: equation of motion, conserved energy, Lyapunov stability and the frequency $\sqrt{g / l}$