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The inverted pendulum about its upward equilibrium: the negative modal stiffness, the exponential quadratic mode and the nonlinear instability

Physical scope

A worked instance of an unstable quadratic direction: the planar inverted pendulum of mass m and length l about its upward equilibrium, with chart data M = m l^2 and U(theta) = m g l cos(theta) measured from the upward vertical, negative Hessian -m g l, the exact exponential motions of the small-oscillation model and the exponential escaping family of the quadratic model, together with the nonlinear Lyapunov instability of the upward equilibrium obtained from the hyperbolic-saddle theorem. The example is a conditional statement about the declared mathematical model; g, m and l are model parameters and no measured period or escape time is asserted.

Example

The algebra below uses the fixed-unit numerical convention of The coupled quadratic oscillator model: mass metric, stiffness, generalized force, modal vocabulary and the small-oscillation model at a mechanical equilibrium, with coordinate units radians, time unit $1 s$ and energy unit $1 J$ . Physical model parameters are given their stated units; in matrix products, initial-vector identities and coefficient formulas their numerical coefficients in these fixed units are used. Modal amplitudes have common unit $a_{*} = (1 J)^{1 / 2} (1 s)$ ; the $j$ -th component of a mass-normalized modal vector has unit $s_{j} / a_{*}$ . Modal oscillatory and hyperbolic coefficients have unit $a_{*}$ , and zero-mode slopes have unit $a_{*} / (1 s)$ . Physical coordinates and energies are reconstructed with the coordinate units and $1 J$ , respectively. In particular a numerical single-mode datum $q (0) = u_{i}$ means a modal amplitude of one in this common amplitude unit.

Global-time, growth and boundedness claims below refer to the unique maximal extension of the autonomous free equation on $R$ ; local motions remain its restrictions. Values at $0$ in coefficient formulas are evaluated on that extension when the original interval omits $0$ .

In the formulas below, $\cosh u = (e^{u} + e^{- u}) / 2$ and $\sinh u = (e^{u} - e^{- u}) / 2$ , as in the hyperbolic family of The free coupled quadratic model: global existence, the complete modal classification, mode energies, intrinsic degenerate mode subspaces, boundedness and the unstable direction.

Let a planar pendulum consist of one particle label of mass $m > 0$ attached to a fixed pivot by a massless rod of length $l > 0$ , and let $θ$ be its angle from the upward vertical, read in radians (dimensionless under Dimensions and SI unit conventions of the non-relativistic classical mechanics framework); let $g > 0$ be the uniform gravitational acceleration of the model. In the chart with kinetic metric field $M (θ) = m l^{2}$ and potential $U (θ) = m g l \cos θ$ , the mechanical system of The mechanical first-order field, Lagrangian and energy of a configuration-dependent kinetic metric in a chart has energy $E = \frac{1}{2} m l^{2} θ^{' 2} + m g l \cos θ$ and equation $m l^{2} θ^{''} = m g l \sin θ$ , that is $θ^{''} = (g / l) \sin θ$ . Then:

$θ = 0$ is a mechanical equilibrium and a strict local maximum of $U$ on the chart, with $K : = D^{2} U (0) = - m g l < 0$ ; the small-oscillation model of clause (d) of The coupled quadratic oscillator model: mass metric, stiffness, generalized force, modal vocabulary and the small-oscillation model at a mechanical equilibrium is the one-dimensional quadratic model $m l^{2} ξ^{''} - m g l ξ = 0$ , whose modal stiffness is $λ = - g / l < 0$ ;
with $κ : = \sqrt{g / l} > 0$ , physical frequency unit $s^{- 1}$ , every motion of that quadratic model on an interval with nonempty interior is $ξ (t) = a \cosh (κ t) + d \sinh (κ t), a, d \in R,$ Here $a, d$ are coefficients of the angle coordinate, read in radians; $b$ below denotes its initial angular velocity in radians per second. The datum $ξ (0) = 0$ , $ξ^{'} (0) = b \neq 0$ gives $ξ (t) = (b / κ) \sinh (κ t)$ , unbounded on $[0, \infty)$ ;
the rest point of the quadratic model is not Lyapunov stable (Equilibria, Lyapunov stability and asymptotic stability of an autonomous first-order system): with $ε_{0} = 1$ and every $δ > 0$ , a datum of size $< δ$ leads to a motion reaching norm $1$ ;
the nonlinear upward equilibrium $(θ, θ^{'}) = (0, 0)$ of the pendulum chart system is not Lyapunov stable either: the field $F (θ, v) = (v, (g / l) \sin θ)$ is $C^{1}$ with $F (0, 0) = 0$ and $D F (0, 0) = (\begin{array}{cc} 0 & 1 \\ g / l & 0 \end{array})$ , whose eigenvalues $\pm \sqrt{g / l}$ are real, nonzero and include the positive value $\sqrt{g / l}$ , so the hyperbolic-saddle theorem applies.

Facts & Assumptions

Given: The parameters $m, l, g > 0$ , the chart metric field and potential $M (θ) = m l^{2}$ , $U (θ) = m g l \cos θ$ , the field $F (θ, v) = (v, (g / l) \sin θ)$ , and the numbers $κ = \sqrt{g / l}$ and $ε_{0} = 1$ .

[F1]

Mechanical chart model (The mechanical first-order field, Lagrangian and energy of a configuration-dependent kinetic metric in a chart): $θ = 0$ satisfies $\nabla U (0) = U^{'} (0) = 0$ and is a mechanical equilibrium; $U^{''} (0) = - m g l$ , so $K = D^{2} U (0)$ is the symmetric matrix $- m g l < 0$ ; the mechanical vector field is $G (θ, v) = (v, - U^{'} (θ) / m l^{2}) = (v, (g / l) \sin θ)$ .

[F2]

Small-oscillation model (The coupled quadratic oscillator model: mass metric, stiffness, generalized force, modal vocabulary and the small-oscillation model at a mechanical equilibrium clause (d)): the model at the equilibrium is the coupled quadratic model with mass metric $m l^{2}$ , stiffness $K = - m g l$ and zero generalized force, in the displacement coordinate $ξ$ ; its modal stiffness is $λ = K / M = - g / l$ .

[F3]

Free-mode theory (The free coupled quadratic model: global existence, the complete modal classification, mode energies, intrinsic degenerate mode subspaces, boundedness and the unstable direction clauses 2 and 6): for $λ < 0$ the free motions are $a \cosh (κ t) + d \sinh (κ t)$ with $κ = \sqrt{- λ}$ , and the rest point of the free model is not Lyapunov stable, for every requested initial size, with an escaping datum of smaller size and an explicit growing family. Stable decaying families are not asserted to escape.

[F4]

Hyperbolic instability (A hyperbolic equilibrium whose linearisation has an eigenvalue with positive real part is unstable): for a $C^{1}$ field $F$ on an open set with $F (0) = 0$ , if no complex eigenvalue of $D F (0)$ has real part $0$ and at least one eigenvalue has positive real part, then $0$ is not a Lyapunov stable equilibrium (Equilibria, Lyapunov stability and asymptotic stability of an autonomous first-order system).

[F5]

Hyperbolic solutions and growth (The degenerate and repulsive oscillator: solutions for k=0 and the explicit growing solution for k<0, The exponential function is strictly increasing): the scalar proposition gives $e^{κ t}$ as a solution of $ξ^{''} - κ^{2} ξ = 0$ with initial data $(1, κ)$ and proves $e^{κ t} \to + \infty$ ; for $t \geq 0$ the strictly increasing exponential also gives $0 < e^{- κ t} \leq 1$ . By the definitions in the Statement, $\sinh (κ t) = (e^{κ t} - e^{- κ t}) / 2$ is therefore unbounded as $t \to + \infty$ .

[F6]

Trigonometric derivative and continuity (The derivatives of sine and cosine are cosine and minus sine, A function differentiable at $c$ is continuous at $c$ , $C^{k}$ Euclidean maps and diffeomorphisms): $D F (θ, v) = (\begin{array}{cc} 0 & 1 \\ (g / l) \cos θ & 0 \end{array}),$ so $D F (0, 0) = (\begin{array}{cc} 0 & 1 \\ g / l & 0 \end{array})$ and its eigenvalues are $\pm \sqrt{g / l}$ . The sine derivative is cosine, and cosine is continuous because it is differentiable; hence $F$ is $C^{1}$ . The square root is positive since $g / l > 0$ (Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ ).

[F7]

Intermediate value theorem (Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on $[a, b]$ takes every value between $f (a)$ and $f (b)$ ): a continuous real-valued function on a closed interval takes every value between its endpoint values.

Verification

technique · direct; read the equilibrium data from the chart, solve the repulsive scalar equation, and apply the free-mode instability clause and the hyperbolic-saddle theorem

1.1F1F2algebra

Equilibrium and stiffness: $U^{'} (θ) = - m g l \sin θ$ vanishes at $θ = 0$ , and $U (θ) < U (0)$ for $0 < ∣ θ ∣ < π$ on the chart, so $θ = 0$ is a strict local maximum and $K = U^{''} (0) = - m g l < 0$ ; the small-oscillation model has data $M = m l^{2}$ , $K = - m g l$ , i.e. modal stiffness $λ = - g / l$ . This is clause 1.

2.1F3F5step 1.1

Exponential families: by [F3] and [F5] the motions of the quadratic model are exactly $ξ (t) = a \cosh (κ t) + d \sinh (κ t)$ with $κ = \sqrt{g / l}$ ; for the datum $ξ (0) = 0$ , $ξ^{'} (0) = b \neq 0$ the hyperbolic coefficients are $a = 0$ and $d = b / κ$ , so the motion is $(b / κ) \sinh (κ t)$ , which is unbounded on $[0, \infty)$ . This is clause 2.

2.2F4F6step 1.1

Nonlinear instability: the field $F (θ, v) = (v, (g / l) \sin θ)$ is $C^{1}$ on $R^{2}$ with $F (0, 0) = 0$ ; by [F6] its derivative at the origin has the real eigenvalues $\pm \sqrt{g / l}$ , none on the imaginary axis and one positive, so the hypotheses of [F4] hold and $(0, 0)$ is not a Lyapunov stable equilibrium of the pendulum chart system. This is clause 4.

3.1F3F5F7step 2.1algebra

Quadratic instability: by clause 6 of [F3] applied to the one-dimensional free model with negative modal stiffness, the rest point is not Lyapunov stable, with the displayed escaping family; explicitly, for every $δ > 0$ choose $0 < ∣ b ∣ < \min (δ, 1)$ . The motion $(b / κ) \sinh (κ t)$ has initial phase norm $∣ b ∣ < 1$ and its position norm is unbounded by [F5]. Thus its continuous phase norm starts below $1$ and exceeds $1$ at some finite time; [F7] gives a time $t > 0$ at which the phase norm is exactly $1 = ε_{0}$ . This is clause 3.

4.1step 1.1step 2.1step 3.1step 2.2∎

Clauses 1-4 are verified on the explicit chart data; clause 4 is the nonlinear conclusion supplied by the hyperbolic-saddle theorem, whereas clause 3 concerns the quadratic model alone. The parameters $m, l, g$ are model data, and no escape time, measured period or apparatus behaviour is asserted.

Remarks

Why the linear and nonlinear conclusions agree here. The linearization at the upward equilibrium is hyperbolic — its spectrum $\pm \sqrt{g / l}$ is disjoint from the imaginary axis — so the saddle theorem converts the positive eigenvalue into a genuine nonlinear instability. This is exactly the conditional route recorded in Scope of the coupled quadratic model: zero modes, degenerate eigenspaces, unstable quadratic directions and the limits of the approximation statements; for a nonhyperbolic critical point (in particular one with a zero mode) the linear calculation would not be sufficient.
The energy picture. The mechanical energy $E = \frac{1}{2} m l^{2} θ^{' 2} + m g l \cos θ$ has a saddle at the upward equilibrium: its potential part has a strict local maximum while its velocity part is positive quadratic; energy conservation therefore cannot trap nearby data in a small neighbourhood, in contrast with the strict-minimum case where the same energy argument gives stability (Small oscillations at a strict minimum: the linearized model, its normal modes and the Lyapunov stability of the nonlinear equilibrium clause 4).

The inverted pendulum about its upward equilibrium: the negative modal stiffness, the exponential quadratic mode and the nonlinear instability

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources