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Small oscillations at a strict minimum: the linearized model, its normal modes and the Lyapunov stability of the nonlinear equilibrium

Physical scope

A mechanical system in a chart at a strict local minimum of its potential: the assigned small-oscillation quadratic model with mass metric M(q_0) and stiffness the Hessian D^2 U(q_0) (symmetric and non-negative), the identification of that model with the derivative linearization of the mechanical field, the complete normal-mode classification of the linearized motions with frequencies sqrt(lambda_i) and no exponential directions, and the Lyapunov stability of the nonlinear equilibrium by exact energy conservation. The stability conclusion is a statement about the declared chart model; it asserts no asymptotic attraction, no measured frequency, no particular physical system and no empirical claim, and the accuracy of the linear model as an approximation of the nonlinear motion is a separate finite-time statement on this page.

Statement

Use the fixed coordinate, time and energy units and numerical matrix convention of The coupled quadratic oscillator model: mass metric, stiffness, generalized force, modal vocabulary and the small-oscillation model at a mechanical equilibrium. Every displayed matrix identity and phase norm is in those numerical coordinates; physical positions, velocities, modal amplitudes, frequencies and energies are recovered by the stated fixed scale factors. This does not change the set of physical motions.

Adopt the mechanical system in a chart of The mechanical first-order field, Lagrangian and energy of a configuration-dependent kinetic metric in a chart with kinetic metric field $M : Ω \to M_{n} (R)$ of class $C^{2}$ , symmetric positive definite at every point of the open set $Ω \subseteq R^{n}$ , and potential $U : Ω \to R$ of class $C^{2}$ ; write $G$ for its mechanical vector field on $Ω \times R^{n}$ . Let $q_{0} \in Ω$ be a strict local minimum of $U$ : $\nabla U (q_{0}) = 0$ and there is $r_{0} > 0$ with $\overline{B} (q_{0}, r_{0}) \subseteq Ω$ and $U (q) > U (q_{0})$ for $0 < ∥ q - q_{0} ∥_{2} < r_{0}$ . Let $M_{0} : = M (q_{0})$ and $K : = D^{2} U (q_{0})$ . Then:

$K$ is symmetric and non-negative, and the small-oscillation model at $q_{0}$ 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 — the coupled quadratic model with mass metric matrix $M_{0}$ , stiffness $K$ and zero generalized force, in the displacement coordinates $ξ = q - q_{0}$ — is an oscillatory model in the sense of clause (e) of that definition;
the linearization of $G$ at $(q_{0}, 0)$ is $D G (q_{0}, 0) (δ q, δ v) = (δ v, - M_{0}^{- 1} K δ q) (δ q, δ v \in R^{n}),$ and for a curve $ξ : J \to R^{n}$ of class $C^{2}$ on an interval $J$ with nonempty interior the following are equivalent: $ξ$ is a motion of the small-oscillation model, $M_{0} ξ^{''} + K ξ = 0$ on $J$ , and $t \mapsto (ξ (t), ξ^{'} (t))$ solves the linear system $y^{'} = D G (q_{0}, 0) y$ on $J$ ;
the motions of the small-oscillation model are exactly the modal superpositions of clause 2 of The free coupled quadratic model: global existence, the complete modal classification, mode energies, intrinsic degenerate mode subspaces, boundedness and the unstable direction with every modal stiffness $λ_{i} \geq 0$ : there are no exponential pure- $λ$ families, every oscillatory frequency is $ω_{i} = \sqrt{λ_{i}}$ , and the pure- $λ$ motions of a degenerate stiffness are the solutions of $q^{''} + λ q = 0$ with values in the intrinsic subspace $N_{λ} = M_{0}^{- 1 / 2} {u : M_{0}^{- 1 / 2} K M_{0}^{- 1 / 2} u = λ u}$ ;
$(q_{0}, 0)$ is a Lyapunov stable equilibrium of the nonlinear mechanical system: for every $ε > 0$ with $\overline{B} ((q_{0}, 0), ε) \subseteq Ω \times R^{n}$ there is $δ > 0$ such that every maximal solution of $z^{'} = G (z)$ with $∥ z (0) - (q_{0}, 0) ∥_{2} < δ$ exists for all $t \geq 0$ and stays within distance $ε$ of $(q_{0}, 0)$ .

No asymptotic attraction, no all-time validity of the linear model as an approximation of the nonlinear motion, and no measured frequency is asserted.

Facts & Assumptions

Given: The metric field $M$ , the potential $U$ , the strict local minimum $q_{0}$ , the mechanical field $G$ , and the data $M_{0} = M (q_{0})$ , $K = D^{2} U (q_{0})$ of the statement.

[F1]

Hessian at a strict minimum (The Hessian matrix and critical points of a scalar field, The Hessian of a $C^{2}$ scalar field is symmetric, A C² local minimum has vanishing gradient and positive semidefinite Hessian): $K$ is the symmetric Hessian matrix of $U$ at the critical point $q_{0}$ and satisfies $w^{T} K w \geq 0$ for all $w \in R^{n}$ .

[F2]

Derivative linearization (Linearisation of the mechanical system at an equilibrium: M(q_0) xi'' + D²U(q_0) xi = 0, and the small-oscillation quadratic model clauses 1-3): $G$ is totally differentiable at $(q_{0}, 0)$ with the displayed derivative $D G (q_{0}, 0)$ ; the linearised equation at the equilibrium is $M_{0} ξ^{''} + K ξ = 0$ ; and the linearized system is exactly the constant-coefficient quadratic system with symmetric positive definite mass matrix $M_{0}$ and symmetric non-negative stiffness $K$ of Decoupling of the constant-coefficient quadratic system $M q^{''} + K q = 0$ into normal modes, with the zero-mode and energy identities.

[F3]

Model dictionary (The coupled quadratic oscillator model: mass metric, stiffness, generalized force, modal vocabulary and the small-oscillation model at a mechanical equilibrium clauses (b), (d), (e)): the small-oscillation model at $q_{0}$ is the coupled quadratic model with mass metric $M_{0}$ , stiffness $K$ and $J \equiv 0$ in the coordinates $ξ$ ; its motions are the $C^{2}$ curves with $M_{0} ξ^{''} + K ξ = 0$ , and its phase system is $y^{'} = A y$ with $A = (\begin{array}{cc} 0 & I_{n} \\ - M_{0}^{- 1} K & 0 \end{array})$ ; it is oscillatory when $K$ is non-negative, and its modal vocabulary is that of the definition's clause (e) with $M = M_{0}$ .

[F4]

Free modal classification (The free coupled quadratic model: global existence, the complete modal classification, mode energies, intrinsic degenerate mode subspaces, boundedness and the unstable direction clauses 1-4): for the free model with symmetric positive definite $M_{0}$ and symmetric $K$ the motions are classified by the modal families of its clause 2; the pure- $λ$ motions form the intrinsic family described in its clause 4.

[F5]

Energy stability (A strict local minimum of the potential gives a Lyapunov stable equilibrium of the mechanical system, by energy conservation): for the $C^{2}$ positive definite metric field $M$ , the $C^{2}$ potential $U$ and the strict local minimum $q_{0}$ , the rest point $(q_{0}, 0)$ of the mechanical field is Lyapunov stable in the sense of Equilibria, Lyapunov stability and asymptotic stability of an autonomous first-order system.

Proof

Proof technique: direct; assemble the assigned model from the definition, identify the derivative linearization with the model's phase system, read the modal classification at a strict minimum, and invoke the energy-stability theorem for the nonlinear statement.

1.1F1F3given

Clause 1: by [F1] the stiffness $K = D^{2} U (q_{0})$ is symmetric and non-negative; by [F3] the small-oscillation model at $q_{0}$ is the coupled quadratic model with data $(M_{0}, K, 0)$ , and a coupled quadratic model whose stiffness is non-negative is oscillatory in the sense of the definition's clause (e).

1.2F2F3givenalgebra

Clause 2: by [F2] the derivative of $G$ at $(q_{0}, 0)$ is $(δ q, δ v) \mapsto (δ v, - M_{0}^{- 1} K δ q)$ ; by [F3] the phase system of the small-oscillation model is $y^{'} = A y$ for the same block matrix $A$ , and the model equation $M_{0} ξ^{''} + K ξ = 0$ is equivalent to $(ξ, ξ^{'})^{'} = A (ξ, ξ^{'})$ by the product rule and invertibility of $M_{0}$ . This proves clause 2.

1.3F5given

Clause 4: the hypotheses of [F5] are the $C^{2}$ positive definite metric field, the $C^{2}$ potential and the strict local minimum of the statement; its conclusion is exactly the Lyapunov stability assertion of clause 4 for the field $G$ and the rest point $(q_{0}, 0)$ , whose equilibrium property is recorded in The mechanical first-order field, Lagrangian and energy of a configuration-dependent kinetic metric in a chart.

2.1F1F3F4step 1.1

Clause 3: apply the classification of [F4] to the data $(M_{0}, K, 0)$ of the small-oscillation model; since $K$ is non-negative by [F1], every modal stiffness satisfies $λ_{i} \geq 0$ , so no exponential family occurs and the oscillatory and zero families of the free-mode theorem exhaust the motions, with frequencies $ω_{i} = \sqrt{λ_{i}}$ and the intrinsic pure- $λ$ families described there.

3.1step 1.1step 1.2step 2.1step 1.3∎

Clauses 1-4 are proved. Clause 4 is a nonlinear conclusion obtained from exact energy conservation, not from the linearization; clauses 2 and 3 describe the linearized model only, and no accuracy of that model as an approximation of the nonlinear motion, and no attraction, is claimed here.

Remarks

Stability is not asymptotic. The proof of clause 4 uses exact conservation of the mechanical energy, so the equilibrium is stable but not attracting: there is no mechanism in an autonomous conservative model that forces nearby motions to converge to the rest point, and none is asserted.
The linear model and the nonlinear motion. Clause 2 identifies the derivative linearization with the small-oscillation model; this makes the normal-mode calculation a statement about the first-order Taylor model at the equilibrium. The difference between that model and the nonlinear motion is quantified, at finite times and on explicitly small data, by Finite-time error control for the small-oscillation approximation: the quadratic remainder of the mechanical field and the variation-of-constants bound; no all-time claim follows from either theorem.

Small oscillations at a strict minimum: the linearized model, its normal modes and the Lyapunov stability of the nonlinear equilibrium

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources