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Finite-time error control for the small-oscillation approximation: the quadratic remainder of the mechanical field and the variation-of-constants bound

Physical scope

Finite-time error control for the small-oscillation model of a mechanical system in a chart at a mechanical equilibrium: for a C^3 kinetic metric field and a C^3 potential with vanishing gradient at q_0, the mechanical field is C^2 near the equilibrium, its Taylor remainder at the linearization is quadratic with constants obtained from compactness, and the finite-time variation-of-constants estimate bounds the difference between the nonlinear motion and the linearized (normal-mode) motion by C delta^2 t exp((||A||+2L)t) on every horizon T with delta exp(LT)<r_0. The statement is a conditional finite-time estimate for the declared chart model under explicit regularity and smallness hypotheses; it asserts no all-time validity of the linear approximation, no stability conclusion, no large-data statement and no empirical claim, and the constants depend on the model data only through its behaviour on the stated ball.

Statement

Fix positive coordinate units $s_{i}$ , a time unit $t_{*}$ and an energy unit $e_{*}$ as in The coupled quadratic oscillator model: mass metric, stiffness, generalized force, modal vocabulary and the small-oscillation model at a mechanical equilibrium. Throughout this theorem $q_{i} = Q_{i} / s_{i}$ , $t = t_{p h y s} / t_{*}$ and $v_{i} = (t_{*} / s_{i}) {\dot{Q}}_{i}$ are dimensionless numerical chart and phase coordinates. $M$ and $U$ are the corresponding normalized metric and potential specified in that definition. In particular $(δ q, δ v)$ is a vector in one Euclidean numerical space; $r_{0}, δ, C, L, A, T$ below refer to that space and numerical time. All constants are calculated after this normalization. If the numerical error bound is $b (t)$ , its physical position and velocity components are bounded respectively by $s_{i} b (t)$ and $(s_{i} / t_{*}) b (t)$ at physical time $t_{*} t$ . Fixed positive linear coordinate/time changes preserve the stated regularity and mechanical equation, so no original chart model is excluded.

Let $n \geq 1$ , let $Ω \subseteq R^{n}$ be open, let $M : Ω \to M_{n} (R)$ be a field of class $C^{3}$ with $M (q)$ symmetric positive definite for every $q \in Ω$ , and let $U : Ω \to R$ be of class $C^{3}$ with $\nabla U (q_{0}) = 0$ at a point $q_{0} \in Ω$ . Let $G$ be the mechanical vector field of The mechanical first-order field, Lagrangian and energy of a configuration-dependent kinetic metric in a chart on $Ω \times R^{n}$ , let $r_{0} > 0$ satisfy $\overline{B} (q_{0}, r_{0}) \subseteq Ω$ and define $F : {ζ = (δ q, δ v) : q_{0} + δ q \in Ω} \to R^{2 n}$ by $F (δ q, δ v) : = G (q_{0} + δ q, δ v)$ . Put $A : = D F (0) = D G (q_{0}, 0)$ . Then:

$F$ is of class $C^{2}$ , $F (0) = 0$ , $F$ is locally Lipschitz in the state variable, and $D G (q_{0}, 0) (δ q, δ v) = (δ v, - M (q_{0})^{- 1} D^{2} U (q_{0}) δ q);$ moreover there are constants $C, L \geq 0$ with $∥ F (ζ) - A ζ ∥_{2} \leq C ∥ ζ ∥_{2}^{2}, ∥ F (ζ) ∥_{2} \leq L ∥ ζ ∥_{2} (∥ ζ ∥_{2} \leq r_{0});$ one may take $L = \sup_{∥ η ∥_{2} \leq r_{0}} ∥ D F (η) ∥$ (The operator norm as the least bound and as the unit-sphere or unit-ball supremum) and $C = Λ / 2$ for any Lipschitz constant $Λ$ of $D F$ on the closed ball $\overline{B} (0, r_{0}) \subseteq R^{2 n}$ (Open ball, closed ball and sphere in a metric space).
For every datum $z_{0} = (q_{0} + δ q_{0}, v_{0})$ with $δ : = ∥ (δ q_{0}, v_{0}) ∥_{2} > 0$ and every $T > 0$ with $δ e^{L T} < r_{0}$ , the maximal solution $z = (q, v)$ of $z^{'} = G (z)$ with $z (0) = z_{0}$ is defined on $[0, T]$ (Extensions, maximal solutions, maximal intervals, and global solutions) and satisfies, for every $t \in [0, T]$ , $∥ z (t) - y (t) ∥_{2} \leq C δ^{2} t e^{(∥ A ∥ + 2 L) t},$ where $y (t) = (q_{0} + ξ (t), η (t))$ and $\tilde{y} = (ξ, η) : R \to R^{2 n}$ solves ${\tilde{y}}^{'} = A \tilde{y}$ , $\tilde{y} (0) = (δ q_{0}, v_{0})$ . Thus $y$ is the linear approximation translated back to the original chart.
Writing $\tilde{y} = (ξ, η)$ for the displacement linear solution of clause 2, the first component $ξ$ is exactly the unique motion of 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 with datum $(δ q_{0}, v_{0})$ : $ξ$ is $C^{2}$ on $R$ with $M (q_{0}) ξ^{''} + D^{2} U (q_{0}) ξ = 0$ , $ξ (0) = δ q_{0}$ and $ξ^{'} (0) = v_{0}$ .

No all-time statement, no statement for data violating the smallness condition $δ e^{L T} < r_{0}$ , and no value of any measured quantity is asserted.

Facts & Assumptions

Given: The $C^{3}$ metric field $M$ , the $C^{3}$ potential $U$ with $\nabla U (q_{0}) = 0$ , the mechanical field $G$ , the radius $r_{0}$ , the translated field $F$ , the matrix $A = D F (0)$ and a datum $z_{0} = (q_{0} + δ q_{0}, v_{0})$ with $δ > 0$ .

[F1]

Regularity of the mechanical field (The mechanical first-order field, Lagrangian and energy of a configuration-dependent kinetic metric in a chart): $G (q, v) = (v, M (q)^{- 1} B (q, v))$ with $B (q, v) = \frac{1}{2} \nabla_{q} (v^{T} M (q) v) - (D_{q} M (q) [v]) v - \nabla U (q)$ ; the maps $M$ , $\partial_{k} M$ , $\nabla_{q} (v^{T} M (q) v)$ , $(D_{q} M (q) [v]) v$ , $\nabla U$ and $q \mapsto M (q)^{- 1}$ are respectively $C^{3}$ , $C^{2}$ , $C^{2}$ , $C^{2}$ , $C^{2}$ and $C^{3}$ under the stated hypotheses, because $M^{- 1} = \det (M)^{- 1} a d j (M)$ has entries polynomial in the entries of $M$ divided by the nonvanishing continuous function $\det M$ (If $\det (A)$ is a unit, then $A^{- 1} = \det (A)^{- 1} adj (A)$ , If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative, $C^{k}$ Euclidean maps and diffeomorphisms); hence $G$ and $F$ are $C^{2}$ , so $F$ is locally Lipschitz in the state variable (Local Lipschitz continuity in the state variable, locally uniform in time and parameters).

[F2]

Linearization and equality of derivatives (Linearisation of the mechanical system at an equilibrium: M(q_0) xi'' + D²U(q_0) xi = 0, and the small-oscillation quadratic model clauses 1-2): $G$ is totally differentiable at $(q_{0}, 0)$ and $D G (q_{0}, 0) (δ q, δ v) = (δ v, - M (q_{0})^{- 1} D^{2} U (q_{0}) δ q)$ ; $D^{2} U (q_{0})$ is symmetric (The Hessian of a $C^{2}$ scalar field is symmetric, The Hessian matrix and critical points of a scalar field). Since $F$ is the translation of $G$ to the equilibrium, $A = D F (0) = D G (q_{0}, 0)$ and $F (0) = G (q_{0}, 0) = 0$ because $q_{0}$ is an equilibrium.

[F3]

Compactness and mean value (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, The mean value inequality: if $f : [a, b] \to R^{m}$ is continuous and differentiable on $(a, b)$ with $∥ f^{'} ∥_{2} \leq M$ , then $∥ f (b) - f (a) ∥_{2} \leq M (b - a)$ , Open ball, closed ball and sphere in a metric space): the closed ball $\overline{B} (0, r_{0})$ is a compact convex subset of the open domain of $F$ ; a continuous real function on it attains its maximum, and a continuous map with $∥ D F ∥ \leq L$ on the ball satisfies $∥ F (η) - F (η^{'}) ∥_{2} \leq L ∥ η - η^{'} ∥_{2}$ there.

[F4]

Fundamental theorem along rays (If $f : [a, b] \to R^{m}$ is differentiable with integrable $f^{'}$ then $\int_{a}^{b} f^{'} = f (b) - f (a)$ ; and a bounded derivative makes $f$ Lipschitz, For $a \leq b$ and $f : [a, b] \to R^{m}$ integrable when $a < b$ , $∥ \int_{a}^{b} f ∥_{2} \leq \int_{a}^{b} ∥ f ∥_{2}$ ; for $a < b$ , $∥ f ∥_{2}$ is integrable, The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral): for $∥ ζ ∥_{2} \leq r_{0}$ the curve $t \mapsto F (t ζ)$ on $[0, 1]$ is $C^{1}$ with derivative $D F (t ζ) ζ$ , and $F (ζ) - F (0) = \int_{0}^{1} D F (t ζ) ζ d t, ∥ \int_{0}^{1} ϕ (t) d t ∥_{2} \leq \int_{0}^{1} ∥ ϕ (t) ∥_{2} d t .$

[F5]

Finite-time linearization bound (Finite-time linearization error bound: quadratic remainder, exponential growth bound and the variation-of-constants estimate with state dimension $2 n$ ): for the open domain of $F$ , the constants $r_{0}, C, L$ and the matrix $A$ of the statement, the maximal solution $ζ$ of $ζ^{'} = F (ζ)$ , $ζ (0) = ζ_{0} : = (δ q_{0}, v_{0})$ is defined on $[0, T]$ whenever $δ e^{L T} < r_{0}$ , it satisfies $∥ ζ (t) ∥_{2} \leq δ e^{L t}$ there, and $∥ ζ (t) - \tilde{y} (t) ∥_{2} \leq C δ^{2} \int_{0}^{t} e^{∥ A ∥ (t - s)} e^{2 L s} d s \leq C δ^{2} t e^{(∥ A ∥ + 2 L) t}$ for $t \in [0, T]$ , where $y$ solves ${\tilde{y}}^{'} = A \tilde{y}$ , $\tilde{y} (0) = ζ_{0}$ ; every initial value problem has a unique maximal solution (Every Picard–Lindelöf initial value problem has one maximal solution on an open interval, Extensions, maximal solutions, maximal intervals, and global solutions).

[F6]

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)): the small-oscillation model at $q_{0}$ is the coupled quadratic model with mass metric matrix $M (q_{0})$ and stiffness $D^{2} U (q_{0})$ , whose phase system is $y^{'} = A y$ with the displayed block matrix $A = (\begin{array}{cc} 0 & I_{n} \\ - M (q_{0})^{- 1} D^{2} U (q_{0}) & 0 \end{array})$ ; a $C^{2}$ curve $ξ$ is a motion of it if and only if $M (q_{0}) ξ^{''} + D^{2} U (q_{0}) ξ = 0$ , with the datum read off $(ξ (0), ξ^{'} (0))$ .

Proof

Proof technique: direct; prove the quadratic-remainder bounds for the translated mechanical field by compactness and the fundamental theorem along rays, then apply the finite-time linearization theorem and identify the linear solution with the small-oscillation motion.

1.1F1F2

Clauses 1 and the derivative identity: by [F1] the translated field $F$ is $C^{2}$ on its open domain and locally Lipschitz; $F (0) = 0$ and $A = D F (0) = D G (q_{0}, 0)$ with the displayed block form are [F2].

1.2F1F3F4algebra

Quantitative remainder and growth bounds: since $F$ is $C^{2}$ , $D F$ is $C^{1}$ , hence continuous on the compact ball $\overline{B} (0, r_{0})$ ; thus $L : = \sup_{∥ η ∥_{2} \leq r_{0}} ∥ D F (η) ∥$ is finite, and $D F$ is Lipschitz on the convex ball with some finite constant $Λ$ (each entry of $D F$ has continuous gradient and is Lipschitz on the ball by [F3] with the supremum of that gradient; a finite sum of such bounds bounds $∥ D F (η) - D F (η^{'}) ∥$ ). For $∥ ζ ∥_{2} \leq r_{0}$ , [F4] applied to the $C^{1}$ curve $t \mapsto F (t ζ) - t A ζ$ , whose derivative is $D F (t ζ) ζ - A ζ = (D F (t ζ) - D F (0)) ζ$ , gives $F (ζ) - A ζ = \int_{0}^{1} (D F (t ζ) - D F (0)) ζ d t,$ hence $∥ F (ζ) - A ζ ∥_{2} \leq \int_{0}^{1} Λ t ∥ ζ ∥_{2}^{2} d t = \frac{Λ}{2} ∥ ζ ∥_{2}^{2}$ ; and $F (ζ) = F (ζ) - F (0)$ with [F3] gives $∥ F (ζ) ∥_{2} \leq L ∥ ζ ∥_{2}$ . This proves the bounds of clause 1 with $C = Λ / 2$ .

2.1F5step 1.2given

The finite-time bound: apply [F5] to $F$ , the constants $r_{0}, C, L$ , the matrix $A$ and the displaced datum $ζ_{0} = (δ q_{0}, v_{0})$ of norm $δ$ . The maximal solution $ζ$ of $ζ^{'} = F (ζ)$ is defined on $[0, T]$ and obeys the error bound against $\tilde{y}$ . Translating back gives $z (t) = (q_{0} + ζ_{q} (t), ζ_{v} (t))$ , the physical phase solution through $z_{0}$ ; $y (t) = (q_{0} + ξ (t), η (t))$ has exactly the same translation, so $∥ z (t) - y (t) ∥_{2} = ∥ ζ (t) - \tilde{y} (t) ∥_{2}$ ; this is clause 2.

3.1F2F6step 2.1algebra

Identification of the linear solution: by [F2] the matrix $A$ is the block matrix $(δ q, δ v) \mapsto (δ v, - M (q_{0})^{- 1} D^{2} U (q_{0}) δ q)$ ; the displacement linear solution $\tilde{y} = (ξ, η)$ of ${\tilde{y}}^{'} = A \tilde{y}$ , $\tilde{y} (0) = (δ q_{0}, v_{0})$ therefore satisfies $η = ξ^{'}$ , hence $ξ^{''} = - M (q_{0})^{- 1} D^{2} U (q_{0}) ξ$ , equivalently $M (q_{0}) ξ^{''} + D^{2} U (q_{0}) ξ = 0$ , with $ξ (0) = δ q_{0}$ , $ξ^{'} (0) = v_{0}$ ; by [F6] this is exactly the motion of the small-oscillation model with that datum, and it is unique by the model's phase-system uniqueness. This is clause 3.

4.1step 1.1step 1.2step 2.1step 3.1∎

Clauses 1-3 are proved. The estimate is a finite-time statement under the explicit smallness hypothesis $δ e^{L T} < r_{0}$ ; the constants $C, L$ exist by compactness and are not asserted to be optimal, and no long-time, stability or empirical conclusion is drawn.

Remarks

Where the exponential factor comes from. The linearized flow itself may grow exponentially at rate $∥ A ∥$ (an unstable equilibrium included), and the nonlinear remainder is controlled by the Lipschitz growth $e^{2 L t}$ of the solution; the product is the displayed rate $∥ A ∥ + 2 L$ . On a fixed horizon this rate is a constant, and the bound is of order $δ^{2}$ in the initial displacement.
Constant versus asymptotic statements. The theorem fixes a horizon $T$ and shrinks the datum; it does not assert that the nonlinear motion stays close to the linear one for all time, nor that a small initial displacement produces a small displacement later. Such all-time statements are separate problems (for the stable case they concern nonlinear stability, for the unstable case nonlinear instability) and are not obtained from this estimate.

Finite-time error control for the small-oscillation approximation: the quadratic remainder of the mechanical field and the variation-of-constants bound

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources