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The free coupled quadratic model: global existence, the complete modal classification, mode energies, intrinsic degenerate mode subspaces, boundedness and the unstable direction

Physical scope

The free coupled quadratic model M q'' + K q = 0 of The coupled quadratic oscillator model: mass metric, stiffness, generalized force, modal vocabulary and the small-oscillation model at a mechanical equilibrium with symmetric positive definite mass metric M and symmetric stiffness K: existence and uniqueness of motions for every datum on the whole real line, the complete classification of motions into oscillatory, affine and exponential modal families with mode frequencies sqrt(lambda_i), the conservation of the mechanical energy with the modal energy decomposition, the intrinsic (basis-free) description of degenerate pure-lambda families in the subspace ker(K - lambda M), boundedness of every motion when K is positive definite, unbounded growth of the affine zero-mode motions and of the exponential directions, and the Lyapunov instability of the rest point of the free model when K has a negative modal stiffness. The result is a conditional statement about the declared model class; it contains no claim that any physical system is described by it, no measured stiffness, mass or frequency, and no empirical qualification beyond the exact idealization of the model.

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.

For real $u$ , define the hyperbolic functions by $\cosh u : = (e^{u} + e^{- u}) / 2$ and $\sinh u : = (e^{u} - e^{- u}) / 2$ .

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 $0$ .

Adopt the free coupled quadratic model of The coupled quadratic oscillator model: mass metric, stiffness, generalized force, modal vocabulary and the small-oscillation model at a mechanical equilibrium with data $(M, K, J)$ in which $J \equiv 0$ , $M$ is symmetric positive definite and $K$ is symmetric, with modal data $M^{1 / 2}$ , $M^{- 1 / 2}$ , $O$ , $Λ = diag (λ_{1}, \dots, λ_{n})$ , modal coordinates $z = O^{T} M^{1 / 2} q$ and mode frequencies $ω_{i} = \sqrt{λ_{i}}$ for $λ_{i} > 0$ . Write $κ_{i} : = \sqrt{- λ_{i}}$ for $λ_{i} < 0$ . Then:

Existence and uniqueness on the whole line. For every $t_{0} \in R$ and every datum $(q_{0}, v_{0}) \in R^{2 n}$ there is exactly one motion $q : R \to R^{n}$ with $q (t_{0}) = q_{0}$ and $q^{'} (t_{0}) = v_{0}$ ; it is the unique maximal motion through that datum.
Complete modal classification. A curve $q : J \to R^{n}$ of class $C^{2}$ on an interval $J \subseteq R$ with nonempty interior is a motion of the free model if and only if its modal coordinates satisfy

$z_{i} (t) = a_{i} \cos (ω_{i} t) + b_{i} \sin (ω_{i} t) (λ_{i} > 0), z_{i} (t) = a_{i} + b_{i} t (λ_{i} = 0), z_{i} (t) = a_{i} \cosh (κ_{i} t) + b_{i} \sinh (κ_{i} t) (λ_{i} < 0)$

for real numbers $a_{i}, b_{i}$ ; the coefficients are unique and are determined by the datum of the unique maximal extension through $a_{i} = z_{i} (0)$ , $b_{i} = z_{i}^{'} (0) / ω_{i}$ for $λ_{i} > 0$ , $b_{i} = z_{i}^{'} (0)$ for $λ_{i} = 0$ and $b_{i} = z_{i}^{'} (0) / κ_{i}$ for $λ_{i} < 0$ . 3. Energy and mode energies. Along every motion the mechanical energy is constant, and with $E_{i} : = \frac{1}{2} (z_{i}^{' 2} + λ_{i} z_{i}^{2})$ one has $E (q, q^{'}) = \sum_{i = 1}^{n} E_{i}$ and every $E_{i}$ is constant; for the families of clause 2,

$E_{i} = \frac{1}{2} ω_{i}^{2} (a_{i}^{2} + b_{i}^{2}) (λ_{i} > 0), E_{i} = \frac{1}{2} b_{i}^{2} (λ_{i} = 0), E_{i} = \frac{1}{2} κ_{i}^{2} (b_{i}^{2} - a_{i}^{2}) (λ_{i} < 0) .$

Degenerate families are intrinsic. Let $λ$ be a modal stiffness, let $E_{λ}^{S} : = {u \in R^{n} : M^{- 1 / 2} K M^{- 1 / 2} u = λ u}$ and let $N_{λ} : = M^{- 1 / 2} E_{λ}^{S}$ , a subspace of dimension equal to the multiplicity of $λ$ . A motion $q$ satisfies $z_{i} \equiv 0$ for every index $i$ with $λ_{i} \neq λ$ if and only if $q$ is a $C^{2}$ curve with $q (t) \in N_{λ}$ for all $t$ in its interval and $q^{''} + λ q = 0$ . Consequently the family of such pure- $λ$ motions is independent of the choice of diagonalization $O$ : for $λ > 0$ it consists exactly of $q (t) = \cos (ω t) q_{0} + ω^{- 1} \sin (ω t) q_{1}$ with $q_{0}, q_{1} \in N_{λ}$ and $ω = \sqrt{λ}$ , for $λ = 0$ of $q (t) = q_{0} + t q_{1}$ with $q_{0}, q_{1} \in N_{λ}$ , and for $λ < 0$ of $q (t) = \cosh (κ t) q_{0} + κ^{- 1} \sinh (κ t) q_{1}$ with $κ = \sqrt{- λ}$ . These formulas on a local interval are restrictions of their unique maximal extensions; $q_{0}, q_{1}$ are the position and velocity of that extension at time $0$ , even if the local interval omits $0$ . The individual modal vectors $u_{i}$ are not intrinsic when the multiplicity exceeds one.
Boundedness. If $K$ is positive definite then every motion is bounded on $R$ . If $K$ is non-negative and some $λ_{i} = 0$ , then a motion with $z_{i}^{'} \equiv b_{i} \neq 0$ on a zero mode is unbounded on $[0, \infty)$ : the zero mode is affine, not oscillatory.
Unstable direction. If $K$ is not non-negative, then $(0, 0)$ is not a Lyapunov stable rest point of the free model (Equilibria, Lyapunov stability and asymptotic stability of an autonomous first-order system): there is $ε_{0} > 0$ such that for every $δ > 0$ there are data with $∥ (q_{0}, v_{0}) ∥_{2} < δ$ whose motion reaches $∥ (q (t), q^{'} (t)) ∥_{2} = ε_{0}$ at some time $t > 0$ .

The statement is read for the declared model data; no value of any parameter, no physical realization and no measured frequency is asserted.

Facts & Assumptions

Given: The data $(M, K, 0)$ of the statement, the modal data $M^{1 / 2}, M^{- 1 / 2}, O, Λ$ , the modal coordinates $z = O^{T} M^{1 / 2} q$ , and a curve $q : J \to R^{n}$ as in clause 2.

[F1]

Sign-unrestricted modal construction of The coupled quadratic oscillator model: mass metric, stiffness, generalized force, modal vocabulary and the small-oscillation model at a mechanical equilibrium clause (e): $M^{1 / 2}, M^{- 1 / 2}$ exist by Positive definite symmetric matrices have positive definite square roots, and the congruence M^{-1/2} K M^{-1/2} preserves symmetry and positivity clauses 1-3, and the symmetric matrix $S = M^{- 1 / 2} K M^{- 1 / 2}$ admits $S = O Λ O^{T}$ by Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis, without assuming $K \geq 0$ . With $q = M^{- 1 / 2} O z$ , direct differentiation and multiplication give $M q^{''} + K q = M^{1 / 2} O (z^{''} + Λ z)$ , $q^{' T} M q^{'} = ∣ z^{'} ∣_{2}^{2}$ and $q^{T} K q = \sum_{i} λ_{i} z_{i}^{2}$ . Thus the modal equivalence and energy identities hold for every symmetric $K$ on every interval; the non-negative- $K$ theorem Decoupling of the constant-coefficient quadratic system $M q^{''} + K q = 0$ into normal modes, with the zero-mode and energy identities is not invoked for indefinite $K$ .

[F2]

Scalar classification (The homogeneous damped oscillator: solution classification by the discriminant and uniqueness for initial data, The degenerate and repulsive oscillator: solutions for k=0 and the explicit growing solution for k<0, The damped driven oscillator model: second-order equation, phase-plane system and characteristic data): the solutions of $z_{i}^{''} + λ_{i} z_{i} = 0$ on an interval with nonempty interior are exactly $a \cos (ω_{i} t) + b \sin (ω_{i} t)$ when $λ_{i} > 0$ (the case $m = 1$ , $c = 0$ , $k = λ_{i}$ of the classification, with $ω_{i} = \sqrt{λ_{i}}$ ), exactly $a + b t$ when $λ_{i} = 0$ , and, writing $κ_{i} = \sqrt{- λ_{i}}$ , exactly $a \cosh (κ_{i} t) + b \sinh (κ_{i} t)$ when $λ_{i} < 0$ (the case $k < 0$ , $c = 0$ of the repulsive classification, with $e^{\pm κ_{i} t} = \cosh (κ_{i} t) \pm \sinh (κ_{i} t)$ ); in each case the pair $(a, b)$ ranges over $R^{2}$ bijectively, with $a = z_{i} (0)$ evaluated on the unique global extension and $b = z_{i}^{'} (0) / ω_{i}$ , $b = z_{i}^{'} (0)$ , $b = z_{i}^{'} (0) / κ_{i}$ respectively.

[F3]

Global linear existence and uniqueness (Variation of constants for $z^{'} = A z + b (t)$ via the matrix exponential): for the constant matrix $A$ of the phase system of The coupled quadratic oscillator model: mass metric, stiffness, generalized force, modal vocabulary and the small-oscillation model at a mechanical equilibrium (clause (b)) and every datum $z_{0}$ , the curve $z (t) = E (t - t_{0}) z_{0}$ solves $z^{'} = A z$ on all of $R$ with $z (t_{0}) = z_{0}$ , and any differentiable solution of that system on an interval containing $t_{0}$ with the same datum agrees with it there. Consequently every local solution is a restriction of the unique global solution, and the global solution is the maximal one (Extensions, maximal solutions, maximal intervals, and global solutions, Every Picard–Lindelöf initial value problem has one maximal solution on an open interval).

[F4]

Lyapunov stability (Equilibria, Lyapunov stability and asymptotic stability of an autonomous first-order system): for the autonomous field $F (z) = A z$ of the free phase system, the rest point $0$ is Lyapunov stable when for every $ε > 0$ there is $δ > 0$ such that every maximal solution with $∥ z (0) ∥_{2} < δ$ exists for all $t \geq 0$ and satisfies $∥ z (t) ∥_{2} < ε$ ; it is not stable when some $ε_{0} > 0$ has data with $∥ z (0) ∥_{2} < δ$ for every $δ > 0$ reaching $∥ z (t) ∥_{2} = ε_{0}$ .

[F5]

Calculus (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 $g (c) \neq 0$ , The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, $C^{k}$ Euclidean maps and diffeomorphisms): sums, products and composites of differentiable curves are differentiated by the algebra and chain rules, and a $C^{2}$ curve on an interval whose derivative vanishes identically is constant. Differentiating the hyperbolic definitions in the Statement using $(e^{u})^{'} = e^{u}$ and $(e^{- u})^{'} = - e^{- u}$ gives $\sinh^{'} = \cosh$ and $\cosh^{'} = \sinh$ ; also $\cosh^{2} - \sinh^{2} = 1$ because $e^{u} e^{- u} = 1$ . Finally, $\sinh (κ_{i} t) \to + \infty$ as $t \to + \infty$ for $κ_{i} > 0$ : $e^{κ_{i} t} \to + \infty$ by the repulsive-oscillator case of [F2], while $0 < e^{- κ_{i} t} \leq 1$ by positivity and monotonicity of the exponential (The exponential function is smooth and $(\exp)^{'} = \exp$ , The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ , The exponential function is strictly increasing).

Proof

technique · cases; transport the free system through the modal bijection, solve the scalar equations of the three cases, and handle degeneracy intrinsically through the invariant subspace $\ker(K-\lambda M)$

1.1F1given

Modal equivalence: by [F1] the curve $q$ is a motion if and only if $z_{i}^{''} + λ_{i} z_{i} = 0$ on $J$ for every $i$ .

1.2F3given

Global solution: the free phase system is $z^{'} = A z$ with the constant matrix $A$ of clause (b) of The coupled quadratic oscillator model: mass metric, stiffness, generalized force, modal vocabulary and the small-oscillation model at a mechanical equilibrium; by [F3] the curve $z (t) = E (t - t_{0}) (q_{0}, v_{0})$ solves it on $R$ with the prescribed datum, and every solution with that datum agrees with it on every interval containing $t_{0}$ . Writing $q (t)$ for the first $n$ components and $v (t)$ for the last $n$ , the equivalence of the phase system with the model equation of clause (b) gives that $q$ is a motion with $q (t_{0}) = q_{0}$ , $q^{'} (t_{0}) = v_{0}$ ; conversely every motion's phase curve is a solution, hence agrees with this one. This is clause 1.

1.3F1givenalgebra

Pure- $λ$ motions, intrinsic form: let $λ$ be a modal stiffness and put $S : = M^{- 1 / 2} K M^{- 1 / 2}$ , so that $K = M^{1 / 2} S M^{1 / 2}$ ; for $u \in E_{λ}^{S}$ one has $S u = λ u$ and hence $K (M^{- 1 / 2} u) = M^{1 / 2} S u = λ M^{1 / 2} u = λ M (M^{- 1 / 2} u)$ , so that $N_{λ} = M^{- 1 / 2} E_{λ}^{S} \subseteq {q : K q = λ M q}$ (*). Conversely, if $K q = λ M q$ then with $u : = M^{1 / 2} q$ we get $M^{1 / 2} S u = K q = λ M q = λ M^{1 / 2} u$ and $S u = λ u$ by invertibility of $M^{1 / 2}$ , so $q \in N_{λ}$ ; hence the two sets are equal. The modal coordinates satisfy $z = O^{T} M^{1 / 2} q$ and $M^{1 / 2} q \in E_{λ}^{S}$ is equivalent to the support of $z$ lying in the index set ${i : λ_{i} = λ}$ , because the columns of $O$ are an orthonormal eigenbasis of $S$ .

2.1assume-case oscillatoryassume-case zeroassume-case exponentialF1F2step 1.1algebra

Scalar families: for each $i$ , [F2] gives that the scalar solutions are exactly the displayed families with unique coefficients determined by the global extension's values $z_{i} (0)$ and $z_{i}^{'} (0)$ ; combining with step 1.1 gives clause 2 and its coefficient formulas.

2.2step 1.3F5algebra

Pure- $λ$ motions, equation: let $q : J \to N_{λ}$ be $C^{2}$ on an interval $J$ . By step 1.3, $N_{λ} = \ker (K - λ M)$ , so $(K - λ M) q (t) = 0$ for every $t \in J$ ; differentiating this identity once and twice, since $K - λ M$ is constant, gives $q^{'} (t), q^{''} (t) \in N_{λ}$ . Hence with (*), $M$ invertible, and $q (t), q^{''} (t) \in N_{λ}$ , $M q^{''} + K q = M q^{''} + λ M q = M (q^{''} + λ q),$ so $q$ is a motion if and only if $q^{''} + λ q = 0$ . This proves the equivalence of clause 4 for such curves and, combined with the support statement of step 1.3, identifies the pure- $λ$ motions with the $C^{2}$ solutions of $q^{''} + λ q = 0$ with values in $N_{λ}$ .

2.3F4step 1.2F5givenalgebra

Unstable direction: suppose $λ_{i} < 0$ and let $u_{i} = M^{- 1 / 2} O e_{i}$ satisfy $K u_{i} = λ_{i} M u_{i}$ and $u_{i}^{T} M u_{i} = 1$ (The coupled quadratic oscillator model: mass metric, stiffness, generalized force, modal vocabulary and the small-oscillation model at a mechanical equilibrium clause (e)). For $c > 0$ define $q_{c} (t) : = (c / κ_{i}) \sinh (κ_{i} t) u_{i}$ ; then $q_{c}^{''} = κ_{i}^{2} (c / κ_{i}) \sinh (κ_{i} t) u_{i}$ and $M q_{c}^{''} + K q_{c} = (κ_{i}^{2} + λ_{i}) M c \sinh (κ_{i} t) u_{i} / κ_{i} = 0$ because $κ_{i}^{2} = - λ_{i}$ and $K u_{i} = λ_{i} M u_{i}$ ; so $q_{c}$ is a motion with datum $(0, c u_{i})$ , whose phase norm $∥ (q_{c} (t), q_{c}^{'} (t)) ∥_{2} \geq ∥ q_{c} (t) ∥_{2} = c κ_{i}^{- 1} \sinh (κ_{i} t) ∥ u_{i} ∥_{2}$ is unbounded as $t \to \infty$ . Fix $ε_{0} : = 1$ ; given $δ > 0$ choose $c > 0$ with $c ∥ u_{i} ∥_{2} < \min (δ, 1)$ , so that $∥ (0, c u_{i}) ∥_{2} < δ$ ; by continuity and unboundedness the motion reaches $∥ (q_{c} (t), q_{c}^{'} (t)) ∥_{2} = 1$ at some $t > 0$ . Hence the condition of [F4] for stability fails, which is clause 6.

3.1F1F2F5step 2.1algebra

Energy: along a motion, differentiating $E (q, q^{'}) = \frac{1}{2} q^{' T} M q^{'} + \frac{1}{2} q^{T} K q$ and using $M q^{''} + K q = 0$ together with the symmetry of $M$ and $K$ gives $\frac{d}{d t} E = q^{' T} M q^{''} + \frac{1}{2} (q^{' T} K q + q^{T} K q^{'}) = q^{' T} (M q^{''} + K q) = 0$ , so $E$ is constant; the identities $q^{' T} M q^{'} = ∣ z^{'} ∣_{2}^{2}$ and $q^{T} K q = \sum_{i} λ_{i} z_{i}^{2}$ of [F1] give $E = \sum_{i} E_{i}$ with $E_{i} = \frac{1}{2} (z_{i}^{' 2} + λ_{i} z_{i}^{2})$ , and $E_{i}^{'} = z_{i}^{'} (z_{i}^{''} + λ_{i} z_{i}) = 0$ shows that each $E_{i}$ is constant. Substituting the three families of step 2.1 gives the displayed values, using $\cosh^{2} - \sinh^{2} = 1$ for $λ_{i} < 0$ and $\cos^{2} + \sin^{2} = 1$ with $ω_{i}^{2} = λ_{i}$ for $λ_{i} > 0$ . This is clause 3.

3.2step 2.2F1F5algebra

Explicit pure- $λ$ families: a $C^{2}$ curve with values in the subspace $N_{λ}$ and $q^{''} + λ q = 0$ has a unique maximal extension and is uniquely determined by that extension's $q (0), q^{'} (0) \in N_{λ}$ , and the displayed formulas $\cos (ω t) q_{0} + ω^{- 1} \sin (ω t) q_{1}$ for $λ > 0$ , $q_{0} + t q_{1}$ for $λ = 0$ and $\cosh (κ t) q_{0} + κ^{- 1} \sinh (κ t) q_{1}$ for $λ < 0$ define such curves for all $q_{0}, q_{1} \in N_{λ}$ : they take values in $N_{λ}$ , and the scalar identities $\cos^{''} = - \cos$ , $\sin^{''} = - \sin$ , $\cosh^{''} = \cosh$ , $\sinh^{''} = \sinh$ verify the equation. The displayed curves take the given data at $0$ and solve the same constant-coefficient phase system, so uniqueness in [F3] makes every local curve with those extension data their restriction; hence the families are exactly the pure- $λ$ motions. The statements of clause 4 about independence of $O$ and the non-intrinsic modal vectors follow because $N_{λ}$ and the equations above involve only $M$ and $K$ : the admissible decouplings differ by an orthogonal change of coordinates preserving $N_{λ}$ , and when the multiplicity of $λ$ exceeds one, any $q_{0} \in N_{λ}$ not parallel to a chosen $u_{i}$ exhibits a pure- $λ$ motion not covered by that single modal vector.

3.3F1F2step 2.1algebra

Boundedness: if $K$ is positive definite then every modal stiffness is $> 0$ : for $u \neq 0$ with $S u = λ u$ one has $0 < q^{T} K q = λ ∣ M^{1 / 2} q ∣^{2}$ with $q = M^{- 1 / 2} u \neq 0$ , so $λ > 0$ . By clause 2 every modal coordinate satisfies $∣ z_{i} (t) ∣ \leq \sqrt{a_{i}^{2} + b_{i}^{2}}$ and $∣ z_{i}^{'} (t) ∣ \leq ω_{i} \sqrt{a_{i}^{2} + b_{i}^{2}}$ on $R$ ; since $q = M^{- 1 / 2} O z$ and $q^{'} = M^{- 1 / 2} O z^{'}$ are bounded linear functions of $(z, z^{'})$ , every motion is bounded on $R$ . If instead $λ_{i} = 0$ for some $i$ and a motion has $b_{i} \neq 0$ in the affine family, then $∣ z_{i} (t) ∣ = ∣ a_{i} + b_{i} t ∣ \to \infty$ as $t \to \infty$ . The map $z \mapsto q = M^{- 1 / 2} O z$ is a linear bijection, so if $(q, q^{'})$ were bounded then $(z, z^{'})$ would be bounded — its components are the coordinates of $(z, z^{'})$ in the basis given by the columns of the invertible block-diagonal matrix $diag (O^{T} M^{1 / 2}, O^{T} M^{1 / 2})$ , and coordinate functionals of a bounded set are bounded; contrapositively the unbounded modal component forces $(q, q^{'})$ unbounded. This is clause 5.

4.1cases-exhaustive:three-scalar-index-typesstep 1.2step 2.1step 3.1step 3.2step 3.3step 2.3∎

Clauses 1-6 are proved: the free model has globally unique motions classified by the three scalar families, the energy decomposition holds, degenerate families are intrinsic subspaces, positive definite stiffness bounds every motion, zero modes are affine and negative modal stiffnesses give an unstable rest point.

Remarks

The three modal types are mutually exclusive per index. Each index is oscillatory, zero or exponential, and the model is oscillatory exactly when $K$ is non-negative. A zero stiffness direction arises, for instance, when the potential has a flat second-order direction at the equilibrium (an isolated strict minimum can still have zero stiffness, for example the scalar potential $U (q) = q^{4}$ at $0$ ); the affine family is then not an oscillation at all, as the remark item of this page records in detail.
What the intrinsic form does not say. Clause 4 shows that the family of pure- $λ$ motions is intrinsic; it does not select a preferred basis of $N_{λ}$ , and no physical meaning attaches to an individual $u_{i}$ beyond being one convenient basis vector. The frequency $ω = \sqrt{λ}$ and the subspace $N_{λ}$ are the intrinsic data of a degenerate mode.

The free coupled quadratic model: global existence, the complete modal classification, mode energies, intrinsic degenerate mode subspaces, boundedness and the unstable direction

Statement

Facts & Assumptions

Proof

Remarks

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