Linearisation along a motion of the mechanical model: the flow derivative solves the variational equation, with Lipschitz dependence, the constant-metric second-order form, and the conditional finite-time error bound of the linear approximation

Let $n\ge 1$, let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be open, let $M:\mathrm{\Omega }\to M_n(\mathbb{R})$ be a $C^2$ field of symmetric positive definite matrices, let $U:\mathrm{\Omega }\to \mathbb{R}$ be of class $C^2$, and let $G$ be the mechanical vector field on the state domain $S_0:=\mathrm{\Omega }\times {\mathbb{R}}^n$ (The mechanical first-order field, Lagrangian and energy of a configuration-dependent kinetic metric in a chart). Require the fixed numerical chart convention of Stability vocabulary of the model: Lyapunov, asymptotic and orbital stability, linear stability of equilibria and map fixed points, and hyperbolicity of equilibria and fixed points: for declared positive component references $a_j$ and duration reference $t_{\mathrm{r}\mathrm{e}\mathrm{f}}$, physical coordinates and velocities are $Q_j=a_j{q}_j$ and $V_j=(a_j\mathrm{/}{t}_{\mathrm{r}\mathrm{e}\mathrm{f}})v_j$, with physical time $t_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=t_{\mathrm{r}\mathrm{e}\mathrm{f}}t$. All data $M,U,G$, balls, vector norms and induced operator norms below are in this dimensionless numerical state chart, so $q^{\mathrm{\prime }}=v$. This field is $C^1$, hence locally Lipschitz; let $\mathrm{\Phi }$ be its local flow as defined in The local flow of an autonomous first-order system, and write $z(t;a)$ for the maximal solution through $a\in S_0$. Fix a state $z_0=(q_0,v_0)\in S_0$, a time $T>0$ and a radius $R>0$ such that $\bar{B}(z_0,R)\subseteq S_0$, put $M_R:={\sup }_{\bar{B}(z_0,R)}\parallel G{\parallel }_2,L_R:={\sup }_{\bar{B}(z_0,R)}\parallel DG\parallel ,$ and let $0<\rho <R-M_{R}T$ (Open ball, closed ball and sphere in a metric space, The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case). Then:

1. (Flow derivative and variational equation.) For every $a\in \bar{B}(z_0,\rho )$ the maximal solution $z(\cdot ;a)$ is defined on $[0,T]$ and stays in $B(z_0,R)$; the map $a\mapsto z(t;a)$ is differentiable at every $a\in B(z_0,\rho )$ for $t\in [0,T]$, and for every $u\in {\mathbb{R}}^{2n}$ $D_{a}z(t;a)u=Y_a(t)u,$ where $Y_a:[0,T]\to M_{2n}(\mathbb{R})$ is the unique solution of the variational equation along the reference motion $z(\cdot ;a)$, $Y^{\mathrm{\prime }}(t)=DG(z(t;a))Y(t),Y(0)=I_{2n},$ in the sense of The variational equation along an ODE solution; moreover $\parallel z(t;a)-z(t;b){\parallel }_2\le e^{L_{R}t}\parallel a-b{\parallel }_2$ for all $a,b\in \bar{B}(z_0,\rho )$ and $t\in [0,T]$.
2. (Constant-metric second-order form.) Suppose in addition that $M$ is constant on $\mathrm{\Omega }$, say $M(q)\equiv M$ with $M$ symmetric positive definite. Then, for any $a\in B(z_0,\rho )$, let $q:[0,T]\to \mathrm{\Omega }$ be the position component of its motion $z(t;a)=(q(t),q^{\mathrm{\prime }}(t))$, and write the matrix solution of clause 1 in block form $Y_a(t)=\left(\begin{array}{c}{Y}_{a,q}(t)\\ Y_{a,v}(t)\end{array}\right)$, where each block has $n$ rows and $2n$ columns, and put $\eta (t):=Y_{a,q}(t)$. The variational equation is equivalent, column by column, to the second-order equation $M{\eta }^{\mathrm{\prime }\mathrm{\prime }}(t)=-D^{2}U(q(t))\eta (t)(0\le t\le T),$ with $D^{2}U$ the Hessian of $U$ (The Hessian matrix and critical points of a scalar field).
3. (Conditional finite-time error bound.) Suppose in addition that $z_0$ is an equilibrium, $G(z_0)=0$, and write $A:=DG(z_0)$; suppose there are constants $C\ge 0$, $L\ge 0$ and $r>0$ with $\bar{B}(z_0,r)\subseteq S_0$ and $\parallel G(z)-A(z-z_0){\parallel }_2\le C\parallel z-z_0{\parallel }_2^2,\parallel G(z){\parallel }_2\le L\parallel z-z_0{\parallel }_2(\parallel z-z_0{\parallel }_2\le r)\mathrm{.}$ Let $T>0$ and $\delta :=\parallel a-z_0{\parallel }_2>0$ with $\delta e^{LT}<r$, let $z(\cdot ;a)$ be the model's motion through $a$, and let $y$ solve the linearised equation $y^{\mathrm{\prime }}=Ay$, $y(0)=a-z_0$. Then $\parallel z(t;a)-z_0-y(t){\parallel }_2\le C{\delta }^2{\int }_0^t{e}^{\parallel A\parallel (t-s)}{e}^{2Ls}\mathrm{d}s\le C{\delta }^{2}t{e}^{(\parallel A\parallel +2L)t}(0\le t\le T)\mathrm{.}$ where $\parallel A\parallel$ is the operator norm (The operator norm as the least bound and as the unit-sphere or unit-ball supremum).
4. (Scope.) The linearised equation approximates the model's motion only for finite times and initial states inside the declared ball with the declared quantitative bounds; the constants $M_R,L_R,C,L,r$ are model-data bounds, not measured quantities. Clause 1 gives no rate of separation beyond the exponential bound, no conclusion when the reference motion leaves $B(z_0,R)$ before time $T$, and no statement about the size of the time interval on which a linearisation is a good approximation; clause 3's hypotheses must be verified for the particular data.

• Interpretation. Clause 1 is the precise sense in which "nearby initial states evolve near each other" in the model: the flow derivative is transported by the variational equation, and clause 3 quantifies when the linearised equation controls the nonlinear motion up to a finite time. A finite-time growth of the deviation is therefore a property of the linearised equation on a stated interval, not a claim about long-time behaviour or chaos; see Finite-time tangent growth rates and Lyapunov exponents of the model's flow: the rate per unit time of a transported deviation, with its existence convention and its limits.

• Regularity. $M$ of class $C^2$ and $U$ of class $C^2$ make $G$ of class $C^1$, which suffices for the variational equation. If $M$ is only continuous, the variational equation need not exist, and clause 1 is not available; the local Lipschitz regularity of the field alone still gives uniqueness of motions.

• Units. All constants and norms in the displayed estimates are numerical in the required chart. The physical rate constants corresponding to $\parallel A\parallel$, $L_R$, $L$ and $C$ are these numbers divided by $t_{\mathrm{r}\mathrm{e}\mathrm{f}}$; each has dimension ${\tau }^{-1}$ relative to the dimensionless state norm. Thus the exponential arguments are dimensionless and clause 3 bounds a dimensionless state difference. A bound $b$ on that norm gives component error bounds $a_{j}b$ for $Q_j$ and $(a_j\mathrm{/}{t}_{\mathrm{r}\mathrm{e}\mathrm{f}})b$ for $V_j$, carrying their respective dimensions (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework).