Cm Math Approximation

Let $m>0$, $c\ge 0$ and $k\in \mathbb{R}$ be given real numbers, let $I\subseteq \mathbb{R}$ be an interval with nonempty interior (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length) and let $f:I\to \mathbb{R}$ be continuous (Continuity of $f:A\to \mathbb{R}$ at a point of $A$ and on $A$: the $\epsilon$-$\delta$ condition, its agreement with ${\lim }_{x\to c}f(x)=f(c)$ at a limit point, and continuity at an isolated point). The damped driven oscillator model with data $(m,c,k,f)$ is the second-order equation

$m{x}^{\mathrm{\prime }\mathrm{\prime }}(t)+c{x}^{\mathrm{\prime }}(t)+kx(t)=f(t),t\in I\mathrm{.}$

A solution of the model on an interval $J\subseteq I$ with nonempty interior is a twice differentiable $x:J\to \mathbb{R}$ such that the identity holds for every $t\in J$, derivatives being ordinary real derivatives (The derivative $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ of $f:A\to \mathbb{R}$ at a point $c\in A$ that is a limit point of $A$, and differentiability on a set, The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral). The model is called homogeneous when $f\equiv 0$, undamped when $c=0$, and strictly damped when $c>0$.

The phase-plane equation of the model has right-hand side

$G_f:I\times {\mathbb{R}}^2⟶{\mathbb{R}}^2,G_f(t,x,v):=(v,\frac{f(t)-cv-kx}{m})\mathrm{.}$

Here ${\mathbb{R}}^2$ is the phase plane, with coordinates $(x,v)$, where $x$ is the displacement and $v$ the velocity. For an interval $J\subseteq I$ with nonempty interior, a differentiable curve $z=(x,v):J\to {\mathbb{R}}^2$ solves the phase-plane equation when $z^{\mathrm{\prime }}(t)=G_f(t,z(t))$ for every $t\in J$, using the domain-relative derivative at any included endpoint (The derivative $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ of $f:A\to \mathbb{R}$ at a point $c\in A$ that is a limit point of $A$, and differentiability on a set). If $x$ is a solution of the model on $J$, then $t\mapsto (x(t),x^{\mathrm{\prime }}(t))$ solves this equation; conversely, if $t\mapsto (x(t),v(t))$ solves it, then $v=x^{\mathrm{\prime }}$ and $x$ is a solution of the model on $J$. When $J$ is open, this is the first-order system of First-order systems, initial value problems, and solutions on intervals on the open domain $J\times {\mathbb{R}}^2$. For the homogeneous model the right-hand side is written $G_0(x,v)=(v,-(cv+kx)\mathrm{/}m)$.

The characteristic polynomial of the model is

$p(\lambda ):=m{\lambda }^2+c\lambda +k,$

with discriminant $\mathrm{\Delta }:=c^2-4mk$. When $k>0$ define the undamped frequency ${\omega }_0:=\sqrt{k\mathrm{/}m}$, the damping rate $\gamma :=c\mathrm{/}(2m)$ and, when $4mk>c^2$, the damped frequency $\omega :=\sqrt{k\mathrm{/}m-{\gamma }^2}=\sqrt{4mk-c^2}\mathrm{/}(2m)$; the corresponding square roots are the nonnegative real ones.

• Equilibrium of the homogeneous phase system. A point $(x,v)\in {\mathbb{R}}^2$ satisfies $G_0(x,v)=(0,0)$ if and only if $v=0$ and $kx=0$; hence the zero set of $G_0$ is $\{(0,0)\}$ when $k\mathrm{\ne }0$ and the whole line $\{(x,0):x\in \mathbb{R}\}$ when $k=0$. For the forced model, $G_f(t,x,v)=(0,0)$ holds if and only if $v=0$ and $kx=f(t)$.

• Interpretation. This item is a purely mathematical model: $m,c,k$ are real parameters, $f$ an arbitrary continuous real function, and no physical units, force law, constitutive assumption or empirical claim is made. The classical-mechanical reading of $m,c,k,f$ and any identification with measured data belong to the consuming physics item, not to this definition.

• Regularity. $G_f$ is continuous relative to $I\times {\mathbb{R}}^2$ and, for each $t$, Lipschitz in $(x,v)$ with constant $L:=\max (1,(c+\mathrm{\mid }k\mathrm{\mid })\mathrm{/}m)\sqrt{2}$, since $\mathrm{\mid }-c\mathrm{\Delta }v-k\mathrm{\Delta }x\mathrm{\mid }\le c\mathrm{\mid }\mathrm{\Delta }v\mathrm{\mid }+\mathrm{\mid }k\mathrm{\mid }\mathrm{\mid }\mathrm{\Delta }x\mathrm{\mid }$ and $\mathrm{\mid }a\mathrm{\mid }+\mathrm{\mid }b\mathrm{\mid }\le \sqrt{2}\sqrt{a^2+b^2}$. At every interior time $t\in I^{\circ }$, its restriction to a sufficiently small open time interval about $t$ satisfies the Picard-Lindelof hypotheses of Picard-Lindelöf local existence and uniqueness for first-order systems. If $I$ includes an endpoint, this item does not assert existence across that endpoint unless $f$ is also defined on an open time neighborhood there.

Let $n\ge 1$, let $P>0$, let $O\subseteq {\mathbb{R}}^n$ be open, and let $f:\mathbb{R}\times O\to {\mathbb{R}}^n$ be continuous (Vector-valued functions $f:A\to {\mathbb{R}}^m$, their limits and continuity, with the dictionary to the metric notions). Assume that for each standard coordinate vector $e_j$ of ${\mathbb{R}}^n$, $0\le j<n$, and every $(t,x)\in \mathbb{R}\times O$, the vector-valued derivative at $r=0$ of $r\mapsto f(t,x+r{e}_j)$ exists, and that each resulting partial derivative $(t,x)\mapsto {\mathrm{\partial }}_{x_j}f(t,x)$ is continuous on $\mathbb{R}\times O$ (Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$). This is the meaning of continuously partially differentiable in the state variable here. Suppose also that $f$ is $P$-periodic in time, that is

$f(t+P,x)=f(t,x)\text{for all }(t,x)\in \mathbb{R}\times O\mathrm{.}$

The average of such an $f$ is the map

$\bar{f}:O⟶{\mathbb{R}}^n,\bar{f}(x):=\frac{1}{P}{\int }_0^{P}f(s,x)ds,$

where the integral is the componentwise Riemann integral of the continuous curve $s\mapsto f(s,x)$ over the compact interval $[0,P]$ (The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length); it is well defined and finite for every $x\in O$ by componentwise application of A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion. The oscillatory part of $f$ is $f-\bar{f}$, and the deviation of $f$ is

$g:[0,P]\times O⟶{\mathbb{R}}^n,g(t,x):={\int }_0^t(f(s,x)-\bar{f}(x))ds\mathrm{.}$

Because ${\int }_0^{P}f(s,x)-\bar{f}(x)ds=P\bar{f}(x)-P\bar{f}(x)=0$ by the linearity clause of The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral and the componentwise constant-integral formula of If $m\le f\le M$ on $[a,b]$ then $m(b-a)\le L(f,P)\le \underset{‾}{{\int }_a^b}f\le \overline{{\int }_a^b}f\le U(f,P)\le M(b-a)$ for every partition $P$; in particular every constant function is integrable, with ${\int }_a^{b}c=c(b-a)$, the formula $g(t+kP,x):=g(t,x)$ for $k\in \mathbb{Z}$ and $t\in [0,P)$ defines a unique extension, again written $g$, to all of $\mathbb{R}\times O$; it satisfies $g(t+P,x)=g(t,x)$ and $g(0,x)=0$ for all $(t,x)$, and on $[0,P]$ it is the integral displayed above.

The averaged (or reduced) system of $f$ for a parameter $\epsilon >0$ is the autonomous first-order system on $O$

$y^{\mathrm{\prime }}(t)=\epsilon \bar{f}(y(t))\mathrm{.}$

• Scope. The objects $\bar{f}$, $g$ and the averaged system are pure mathematics: $f$ is an arbitrary continuous, $P$-periodic, continuously differentiable-in-state map on $\mathbb{R}\times O$. No physical reading of $t$, $x$, $\epsilon$ or $f$ is asserted here; a physical model that identifies them with time, state, a small parameter and a force field is the consumer's responsibility.

• The deviation is bounded on one period. For nonempty compact $K\subseteq O$, $[0,P]\times K$ is compact by Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ 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, so $f$ is uniformly continuous there by Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous. The integral norm inequality and monotonicity (For $a\le b$ and $f:[a,b]\to {\mathbb{R}}^m$ integrable when $a<b$, $\parallel {\int }_a^{b}f{\parallel }_2\le {\int }_a^b\parallel f{\parallel }_2$; for $a<b$, $\parallel f{\parallel }_2$ is integrable, If $f\le g$ on $[a,b]$ and both are integrable then ${\int }_a^{b}f\le {\int }_a^{b}g$; and $m(b-a)\le {\int }_a^{b}f\le M(b-a)$) give $\mathrm{\parallel }\bar{f}(x)-\bar{f}(y){\mathrm{\parallel }}_2\le {\sup }_{s\in [0,P]}\mathrm{\parallel }f(s,x)-f(s,y){\mathrm{\parallel }}_2$, hence continuity of $\bar{f}$ on $K$. Put $u(s,x)=f(s,x)-\bar{f}(x)$. Then $u$ is continuous on the compact product and $C_K:={\max }_{[0,P]\times K}\mathrm{\parallel }u{\mathrm{\parallel }}_2$ is finite by A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value. For $t,r\in [0,P]$ and $x,y\in K$, splitting the integral at $r$ and using its orientation gives $\mathrm{\parallel }g(t,x)-g(r,y){\mathrm{\parallel }}_2\le P{\sup }_{s\in [0,P]}\mathrm{\parallel }u(s,x)-u(s,y){\mathrm{\parallel }}_2+C_K\mathrm{\mid }t-r\mathrm{\mid }$. Uniform continuity of $u$ proves joint continuity of $g$ on that product, so its norm has a finite maximum there. The periodic extension preserves this bound; for $K=\mathrm{\varnothing }$ boundedness is vacuous. Quantitative bounds with explicit constants are supplied separately.

Let $\epsilon >0$, let $\widehat{J},\widehat{S}\subseteq \mathbb{R}$ be open intervals, and let $J\subseteq \widehat{J}$ and $S\subseteq \widehat{S}$ be intervals with nonempty interiors (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length). Let $h:\widehat{J}\times \widehat{S}\to \mathbb{R}$ and $k:\widehat{J}\times \mathbb{R}\times \widehat{S}\to \mathbb{R}$ be of class $C^2$ on these open Euclidean domains ($C^k$ Euclidean maps and diffeomorphisms), and suppose $k$ is $2\pi$-periodic in its second variable on $J\times \mathbb{R}\times S$:

$k(I,\theta +2\pi ,s)=k(I,\theta ,s)\text{for all }(I,\theta ,s)\in J\times \mathbb{R}\times S\mathrm{.}$

The transformed Hamiltonian of the datum is

$K(I,\theta ,s):=h(I,s)+\epsilon k(I,\theta ,s),$

its frequency function is

$\omega (I,s):={\mathrm{\partial }}_{I}h(I,s),$

its angle average is $⟨k⟩(I,s):=\frac{1}{2\pi }{\int }_0^{2\pi }k(I,\theta ,s)d\theta$, and, on the set where $\omega \mathrm{\ne }0$, its drift generator is

$g(I,\theta ,s):=\frac{k(I,\theta ,s)-⟨k⟩(I,s)}{\omega (I,s)}\mathrm{.}$

The branch vector field of the datum is the restriction to $J\times \mathbb{R}\times S$ of the map $\widehat{\mathrm{\Phi }}:\widehat{J}\times \mathbb{R}\times \widehat{S}\to {\mathbb{R}}^3$,

$\widehat{\mathrm{\Phi }}(I,\theta ,s):=(-\epsilon {\mathrm{\partial }}_{\theta }k(I,\theta ,s),{\mathrm{\partial }}_{I}h(I,s)+\epsilon {\mathrm{\partial }}_{I}k(I,\theta ,s),\epsilon ),$

and a branch solution on an interval $T\subseteq \mathbb{R}$ with nonempty interior is a differentiable $(I,\theta ,s):T\to J\times \mathbb{R}\times S$ satisfying $(I,\theta ,s{)}^{\mathrm{\prime }}=\widehat{\mathrm{\Phi }}(I,\theta ,s)$ at every $t\in T$, with derivatives relative to $T$ at included endpoints. On open time intervals this is the first-order system of First-order systems, initial value problems, and solutions on intervals on the open domain $\widehat{J}\times \mathbb{R}\times \widehat{S}$ (Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$). Partial derivatives are those of the ambient Euclidean coordinates.

• Angle interpretation. Since $k$ is $2\pi$-periodic in $\theta$, so are ${\mathrm{\partial }}_{\theta }k$ and ${\mathrm{\partial }}_{I}k$ on $J\times \mathbb{R}\times S$, hence the restriction of $\widehat{\mathrm{\Phi }}$ to $J\times \mathbb{R}\times S$ is $2\pi$-periodic in the second variable: reading the second coordinate in $\mathbb{R}\mathrm{/}2\pi \mathbb{Z}$ gives a well-defined vector field on $J\times (\mathbb{R}\mathrm{/}2\pi \mathbb{Z})\times S$, and if $(I,\theta ,s)$ is a branch solution, so is $(I,\theta +2\pi n,s)$ for every $n\in \mathbb{Z}$. The definition keeps the covering description on $\mathbb{R}$ to avoid quotient notation; all statements about $\theta$ below are invariant under $2\pi$-translations.

• Reading. A datum of this definition is the normal form of a one-frequency Hamiltonian problem in action-angle coordinates after the slow parameter is promoted to a state variable $s$ with $s^{\mathrm{\prime }}=\epsilon$: $h$ is the angle-independent part, $\epsilon k$ the angle-dependent remainder including any contribution of the parameter-dependent chart, and $\omega ={\mathrm{\partial }}_{I}h$ the fast frequency. The definition asserts no existence theorem: whether a given mechanical family admits such a chart with the stated regularity, on which domain, and with $\omega \mathrm{\ne }0$, is a separate hypothesis of any consumer, and the adiabatic statements below are conditional on it.

• Scope. This is pure mathematics: no physical identification of $I$, $\theta$, $s$, $h$, $k$, $\epsilon$ or of the parameter variation with measured data is made here.

Let $n\ge 1$, equip ${\mathbb{R}}^n$ with the Euclidean inner product and norm of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$, and let $I\subseteq \mathbb{R}$ be an interval with nonempty interior (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length), let $t_0\in I$, let $U\subseteq \mathbb{R}$ be an open interval with $I-I:=\{t-s:t,s\in I\}\subseteq U$, let $g:U\to M_n(\mathbb{R})$ have $C^1$ entries (The vector space $M_{m\times n}(F):=F^{m\times n}$ of $m$ by $n$ matrices over a field, with entrywise operations, $C^k$ Euclidean maps and diffeomorphisms) and let $f:I\to {\mathbb{R}}^n$ be continuous (Vector-valued functions $f:A\to {\mathbb{R}}^m$, their limits and continuity, with the dictionary to the metric notions). For $B=(b_{ij})\in M_n(\mathbb{R})$ and $v\in {\mathbb{R}}^n$, define $(Bv{)}_i:={\sum }_{j<n}{b}_{ij}{v}_j$. This is a linear map by the coordinate formula; the Euclidean norm is a norm by Cauchy-Schwarz $\mid ⟨x,y⟩\mid \le \parallel x{\parallel }_2\parallel y{\parallel }_2$ with its equality case, the triangle inequality for $\parallel \cdot {\parallel }_2$, the parallelogram law and polarisation, and the standard coordinate vectors form an ordered basis by The standard list $e:n\to F^n$ with $e_i(i)=1_F$ and $e_i(j)=0_F$ for $j\mathrm{\ne }i$ is an ordered basis of $F^n$; hence ${\dim }_F{F}^n=n$, and $F^0$ is the zero space with basis $\mathrm{\varnothing }$ and dimension $0$, so the map is bounded by A linear map from a finite-dimensional normed space is bounded. Let $\mathrm{\parallel }B\mathrm{\parallel }$ denote its induced operator norm (The operator norm as the least bound and as the unit-sphere or unit-ball supremum). All matrix actions in the integrals use this convention. Since $0\in I-I\subseteq U$, the value $g(0)$ is defined. Define

$F:I⟶{\mathbb{R}}^n,F(t):={\int }_{t_0}^{t}g(t-s)f(s)ds,$

with the oriented integral of The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral and the matrix action just defined; the integrand is continuous in $s$, hence integrable on every compact subinterval. Then:

1. $F$ is differentiable at every interior point $t$ of $I$ (The derivative $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ of $f:A\to \mathbb{R}$ at a point $c\in A$ that is a limit point of $A$, and differentiability on a set) and

$F^{\mathrm{\prime }}(t)=g(0)f(t)+{\int }_{t_0}^t{g}^{\mathrm{\prime }}(t-s)f(s)ds\mathrm{.}$

2. If moreover $g^{\mathrm{\prime }}$ is bounded on compact subintervals of $U$ (which holds automatically because $g$ is $C^1$), then $F$ is of class $C^1$ on the interior of $I$.

Let $m>0$, $c\ge 0$ and $k>0$, and consider the homogeneous damped oscillator model $m{x}^{\mathrm{\prime }\mathrm{\prime }}+c{x}^{\mathrm{\prime }}+kx=0$ of The damped driven oscillator model: second-order equation, phase-plane system and characteristic data with time domain $I=\mathbb{R}$ and forcing $f:\mathbb{R}\to \mathbb{R}$ identically zero, with characteristic polynomial $p(\lambda )=m{\lambda }^2+c\lambda +k$, discriminant $\mathrm{\Delta }=c^2-4mk$, damping rate $\gamma :=c\mathrm{/}(2m)$, and, when applicable, roots ${\lambda }_{\pm }:=\frac{-c\pm \sqrt{\mathrm{\Delta }}}{2m}(\mathrm{\Delta }\ge 0),\lambda :=-\frac{c}{2m},\omega :=\frac{\sqrt{4mk-c^2}}{2m}(\mathrm{\Delta }<0)\mathrm{.}$

A local solution here may have any interval $J\subseteq \mathbb{R}$ with nonempty interior, with domain-relative derivatives at included endpoints. It has a unique extension to the global family below. A maximal solution means one that has no solution extension to a strictly larger interval; the maximal solutions of this global autonomous equation are precisely the family members on $\mathbb{R}$. For a homogeneous model restricted to a time domain $I$, its solutions have $J\subseteq I$ and are restrictions of these same global members; their extensions solve the all-$\mathbb{R}$ equation.

1. Distinct real roots. If $\mathrm{\Delta }>0$, then ${\lambda }_{+}>{\lambda }_{-}$ are exactly the real roots of $p$, and a function $x:J\to \mathbb{R}$ on an interval $J\subseteq \mathbb{R}$ with nonempty interior is a solution of the model if and only if $x(t)=A{e}^{{\lambda }_{+}t}+B{e}^{{\lambda }_{-}t}(t\in J)$ for some $A,B\in \mathbb{R}$.

2. Double real root. If $\mathrm{\Delta }=0$, then $\lambda =-c\mathrm{/}(2m)$ is the only root of $p$, and $x:J\to \mathbb{R}$ is a solution if and only if $x(t)=(A+Bt)e^{\lambda t}(t\in J)$ for some $A,B\in \mathbb{R}$.

3. Complex conjugate roots. If $\mathrm{\Delta }<0$, then $\omega >0$, the roots of $p$ are $-\gamma \pm i\omega$, and $x:J\to \mathbb{R}$ is a solution if and only if $x(t)=e^{-\gamma t}(A\cos (\omega t)+B\sin (\omega t))(t\in J)$ for some $A,B\in \mathbb{R}$.

4. Initial data. In each of the three cases and for every $(x_0,v_0)\in {\mathbb{R}}^2$ there is exactly one solution $x:\mathbb{R}\to \mathbb{R}$ with $x(0)=x_0$ and $x^{\mathrm{\prime }}(0)=v_0$. Its coefficients are $A=\frac{v_0-{\lambda }_{-}{x}_0}{{\lambda }_{+}-{\lambda }_{-}},B=\frac{{\lambda }_{+}{x}_0-v_0}{{\lambda }_{+}-{\lambda }_{-}}(\mathrm{\Delta }>0),A=x_0,B=v_0-\lambda x_0(\mathrm{\Delta }=0),$ $A=x_0,B=\frac{v_0+\gamma x_0}{\omega }(\mathrm{\Delta }<0)\mathrm{.}$