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Classification of the motions of the damped oscillator by the discriminant, and nonincreasing energy $E^{'} = - c x^{' 2}$

Physical scope

The motion classification of the damped oscillator model with m>0, k>0, c>=0: the three solution families determined by the sign of the discriminant c^2-4mk (distinct real roots, double root, oscillatory roots), uniqueness and global existence for every datum, the identity E'=-c x-dot^2 for the mechanical energy E=m x-dot^2/2+k x^2/2, nonincrease of the energy, strict negativity of its derivative when c>0 and x-dot is nonzero, and the characterisation of the constant-energy motions as the rest motion when c>0. The result is a conditional deduction from the declared model data; it asserts no value of the parameters, no measured decay and no behaviour of any real damper.

Statement

Adopt the damped oscillator model of The damped oscillator model: a harmonic restoring force with a phenomenological linear drag $- c \dot{x}$ , $c \geq 0$ with $m > 0$ , $k > 0$ , $c \geq 0$ , force law $F (x, v) = - k x - c v$ and mechanical energy $E (x, v) = \frac{1}{2} m v^{2} + \frac{1}{2} k x^{2}$ . Write $Δ : = c^{2} - 4 m k, γ : = \frac{c}{2 m}, λ_{\pm} : = \frac{- c \pm \sqrt{Δ}}{2 m} (Δ \geq 0), ω : = \frac{\sqrt{4 m k - c^{2}}}{2 m} (Δ < 0),$ and note that $Δ = - 4 m k < 0$ and $ω = \sqrt{k / m}$ when $c = 0$ . Then:

Classification and uniqueness. Let $J \subseteq R$ $J \subseteq R$ be an interval with nonempty interior. A curve $x : J \to R$ $x : J \to R$ is a motion of the model if and only if it is one of the following:
- $Δ > 0$ : $x (t) = A e^{λ_{+} t} + B e^{λ_{-} t}$ for reals $A, B$ ;
- $Δ = 0$ : $x (t) = (A + B t) e^{- c t / (2 m)}$ for reals $A, B$ ;
- $Δ < 0$ : $x (t) = e^{- γ t} (A \cos (ω t) + B \sin (ω t))$ for reals $A, B$ . For every datum $(t_{0}, x_{0}, v_{0})$ there is exactly one maximal motion, it is defined on all of $R$ , and every motion with that datum is its restriction; when its family is written in $s = t - t_{0}$ instead of $t$ , the coefficients are those supplied by clause 4 of The homogeneous damped oscillator: solution classification by the discriminant and uniqueness for initial data for the shifted curve $s \mapsto x (t_{0} + s)$ . In particular, for $c = 0$ the motions are exactly those of the harmonic model of The harmonic (linear restoring) model: a declared linear force law with parameter $k > 0$ with the same $m$ and $k$ .
Energy dissipation. For every motion $x : J \to R$ the energy $e (t) : = E (x (t), x^{'} (t))$ is differentiable at every interior point of $J$ with $e^{'} (t) = - c x^{'} (t)^{2} \leq 0,$ and for $t_{1} < t_{2}$ in $J$ one has $e (t_{2}) - e (t_{1}) = - c \int_{t_{1}}^{t_{2}} x^{'} (s)^{2} d s \leq 0$ : the energy is nonincreasing on $J$ . If $c > 0$ , then the following are equivalent: (i) $e$ is constant on $J$ ; (ii) $x^{'} (t) = 0$ at every interior point of $J$ ; (iii) $x$ is constant on $J$ ; (iv) $x \equiv 0$ on $J$ , the rest motion of the model. Moreover, at every interior point with $x^{'} (t) \neq 0$ one has $e^{'} (t) < 0$ , so the energy strictly decreases there.
Rest points. The rest points of the model are exactly the pairs $(x, 0)$ with $k x = 0$ , that is $(0, 0)$ , and the constant curve $x \equiv 0$ is its maximal motion with datum $(t_{0}, 0, 0)$ .

The statement is read for the declared data; it asserts no value of $m, k, c$ and no empirical decay law.

Facts & Assumptions

Given: The damped model with data $(m, c, k, 0)$ , $m > 0$ , $k > 0$ , $c \geq 0$ , its motions $x : J \to R$ , its energy $E$ , and the constants $Δ, γ, λ_{\pm}, ω$ .

[L1]

Model vocabulary (The damped oscillator model: a harmonic restoring force with a phenomenological linear drag $- c \dot{x}$ , $c \geq 0$ , The one-dimensional force models: data, motions, phase system, potential presentation, energy and equilibria clauses (b), (d), (f), The damped driven oscillator model: second-order equation, phase-plane system and characteristic data): the motion equation is $m x^{''} + c x^{'} + k x = 0$ , which is the homogeneous damped oscillator model of The damped driven oscillator model: second-order equation, phase-plane system and characteristic data with the same data; the force law is presented through $U (x) = \frac{1}{2} k x^{2}$ with residual force $d = - c v$ ; the mechanical energy is $E = \frac{1}{2} m v^{2} + \frac{1}{2} k x^{2}$ .

[L2]

The homogeneous damped oscillator: solution classification by the discriminant and uniqueness for initial data with $m > 0$ , $c \geq 0$ , $k > 0$ : the model's solutions on an interval with nonempty interior are exactly the three displayed families by the sign of $Δ$ ; for every datum at $0$ there is exactly one solution $R \to R$ , with the displayed coefficients; and every solution extends to $R$ . For $c = 0$ one has $Δ = - 4 m k < 0$ and $ω = \sqrt{4 m k} / (2 m) = \sqrt{k / m}$ because $m > 0$ (Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ ).

[L3]

Differentiation rules (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, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set): sums and constant multiples of differentiable functions are differentiated termwise; the product $x x^{'}$ has derivative $x^{' 2} + x x^{''}$ .

[L4]

Integration (Newton–Leibniz needs only continuity on $[a, b]$ , differentiability on $(a, b)$ , and a Riemann-integrable extension of the interior derivative, A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, If $f \leq g$ on $[a, b]$ and both are integrable then $\int_{a}^{b} f \leq \int_{a}^{b} g$ ; and $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ ): if $G$ is continuous on $[t_{1}, t_{2}]$ and differentiable on $(t_{1}, t_{2})$ with integrable derivative, then $\int_{t_{1}}^{t_{2}} G^{'} = G (t_{2}) - G (t_{1})$ ; a continuous real function on $[t_{1}, t_{2}]$ is Riemann integrable; and if $h_{1} \leq h_{2}$ are integrable on $[t_{1}, t_{2}]$ with $h_{1} \leq h_{2}$ pointwise, then $\int h_{1} \leq \int h_{2}$ . In particular a nonpositive continuous integrand has nonpositive integral.

[L5]

A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant: a continuous real function on an interval whose derivative vanishes at every interior point is constant. Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length: intervals are order-convex and have the stated interior points.

Proof

technique · direct; transport the mathematical classification of the cited supplier (whose case split by the discriminant is performed there) to the model, then differentiate and integrate the energy along a motion

1.1givenL1L2

Clause 1: by [L1] the motions of the damped model are the solutions of the homogeneous damped oscillator equation with data $(m, c, k)$ under the identification clause (f) of The one-dimensional force models: data, motions, phase system, potential presentation, energy and equilibria; [L2] gives the three unshifted families. For a datum at $t_{0}$ , the curve $y (s) = x (t_{0} + s)$ solves the same autonomous equation with $y (0) = x_{0}$ , $y^{'} (0) = v_{0}$ ; clause 4 of its supplier fixes the coefficients of $y$ in $s$ . Translating that unique all-real solution back gives existence, uniqueness and the restriction property for the datum at $t_{0}$ , and the special case $c = 0$ with $Δ = - 4 m k < 0$ , $γ = 0$ and $ω = \sqrt{k / m}$ . In that case the family of the classification is $x (t) = A \cos (ω t) + B \sin (ω t)$ , which is exactly the motion family of the harmonic model in clause 1 of The motions of the harmonic model are the sinusoids $A \cos (ω (t - t_{0})) + B \sin (ω (t - t_{0}))$ : amplitude and phase, the least period, the conserved energy $k R^{2} / 2$ and the elliptical phase curves; hence the motions of the model with the parameter value $c = 0$ are exactly those of the harmonic model with the same $m$ and $k$ .

1.2givenL1L3algebra

Clause 2: let $x : J \to R$ be a motion and $e (t) : = E (x (t), x^{'} (t))$ . At every interior point $t$ of $J$ , $x$ is twice differentiable and $x^{'}$ is differentiable, so by [L3] $e^{'} (t) = m x^{'} (t) x^{''} (t) + k x (t) x^{'} (t) = x^{'} (t) (m x^{''} (t) + k x (t)) = - c x^{'} (t)^{2},$ where the equation of motion was used in the last step. This is nonpositive, and it is strictly negative whenever $c > 0$ and $x^{'} (t) \neq 0$ .

2.1step 1.2L4

For $t_{1} < t_{2}$ in $J$ the function $e$ is continuous on $[t_{1}, t_{2}]$ and differentiable on $(t_{1}, t_{2})$ with continuous derivative by step 1.2. The derivative is Riemann integrable on this closed interval by [L4], so Newton–Leibniz gives $e (t_{2}) - e (t_{1}) = \int_{t_{1}}^{t_{2}} e^{'} (s) d s = - c \int_{t_{1}}^{t_{2}} x^{'} (s)^{2} d s \leq 0,$ the inequality by the pointwise nonnegativity of the integrand and the monotonicity of the integral in [L4]. Hence $e$ is nonincreasing on $J$ .

2.2step 1.2L1L5algebra

Let $c > 0$ . If $e$ is constant on $J$ , then $e^{'} (t) = 0$ at every interior point, so $x^{'} (t)^{2} = 0$ there by step 1.2, giving (ii); if (ii) holds then $x$ is constant on $J$ by [L5], giving (iii); if $x$ is constant on $J$ , then $m x^{''} = 0$ and the equation reads $k x = 0$ on $J$ , so $x \equiv 0$ because $k > 0$ and $J$ is nonempty, giving (iv); and (iv) trivially gives (i) because $E (0, 0) = 0$ . This proves the equivalence of (i)-(iv).

3.1step 1.1L1algebra∎

Clause 3: substituting the constant curve $x \equiv x_{*}$ into the equation gives $k x_{*} = 0$ , that is $x_{*} = 0$ because $k > 0$ ; conversely $x \equiv 0$ solves the equation, so $(0, 0)$ is a rest point and, by uniqueness in clause 1, the maximal motion with datum $(t_{0}, 0, 0)$ .

Remarks

Two time scales in the heavily damped case. For $c > 0$ the roots $λ_{\pm}$ are real and negative when $Δ > 0$ , with $λ_{-} < - c / (2 m) < λ_{+} < 0$ ; the source reads the reciprocal rates as a short momentum-relaxation time and a long coordinate-relaxation time. Those readings are remarks about the explicit families, not additional claims of this item.
Energy is not the only decreasing quantity. Clause 2 identifies the exact rate of energy loss at every instant; the monotone decrease on $J$ follows by integration. Whether the motion tends to rest as $t \to \infty$ , and how the decay compares with the explicit exponential bounds, is the subject of the following item for $c > 0$ .

Classification of the motions of the damped oscillator by the discriminant, and nonincreasing energy $E^{'} = - c x^{' 2}$

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

Classification of the motions of the damped oscillator by the discriminant, and nonincreasing energy E′=−c x′2E′=−cx′2

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

Classification of the motions of the damped oscillator by the discriminant, and nonincreasing energy $E^{'} = - c x^{' 2}$