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Runaway and bounded-future branches of the Abraham–Lorentz equation

Physical scope

Exact conditional properties of the adopted nonrelativistic prescribed-force ODE for bounded continuous forces: every acceleration solution, the unique solution bounded on a future half-line, and an explicit smooth future force for which that selected branch accelerates before the force begins. The proof concerns the formal effective equation only; it asserts no measured preacceleration, no self-consistent Maxwell–Lorentz trajectory, and no empirical validity outside the stated model regime.

Statement

Use the equation and constants of the preceding postulate, with $m, τ > 0$ . Let $f : R \to R^{3}$ be bounded and continuous, and let a solution mean a continuously differentiable map $a : R \to R^{3}$ satisfying $m (a - τ \dot{a}) = f$ at every real time. Bounded for a vector map means each of its three real coordinates is bounded; in $R^{3}$ this is equivalent to bounded Euclidean norm. The vector improper Riemann integral below is taken componentwise, with $\int_{t}^{\infty} h (s) d s : = \lim_{R \to \infty} \int_{t}^{R} h (s) d s$ when that limit exists. Then the integral converges absolutely, and

$a_{*} (t) : = \frac{1}{m τ} \int_{t}^{\infty} \exp ⁣ (- \frac{s - t}{τ}) f (s) d s$

is a bounded solution. For any fixed $t_{0} \in R$ , every solution has the form

$a (t) = a_{*} (t) + C \exp ⁣ (\frac{t - t_{0}}{τ}), C \in R^{3}$

Here $C$ is a constant acceleration vector with units $m s^{- 2}$ .

The only solution bounded on a future half-line $[T, \infty)$ , for any finite $T$ , is $a_{*}$ . A nonzero $C$ gives an exponentially growing acceleration branch, called a runaway branch in this item. Moreover, there is a bounded smooth force that vanishes for $t \leq 0$ , is nonzero for some $t \in (0, 1)$ , and for which $a_{*} (t) \neq 0$ for every $t < 0$ . Preacceleration means this nonzero acceleration before the force begins; it is a property of the selected ODE branch, not an observation about a physical particle.

Statement references

The nonrelativistic Abraham–Lorentz equation as a restricted effective model

The real exponential function and the number $e$ by a power series

The exponential function is smooth and $(\exp)^{'} = \exp$

The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$

The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$

The first fundamental theorem: if $f$ is integrable on $[a, b]$ and continuous at $c$ , then $F^{'} (c) = f (c)$ ; in particular a continuous $f$ has $F$ as a primitive

A smooth bump between concentric Euclidean balls

Facts & Assumptions

Given: The adopted equation $m (a - τ \dot{a}) = f$ with $m, τ > 0$ , a bounded continuous force $f : R \to R^{3}$ , and the SI laboratory time and spatial vector conventions.

[A1]

The adopted effective model equation is $m (a (t) - τ \dot{a} (t)) = f (t)$ for all $t \in R$ , with $m > 0$ and $τ > 0$ . (The nonrelativistic Abraham–Lorentz equation as a restricted effective model)

[F1]

The real exponential is defined by its all-real power series, is smooth with derivative $\exp^{'} = \exp$ , and satisfies $\exp (- x) = 1 / \exp (x) > 0$ . For $x \geq 0$ , its series gives $\exp (x) \geq 1 + x$ . (The real exponential function and the number $e$ by a power series, The exponential function is smooth and $(\exp)^{'} = \exp$ , The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ )

[F2]

Products and scalar multiples differentiate by the product rule; the chain rule applies to $t \mapsto \exp (- t / τ)$ ; continuous functions on compact real intervals are Riemann integrable; and the first fundamental theorem gives $J^{'} (x) = h (x)$ for the integral function $J (x) = \int_{u}^{x} h (s) d s$ of a continuous real function $h$ on a compact interval $[u, v]$ with $u < v$ . (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 $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, The first fundamental theorem: if $f$ is integrable on $[a, b]$ and continuous at $c$ , then $F^{'} (c) = f (c)$ ; in particular a continuous $f$ has $F$ as a primitive)

[F3]

On a compact interval, the Riemann integral is linear and monotone; oriented integrals obey $\int_{b}^{a} h = - \int_{a}^{b} h$ . In particular, if $∣ h ∣ \leq M g$ and $g \geq 0$ , then $∣ \int h ∣ \leq M \int g$ , by applying monotonicity to $- M g \leq h \leq M g$ . (Integrable functions on $[a, b]$ form a set closed under sums and scalar multiples, and $\int_{a}^{b} (λ f + μ g) = λ \int_{a}^{b} f + μ \int_{a}^{b} g$ , If $f \leq g$ on $[a, b]$ and both are integrable then $\int_{a}^{b} f \leq \int_{a}^{b} g$ ; and $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ , The integral with oriented limits: $\int_{a}^{a} f : = 0$ and $\int_{b}^{a} f : = - \int_{a}^{b} f$ )

[F4]

If a real function is continuous on $[u, v]$ , differentiable on $(u, v)$ and $u < v$ , then its increment is its derivative at an interior point times $v - u$ . (The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ )

[F5]

For $0 < r < R$ and $n \geq 1$ there is a smooth map $ρ : R^{n} \to [0, 1]$ that equals $1$ on ${\overline{B}}_{r} (0)$ and has support in $B_{R} (0)$ . (A smooth bump between concentric Euclidean balls)

Proof

technique · direct; construct the future integral, verify its differential equation, classify the homogeneous solutions, and exhibit a smooth force profile with a strictly positive future integral

1.1givenF1F2F3F4algebra

Put $M : = \max_{0 \leq j \leq 2} \sup_{s \in R} ∣ f_{j} (s) ∣ < \infty$ . For $R > t$ , put $k (s) = \exp (- (s - t) / τ)$ and $J (x) = \int_{t}^{x} k (s) d s$ on $[t, R]$ . By [F1]–[F2], $J$ and $P (x) = - τ \exp (- (x - t) / τ)$ are continuous on $[t, R]$ , differentiable in its interior, and $J^{'} = P^{'} = k$ . Thus $P - J$ has zero derivative; [F4] gives $P (R) - J (R) = P (t) - J (t)$ , where $J (t) = 0$ . Consequently $\int_{t}^{R} \exp (- (s - t) / τ) d s = τ (1 - \exp (- (R - t) / τ)) \leq τ$ for $R > t$ . For $x \geq 0$ , [F1] implies $0 < \exp (- x) \leq (1 + x)^{- 1} \to 0$ as $x \to \infty$ . Hence for $S > t$ and $R > S$ , the component tail is bounded by $M τ \exp (- (S - t) / τ)$ , which tends to zero as $S \to \infty$ . Thus each improper integral defining $a_{*} (t)$ converges absolutely, and $∣ a_{*, j} (t) ∣ \leq M / m$ , so $a_{*}$ is bounded.

2.1F1F2step 1.1algebra

For $h \neq 0$ , splitting the oriented integral at $t$ gives $a_{*} (t + h) = \exp (h / τ) (a_{*} (t) - \frac{1}{m τ} \int_{0}^{h} \exp (- v / τ) f (t + v) d v) .$ For each component, the average $h^{- 1} \int_{0}^{h} \exp (- v / τ) f_{j} (t + v) d v$ tends to $f_{j} (t)$ : its difference from $f_{j} (t)$ is bounded by $\sup_{∣ v ∣ \leq ∣ h ∣} ∣ \exp (- v / τ) f_{j} (t + v) - f_{j} (t) ∣$ , which tends to zero by continuity. Taking the difference quotient and using $\lim_{h \to 0} (\exp (h / τ) - 1) / h = 1 / τ$ from [F1] yields ${\dot{a}}_{*} = a_{*} / τ - f / (m τ)$ , so $m (a_{*} - τ {\dot{a}}_{*}) = f$ .

3.1A1F1F2F4step 2.1algebra

If $a$ is any other solution, $b : = a - a_{*}$ satisfies $\dot{b} = b / τ$ . For each coordinate, the product rule gives $(\exp (- t / τ) b_{j} (t))^{'} = 0$ . By [F4] its values at any two real times agree, so $\exp (- t / τ) b (t)$ is a constant vector. Therefore $b (t) = C \exp ((t - t_{0}) / τ)$ after absorbing the fixed factor $\exp (t_{0} / τ)$ into $C$ . Conversely, differentiating this expression verifies that every such $a_{*} + b$ solves the same equation.

4.1F1step 1.1step 3.1algebra

If $C \neq 0$ , at least one coordinate $C_{j} \neq 0$ ; as $t \to \infty$ , $∣ C_{j} ∣ \exp ((t - t_{0}) / τ) \to \infty$ by the lower bound $\exp (x) \geq 1 + x$ for $x \geq 0$ , while $a_{*, j}$ is bounded by step 1.1. Thus that coordinate of $a$ is unbounded on every future half-line. Step 1.1 shows that $C = 0$ gives a bounded solution, proving uniqueness of the bounded-future branch.

5.1F1F2F3F5step 2.1algebra∎

Apply [F5] in dimension one with $r = 1 / 4$ , $R = 1 / 2$ and compose its bump with $t \mapsto 4 (t - 1 / 2)$ . This gives $g \in C_{c}^{\infty} ((0, 1))$ , $g \geq 0$ , with $g = 1$ on $[7 / 16, 9 / 16]$ . Set $u = (1, 0, 0)$ and $f (t) = g (t) u$ . Then $f$ is bounded and smooth, vanishes for $t \leq 0$ , and is nonzero on a future interval. For $t < 0$ , $a_{*} (t) = \frac{\exp (t / τ)}{m τ} (\int_{0}^{1} \exp (- s / τ) g (s) d s) u .$ The scalar integrand is continuous and positive on $[7 / 16, 9 / 16]$ , so by continuity it has a positive lower bound on some nondegenerate subinterval there; [F2]–[F3] make the displayed integral strictly positive. Since $\exp (t / τ) > 0$ , $a_{*} (t) \neq 0$ for every $t < 0$ , before the applied force begins. This is a conditional ODE deduction, not an experimental result.

Runaway and bounded-future branches of the Abraham–Lorentz equation

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources