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Future-boundary solution of the nonrelativistic Abraham–Lorentz equation

Physical scope

Exact solution and sign check for a separately stated one-dimensional nonrelativistic Abraham–Lorentz ODE. Pre-acceleration is a property of this future-boundary model solution, not an empirical finding or a conclusion about finite-body causal dynamics.

Example

Let $m > 0$ and $τ_{e} > 0$ . Let $T > 0$ and let $F_{e x t} \in C_{c}^{\infty} (R)$ be nonnegative, not identically zero, and supported in $[0, T]$ ; $C_{c}^{\infty}$ means smooth with compact support. Here $F_{e x t}$ is a scalar force history in newtons. Consider the separate nonrelativistic Abraham–Lorentz model equation for acceleration $a (t)$ ,

$a (t) = \frac{F_{e x t} (t)}{m} + τ_{e} \frac{d a}{d t} (t),$

where $τ_{e}$ has units of time. A candidate solution is a continuously differentiable map $a : R \to R$ that satisfies this equation pointwise for every $t \in R$ . Impose the future boundary condition $\lim_{t \to + \infty} a (t) = 0$ . Interpret the integral to $+ \infty$ as a finite Riemann integral after a truncation beyond the compact support, with the value independent of the truncation. The unique candidate solution satisfying that boundary condition is

$a (t) = \frac{1}{m τ_{e}} \int_{t}^{\infty} e^{(t - s) / τ_{e}} F_{e x t} (s) d s .$

It vanishes for $t \geq T$ and is strictly positive for every $t < 0$ . Thus this equation with this future boundary condition has pre-acceleration before the pulse support. The example concerns the stated ODE only; it is not a measurement and does not establish the behavior of finite charges.

Facts & Assumptions

Given: $m > 0$ , $τ_{e} > 0$ , the stated smooth compactly supported nonnegative pulse, and the separate ODE and future boundary condition above.

[A1]

Adopt only for this mathematical example the scalar equation $a = F_{e x t} / m + τ_{e} a^{'}$ and the future decay condition.

[F1]

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 and A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion: a continuous integrand on each compact interval has the integral regularity needed for the fundamental theorem of calculus.

[F2]

The exponential function is smooth and $(\exp)^{'} = \exp$ , 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 $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , and 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 $g (c) \neq 0$ : exponentials differentiate by the chain rule, and products obey the product rule.

[F3]

If $f \leq g$ on $[a, b]$ and both are integrable then $\int_{a}^{b} f \leq \int_{a}^{b} g$ ; and $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ and If $m \leq f \leq M$ on $[a, b]$ then $m (b - a) \leq L (f, P) \leq \underline{\int_{a}^{b}} f \leq \overline{\int_{a}^{b}} f \leq U (f, P) \leq M (b - a)$ for every partition $P$ ; in particular every constant function is integrable, with $\int_{a}^{b} c = c (b - a)$ : the integral of a nonnegative continuous function is nonnegative and bounded below by a positive lower bound times interval length.

[F4]

The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ : $e^{x} > 0$ and $e^{- x} = 1 / e^{x}$ .

[F5]

A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value: a continuous real function on a compact interval attains its minimum; if it is positive everywhere then that minimum is positive.

[F6]

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 $f (b) - f (a) = f^{'} (c) (b - a)$ and The exponential function is strictly increasing: the mean value theorem and positive derivative give strict increase of the real exponential.

[F7]

The exponential tends to $+ \infty$ at $+ \infty$ and to $0$ at $- \infty$ : $e^{x} \to 0$ as $x \to - \infty$ .

Verification

1.1givenA1F1F2algebra

Define $I (t) : = \int_{t}^{\infty} e^{(t - s) / τ_{e}} F_{e x t} (s) d s$ , interpreting the integral as a finite Riemann integral after any upper truncation beyond $\max {t, T}$ ; compact support makes the value independent of that truncation. For $t < T$ , write $I (t) = e^{t / τ_{e}} \int_{t}^{T} e^{- s / τ_{e}} F_{e x t} (s) d s$ . By [F1] the integrand is integrable on this compact interval, and [F2] together with the fundamental theorem gives $I^{'} (t) = - F_{e x t} (t) + \frac{1}{τ_{e}} I (t) (t < T) .$ For $a = I / (m τ_{e})$ this identity becomes $τ_{e} a^{'} = - F_{e x t} / m + a$ , which rearranges to the displayed Abraham–Lorentz equation. For $t > T$ , both $I$ and $F_{e x t}$ vanish, so the same equation holds. At $t = T$ , smoothness and support imply $F_{e x t} (T) = 0$ , and the one-sided derivatives agree with the displayed identity.

2.1givenA1F2F3F4F5step 1.1

For $t \geq T$ , the integration interval contains no point where $F_{e x t}$ is nonzero, so $a (t) = 0$ . For $t < 0$ , continuity and nonnegativity of the nonzero pulse give a closed subinterval of $(0, T)$ on which $F_{e x t} \geq d > 0$ . The exponential factor is positive and continuous there by [F2, F4]; its minimum is positive by [F5]. The integral over that subinterval is strictly positive by the lower bound [F3], and all remaining contributions are nonnegative. Thus $a (t) > 0$ . Since $a (t) = 0$ for $t \geq T$ , it satisfies the future boundary condition.

3.1A1F2F4F6F7step 2.1algebra∎

If $a_{1}, a_{2}$ are two solutions, their difference $d = a_{1} - a_{2}$ satisfies $d^{'} = d / τ_{e}$ . The product rule gives $(e^{- t / τ_{e}} d (t))^{'} = 0$ , so [F6] implies this product is a constant $C$ . By [F4], $d (t) = C e^{t / τ_{e}}$ ; if $C \neq 0$ , then [F7] makes $∣ d (t) ∣ \to \infty$ as $t \to + \infty$ , contradicting $d (t) \to 0$ . Hence $C = 0$ and $d = 0$ , proving uniqueness. Together with step 2.1, the unique future-decaying solution is strictly positive for all $t < 0$ : this pre-acceleration is a property of the stated model equation, not an empirical claim.

Future-boundary solution of the nonrelativistic Abraham–Lorentz equation

Example

Facts & Assumptions

Verification

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