Manuscript Em02-E@1 · Em02-E-v1.tex · lemma · f3795050

Lemma: Clock displacement

Clock slices and harmonic spatial labels › Clock-adapted description · lines 412–437 · proof included or linked

Em02-E@1#lem:clock, Em02-E@1#eq:u

The unique formal solution of the clock equation is

u_{1} = - \frac{φ}{v}, u_{2} = - \frac{ψ + 2 u _{1} φ ˙ + v ˙ u _{1} ^{2}}{v} .

All quantities on the right are evaluated at $(s, x)$ .

Proof. Expand $ϕ_{λ} (τ_{λ}, x)$ through second order:

ϕ_{0} (s) + λ (v u_{1} + φ) + \frac{λ ^{2}}{2} (v u_{2} + v ˙ u_{1}^{2} + 2 u_{1} φ ˙ + ψ) .

Setting the coefficients to zero proves the result because $v \neq = 0$ .

Displayed formulas in this passage

Em02-E@1#eq:u · line 415
$u_{1} = - \frac{φ}{v}, u_{2} = - \frac{ψ + 2 u _{1} φ ˙ + v ˙ u _{1} ^{2}}{v} .$
unlabeled · line 429
$ϕ_{0} (s) + λ (v u_{1} + φ) + \frac{λ ^{2}}{2} (v u_{2} + v ˙ u_{1}^{2} + 2 u_{1} φ ˙ + ψ) .$