Suppose we have $2$ vector valued functions of time,$\vec R(t)$ and $\vec r(t)$. We can represent those functions as:- $$ \begin{split} \vec R(t)&=\sum R_i(t)\hat I \\ \vec r(t)&=\sum r_i(t)\hat I \end{split} $$ Furthermore assume that the following relation between them holds:- $$ \vec R\left(t+\frac{|\vec r(t)|}{c}\right)=\vec r(t) $$ where $c$ is a constant. This relation implies the following scalar equations for each vector component:- $$ R_i\left(t+\frac{|\vec r(t)|}{c}\right)=r_i(t)\quad i= 1, \ldots, n. $$ Here $r_i(t)$ is expressed as a function of $t$: my goal is to express $R_i$ as a function of $t$ too. And in order to achieve my objective I applied Taylor's theorem to both sides of the equation, but this tentative was unsuccessful. Can someone help me get an expression for $R_i(t)$ as a function of $t$ only?
1 Answer
I seem to have found the way out.I will first express: $$R_i(t)=R_i(t+\frac{|\vec r(t)|}{c}-\frac{|\vec r(t)|}{c})=R_i(t'-\frac{|\vec r(t)|}{c})$$ Where: $$t'=t+\frac{|\vec r(t)|}{c}$$
Therefore we are now able to apply the taylor series usefully: $$R_i(t)=R_i(t'-\frac{|\vec r(t)|}{c})$$ Meanwhile let's first express some long expressions by single variables.At the end we will replace them by the real expressions again: $$\frac{|\vec r(t)|}{c}=a(t)$$ $$b(t)=\frac{dt}{dt'}=\frac{1}{(\frac{dt'}{dt})}=\frac{1}{1+\frac{\dot {\vec r(t)}\cdot \vec r(t) }{c|\vec r(t)|}}$$ And the operator $$\hat O=b(t)\frac{d}{dt}$$ Now finally the taylor series: $$R_i(t)=\sum_{n=0}^\infty (\frac{d^n}{dt'^n}R_i(t'))\frac{a^n(-1)^n}{n!}$$ $$R_i(t')=r_i(t)$$ $$R_i(t)=\sum_{n=0}^\infty (\frac{d^n}{dt'^n}r_i(t))\frac{a^n(-1)^n}{n!}$$ Now we need deal with the derivative term: $$\frac{d^0}{bt'^0}r_i(t)=r_i(t)$$ $$\frac{d^1}{bt'^1}r_i(t)=(\frac{d}{dt}r_i(t))(\frac{dt}{dt'})=g(t)\frac{d}{dt}r_i(t)=\hat O r_i(t)$$ $$\frac{d^2}{dt'^2}r_i(t)=\frac{d}{dt'}\frac{d}{dt'}r_i(t)=(\frac{d}{dt}(\hat O r_i(t)))(\frac{dt}{dt'})=\hat O^2 r_i(t)$$ $$\frac{d^n}{dt'^n}=\hat O^n$$ Therefore $$R_i(t)=\sum_{n=0}^\infty (\hat O^n r_i(t))\frac{(-1)^n}{n!}\frac{|\vec r(t)|^n}{c^n}$$ Where $$\hat O=b(t)\frac{d}{dt}$$