Ahlan wa Sahlan.
Excuse me ... .
I have a question about scalar quantity ... .
Can a scalar be regarded as a one-dimensional vector ... ?
For example, the continuum electric charge $q$ is a scalar quantity which can be regarded to have two directions in real line $\mathbb{R}$, namely, the positive direction ($q>0$) and the negative direction ($q<0$) ... .
In Lagrangian Mechanics, an electrical system can be analogized as a mechanical system, with the electric charge $q$ as a generalized co-ordinate, and $I:=\dot{q}:=dq/dt$ as a generalized velocity ... .
Suppose that here is an electrical circuit which consists of an inductor of inductance $L$ constant and a voltage source $\varphi$ as function of time $t$ ... .
The Lagrangian of such system is $\mathcal{L}:=\frac{1}{2}L\dot{q}^2+q\varphi$ ... .
The Euler-Lagrange’s equation of this system is
$\displaystyle\frac{\partial\mathcal{L}}{\partial{q}}=\frac{d}{dt}\frac{\partial\mathcal{L}}{\partial\dot{q}}$
as its equation of motion ... .
Thus, the Newton’s second law for this electrical circuit is $\varphi=L\ddot{q}\equiv\,L\,dI/dt$, where $L$ can be analogized as constant mass, and $\varphi$ can be analogized as time-dependent-force ... .
Thank you for the answer ... .
Sekian dan terima kasih.
