Scalar as A One-Dimensional Vector

[trfrm]

Terpujilah Kristus.

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 ... .

Terpujilah Kristus.

[trfrm]

Hosana in excelcis.

In the general case, the magnitude of a vector, in my mind, should depend on the coordinate system chosen, even in one dimension.

Thanks ... .   In my mind, the charge plays a role as a “vector” in “charge-space” but not in configuration-space ... . The charge plays a role as a scalar in configuration space ... .

In the “charge-space”, the magnitude of “vector charge” depends on choosing "charge co-ordinate system" ... . But, in the configuration space, the value of the charge (as a scalar) does not depend on choosing co-ordinate system ... .

The “charge-space” is different from the configuration space ... , as “momentum-space” which is part of “classical-phase-space” ... .

I’m sorry if my opinion was wrong ... .

http://www.thescienceforum.com/physics/35119-scalar-one-dimensional-vector.html#post411784

Gloria in excelsis Deo.