Co-ordinate system and it's rotation.

Co-ordinate system and its rotation:

Right handed and left handed co-ordinate system:

The co-ordinate system in which the reference line is taken as x-axis and rotating to anticlockwise direction is known as right handed co-ordinate system. But it reference line is taken as x-axis and rotated clockwise direction then such co-ordinate system is called left handed co-ordinate system. We use the convention of right handed co-ordinate system.

Transformation of co-ordinate:

Let us consider a co-ordinate system S(x,y,z) which has unit vectors \(\hat i,\hat j\) and \(\hat k\) and also has centrer at origin. Now, let us rotate S(x,y,z) to prime co-ordinate system by some angle to S'(x',y',z') having unit vectors \(\hat i', \hat j',\hat k'\) respectively. Let P(x,y,z) be the position vector in unprimed co-ordinate system which is represented by \(\vec r\).

Since, \(\vec r\) and \(\vec r'\) are same vectors.

So, \(\vec r=\vec r'\)

Or,\(x\hat i+y\hat j+z\hat k=x'\hat i'+ y'\hat j'+z'\hat k'\dotsm(A)\)

Now in S' – frame,

\(\vec A= A_x'\hat i'+A_y' \hat j'+A_z'\hat k'\dotsm(B)\)

Where, \(\hat i' ,\hat j' \) and \(\hat k'\) are unit vectors or new co-ordinate system.

Taking dot product by \(\hat i', \hat j'\) and \(\hat k'\) on both side of equation(B)

$$A_x'=\vec A\cdot\hat i'\dotsm(a)$$

$$A_y'=\vec A\cdot\hat j'\dotsm(a)$$

$$A_z'=\vec A\cdot\hat k' \dotsm(a)$$

Substituting value of (a) in (B) we get,

$$\vec A=(\vec A\cdot\hat i')\hat i' +(\vec A\cdot\hat j')\hat j'+ (\vec A\cdot \hat k')\hat k'$$

$$=(\hat i'\cdot \vec A)\hat i'+ (\hat j' \vec A)\hat j'+(\hat k'\cdot\vec A)\hat k'$$

This is valid for all vector so, substituting \(\hat i\) in above equation we get,

$$or,\; \hat i= (\hat i' \cdot \hat i)\hat i'+(\hat j'\cdot\hat i)\hat j'+(\hat k'\cdot \hat j)\hat k'$$

$$or,\; hat i= l_{11}\hat i'+ l_{21}\hat j'+l_{31}\hat k'$$

Similarly,

$$\hat j=l_{12} \hat i'+l_{22}\hat j'+ l_{32} \hat k$$

$$\hat k= l_{12} \hat i'+ l_{23}\hat j'+l_{33} \hat k$$

Substituting the value of \(\hat i, \hat j\) and \(\hat k\) in equation (A) and comparing the coefficient of \(\hat i',\hat j'\) and \(\hat k'\) we get,

$$x=l_{11} x'+l_{21}+l_{32}z'$$

$$y=l_{12}x'+l_{22}y'+l_{32}z'$$

$$z=l_{13}x'+l_{23}y'+l_{33}z'$$

Note: The complete set of transformation are,

$$\hat i'= l_{11}\hat i+l_{12}\hat j+l_{13}\hat k$$

$$\hat j'=l_{21}\hat i+ l_{22}\hat j+l_{23}\hat k$$

$$\hat k'=l_{31}\hat i+l_{32}\hat j+ l_{33}\hat k$$

$$\hat i=l_{11}\hat i' +l_{21}\hat j'+l_{31}\hat k'$$

$$\hat j= l_{21}\hat i'+ l_{22}\hat j'+l_{32}\hat k'$$

$$\hat k=l_{31}\hat i'+l_{32}\hat j'+l_{33}\hat k'$$

$$x'=l_{11} x+ l_{12}y+l_{12}z$$

$$y'=l_{21}x+l_{22}y+l_{23}z$$

$$z'=l_{31}x+l_{32}y+l_{33}z$$

Prove:

$$\sum_{p=1}^3 l_{pm}l_{pn}=1 if m=n$$

$$\sum_{p=1}^3 l_{pm}l_{pn}=0 if m\ne n$$

Solution: We know the transformation relation for unit vectors are,

$$\hat i= l_{11}\hat i'+l_{21}\hat j'+l_{31}\hat k'$$

$$\hat j= l_{12}\hat i'+l_{22}\hat j'+l_{32}\hat k'$$

$$\hat k=l_{12} \hat i' + l_{23} \hat j'+ l_{33} \hat k'$$

Now, \(\hat i\cdot \hat i=1\)

$$(l_{11}\hat i'+l_{21}\hat j'+l_{31}\hat k')\cdot( l_{11}\hat i'+l_{21}\hat j'+l_{31}\hat k')$$

$$or,\; l_{11}l_{11}+l_{21}l_{21}+l_{31}l_{31}=1$$

$$or,\; \sum_{p=1}^3 l_{p2} l_{p2}=1$$

$$or,\; sum_{p=1}^3 l_{p3} l_{p3}=1$$

Combining all three we get,

$$\sum_{p=1}^3 l_{pm}l{pn}=1\;\; If\; m=n$$

Now, \(\hat i\cdot\hat j=0\)

\(l_{11}\hat i+l_{21}\hat j+ l_{31}\hat k)+(l_{12}\hat i'+l_{22}\hat j'+l_{32}\hat k)=0\)

\(l_{11}l_{12}+l_{21}l_{22}+l{31}l_{32}=0\)

\(\sum_{p=1}^3 l_{pm}l_{pn}=0\) If \(m\ne n\)

Combining above

$$\sum_{p=1}^3 l_{pm}l_{pn}=1\;for\;m=n$$

$$\sum_{p=1}^3 l_{pm} l_{pn}=0\;for m\ne n$$

Show that \(\nabla\) is invariant during the rotation of axes.

Let us consider a co-ordinate system S(x,y,z) fixed at origin 0 having unit vector \(\hat i,\hat j\)and \(\hat l\). Let the co-ordinate system is rotated by some angle so that a new co-ordinate system S'(x',y',z') is formed with unit vectors \(\hat i',\hat j'\) and \(\hat k'\).

Then we know the form of \(\nabla\) in S co-ordinate system is,

$$\nabla=\hat i\frac{\delta}{\delta x}+\hat j\frac{\delta }{\delta y}+\hat k\frac{\delta}{\delta z}$$

Now, the transformation equation relating unit vectors and co-ordinates are

$$\hat i' = l_{11}\hat i+ l_{12}\hat j+l_{13}\hat k$$

$$\hat j'= l_{21}\hat i+l_{22}\hat j+ l_{32}\hat k$$

$$\hat k'=l_{31}\hat i+ l_{32}\hat j+l_{33}\hat k$$

And,

$$x'=l_{11} x+ l_{12}y+l_{12}z$$

$$y'= l_{21}x+l_{22}y+l_{23}z$$

$$z'=l_{31}x+ l_{32}y+l_{33}z$$

Now,

$$\frac{\delta}{\delta x}=\frac{\delta}{\delta x'}\frac{\delta x'}{\delta x}+\frac{\delta}{\delta y'}\frac{\delta y'}{\delta x}+\frac{\delta}{\delta z'}\frac{\delta z'}{\delta x}$$

$$=\frac{\delta}{\delta x'}l_{11}+\frac{\delta}{\delta y'}l_{21}+\frac{\delta}{\delta z'}l_{31}\dotsm(a)$$

Again,

$$\frac{\delta}{\delta y}=\frac{\delta}{\delta x'}\frac{\delta x'}{\delta y}+\frac{\delta}{\delta y'}\frac{\delta y'}{\delta y}+\frac{\delta}{\delta z'}\frac{\delta z'}{\delta y}$$

$$=\frac{\delta}{\delta x'}l_{12}+\frac{\delta}{\delta y'}l_{22}+\frac{\delta}{\delta z'}l_{32}\dotsm(b)$$

$$\frac{\delta}{\delta z}=\frac{\delta}{\delta x'}\frac{\delta x'}{\delta z}+\frac{\delta}{\delta y'}\frac{\delta y'}{\delta z}+\frac{\delta}{\delta z'}\frac{\delta z'}{\delta z}$$

$$=\frac{\delta}{\delta x'}l_{13}+\frac{\delta}{\delta y'}l_{23}+\frac{\delta}{\delta z'}l_{33}\dotsm(c)$$

Now multiplying equation (a) by \(\hat i\), (b) by \(\hat j\) and c by \(\hat k\).

$$\biggl(\hat i\frac{\delta }{\delta x}+\hat j\frac{\delta}{\delta y}+\hat k\frac{\delta}{\delta z}\biggr)=\frac{\delta }{\delta x'}(\hat i l_{11}+ \hat j l_{12}+\hat k l_{13})+ \frac{\delta}{\delta y}(\hat i l_{21} + \hat j l_{22}+\hat k l_{23})+\frac{\delta}{\delta z'}(\hat i l{31}+\hat j l_{32}+\hat k l_{33})$$

$$\nabla=\hat i' \frac{\delta }{\delta x'}+\hat j \frac{\delta}{\delta y'}+k'\frac{\delta }{\delta z'}$$

$$\nabla=\nabla'$$

So, \(\nabla\) s invariant during the rotation of axes.

Reference:

1. Adhikari, Pitri Bhakta, and Dya Nidhi Chhatkuli.A Textbook of Physics. Third Revised Edition ed. Vol. III. Kathmandu: SUKUNDA PUSTAK BHAWAN, 2072.
2. Vaughn, Michael T.Introduction to Mathematical Physics. Weinheim: Wiley-VCH, 2007
3. Butkov, Eugene.Mathematical Physics. Reading, MA: Addison-Wesley Pub., 1968.
4. Carroll, Robert W.Mathematical Physics. Amsterdam: North-Holland, 1988.
5. Dass, H. K.Mathematical Physics. New Delhi: S. Chand, 2005.