Mathematical Physics Vol 1

4.6 Examples

149

4.6.5 Invariant

Exercise 76 Two cartesian coordinate systems xyz and x ′ y ′ z ′ , with a common coordinate origin, rotate relative to each other. Determine the connections between the coordinates of an arbitrary point P expressed in relation to these two coordinate systems (coordinate transformations).

Solution Let r and r ′ be the position vectors of point P in these two systems. Given that these coordinate systems have the same origin, it follows that OA = r = r ′ , and expressed by their components (in relation to the coordinate systems) x ′ i ′ + y ′ j ′ + z ′ k ′ = x i + y j + z k . (4.178) If we multiply equation (4.178) alternately with i ′ , j ′ , k ′ , we obtain

x ′ = ℓ 11 x + ℓ 12 y + ℓ 13 z , y ′ = ℓ 21 x + ℓ 22 y + ℓ 23 z , z ′ = ℓ 31 x + ℓ 32 y + ℓ 33 z ,

(4.179)

where ℓ 11 = i ′ · i , ℓ 12 = i ′ · j , . . . , ℓ 33 = k ′ · k .

Exercise 77

Prove that

i ′ = ℓ 11 i + ℓ 12 j + ℓ 13 k , j ′ = ℓ 21 i + ℓ 22 j + ℓ 23 k , k ′ = ℓ 31 i + ℓ 32 j + ℓ 33 k .

Solution Each vector v ′ in the system S ′ can be expressed by means of unit vectors i , j and k of the system S as follows v ′ = α i + β j + γ k , which is also valid for the unit vectors of the system S ′ , namely i ′ = α 1 i + β 1 j + γ 1 k , (4.180)

j ′ = α 2 i + β 2 j + γ 2 k , k ′ = α 3 i + β 3 j + γ 3 k .