Mathematical Physics Vol 1

Chapter 4. Field theory

108

Give that x = const , we further obtain

1 x Z

− π 2

2 k x

cos α d α = −

X = − k

(4.122)

.

π 2

+

We can now look for the potential. If the potential U exists, it must satisfy the relation

∂ U ∂ x

2 k x

X =

(4.123)

= −

,

from where we finally obtain

U = − 2 k ln x ili U = 2 k ln 1 x .

(4.124)

Theorem10 An arbitrary vector field F , unambiguous, continuous, and bounded, can be decomposed into the sum of a potential and an irrotational vector field in the form F = − ∇ U + ∇ × A , where ∇ · A = 0. The scalar function U is called the scalar potential , and the vector function A is called the vector potential of the vector field F . This theorem is known in literature as Helmholtz’s 16 theorem. We have stated the theorem because of its importance. However, as the proof is relatively complex, we will not present it, but rather refer the reader to references: [ 4 ] (p. 79), [ 29 ] (p. 50). 4.4 Generalized coordinates The position of a point in three-dimensional Euclidean space is determined in relation to a predetermined point O , which is called the pole or origin, by a position vector r . In a Cartesian rectangular coordinate system Oxyz , with origin in pole O , the position of a point is determined by the point coordinates ( x , y , z ) . Orthogonal projections of the end of the position vector on the axes of this coordinate system coincide with the coordinates of the point, and thus the coordinates of the position vector r coincide with these coordinates x , y , z : r = { x , y , z } or r = x i + y j + z k . (4.125) However, the position of a point in space can also be determined using some three mutually inde pendent parameters q 1 , q 2 , q 3 (1, 2 and 3 here are not powers but rather parameter designations !!!), or shorter q i ( i =1,2,3). When the parameter q i receives all possible values, and one and only one ordered set of three numbers ( q 1 , q 2 , q 3 ) corresponds to each point of space, and conversely, one and only one point in space (three-dimensional) corresponds to each set of thw three numbers ( q 1 , q 2 , q 3 ) , then the parameters q i are called general or generalized coordinates of the point. The position vector can now be represented by the generalized coordinates: r = r ( q 1 , q 2 , q 3 ) , or shortly r = r ( q i ) . (4.126) 16 Hermann von Helmholtz (1821–1894), German physicist. He is known for very important works in the field of thermodynamics, hydrodynamics and acoustics.