Mathematical Principles of Theoretical Physics

2.1. ESSENCE OF PHYSICS 41

1) Galilean transformation

(2.1.16) ̃t=t, ̃x=x+vt for x∈R^3 ,

where v is a constant velocity, and

2) translational and rotational transformations

(2.1.17) ̃t=t+t 0 , ̃x=Ax+b for x∈R^3 ,

where A∈SO( 3 )is an orthogonal matrix, b∈R^3 and t 0 ∈R^1 are constant.

By (2.1.12) we know that the invariance of physical laws is equivalent to the covariance
of the corresponding differential equations. We now demonstrate that the Newton Second
Law obeys the Galileo Invariance Principle2.7.
The Newton Second Law reads as

(2.1.18) ma=F,

and the corresponding differential equation describing the motion of a particle is given by

(2.1.19) m

d^2 x

dt^2

=F.

When we investigate the motion in other reference frame( ̃x, ̃t)with the transformation

(2.1.20) ̃t=t+t 0 , ̃x=Ax+vt+b,

by (2.1.20) we derive

(2.1.21)

d^2 ̃x

d ̃t^2

=A

d^2 x

dt^2

.

On the other hand, experiments show that the expressionsF ̃in( ̃x, ̃t)andFin(x,t)of force
satisfy the relation:

(2.1.22) F ̃=AF.

Hence, by (2.1.19) we deduce from (2.1.21) and (2.1.22) that

(2.1.23) m
d^2 ̃x
d ̃t^2

=F ̃.

It is clear that both forms of (2.1.19) and (2.1.23) are the same under the transformation
(2.1.20), i.e. under the transformations (2.1.16) and (2.1.17).
Mathematically, the transformations corresponding to (2.1.16) and (2.1.17) are the Galileo
group andSO( 3 ), and the functions satisfying (2.1.21) and (2.1.22) under the transformation
(2.1.20) are called first-order Cartesian tensors, which are also the usual vectors inR^3.
Also, this example demonstrates that invariance principles must be characterized by cor-
responding tensors. In the later sections, we shall give precise definitions of various types of
tensors.