Mathematical Principles of Theoretical Physics

2.1. ESSENCE OF PHYSICS 35

2.1.2 Phenomenological methods

Theories of physics fall into two categories: first principle theories and phenomenological
theories. The first principle theories are derived based on afew fundamental laws and princi-
ples of Nature, and phenomenological theories are conjectured and synthesized from obser-
vational data.
For example, the Newton’s gravitational law is given by

(2.1.1) F=−

m 1 m 2 G

r^2

.

which is essentially deduced by using the phenomenologicaltechnique based on a large num-
ber of astronomical data. First, one readily conjectures that the gravitational force is propor-
tional to the massesmandMof the two bodies, and is a function of distancer:

F=−mMΦ(r),

whereΦ(r)is an undetermined function.
Then one considers two planets with massesm 1 andm 2 , rotating around the Sun with
velocitiesv 1 andv 2. By the balance between the gravitational and centrifugal forces, we have

v^2 i

ri

=MΦ(ri) fori= 1 , 2 ,

whereMis the mass of the Sun. Consequently,

(2.1.2)

Φ(r 1 )

Φ(r 2 )

=

r 2

r 1

v^21

v^22

.

By the Kepler Third Law, the periodsT 1 andT 2 of the two planets satisfy

T 12

T 22

=

r 13

r 23

, Tivi= 2 πri fori= 1 , 2 ,

which implies that

(2.1.3)

v^21

v^22

=

r 2

r 1

.

Then, it follows from (2.1.2) and (2.1.3) that

Φ(r) =

G

r^2

,

whereGis a constant, called the gravitational constant. Thus, by the Kepler Third Law, one
can postulate naturally the Newtonian gravitational law (2.1.1). In other words, the Newto-
nian Gravitational Law can be regarded as a phenomenological theory.
Albert Einstein was the first who tried to deduce basic physical laws from first principles.
For example, the Einstein theory of general relativity and the Einstein gravitational field
equations:

(2.1.4) Rμ ν−

1

2

gμ νR=−

8 πG

c^4

Tμ ν,

are derived based on the following three basic principles: