Mathematical Principles of Theoretical Physics

36 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS

2.3.1 Principle of general relativity.

2) the principle of equivalence, and

2.5 Principle of Lagrangian Dynamics (PLD)

The Schwarzschild solution of the Einstein gravitational field equations (2.1.4) offers a natu-
ral link between the field equations and the Newtonian gravitational law (2.1.1).

2.1.3 Fundamental principles in physics

Fundamental first principles refer to the laws of Nature thatcannot be derived from other
more basic laws.
Based on Essence of Physics2.2, if we would like to better understand theoretical physics,
it is crucial to know and find out all fundamental laws. In thissubsection, we shall list the
known and important principles in various physical fields, most of which will be introduced
in later chapters.
We start with the introduction of Principle of Lagrangian Dynamics (PLD), which is of
special importance.
In classical mechanics, we know the least action principle.For a mechanical system with
the position and velocity variables

(2.1.5) x= (x 1 ,···,xn), x ̇= (x ̇ 1 ,···,x ̇n),

let the kinetic energyTand the potential energyVbe functions of the position and velocity
variables in (2.1.5). Then the states of the system are the extremum points of thefunctional

(2.1.6) L(x) =

∫t 1

t 0

L(x,x ̇)dt,

where the integrandL

(2.1.7) L(x,x ̇) =T−V.

is called the Lagrange density, and the functional (2.1.6) is called the Lagrange action. The
extremum pointsxsatisfy the variational equation, called the Euler-Lagrange equations of
(2.1.6):

(2.1.8) δL(x) = 0 ,

whereδLis the variational derivative operator ofL. The equation (2.1.8) can be equivalently
expressed as
d
dt

∂L

∂x ̇

−

∂L

∂x

= 0.

The most important point is that the least action principle can be generalized to all phys-
ical systems describing motions, as well as to fundamental interactions of Nature. The gen-
eralization in a motion system is called the PLD, and in an interaction system is called the
Principle of Interaction Dynamics (PID). PID will be introduced in detail In Chapter 4 of this
book, and PLD is stated as follows.