Mathematical Principles of Theoretical Physics

38 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS

• Basic Laws of Thermodynamics,


• Le Chˆaterlier Principle.

5) Nonlinear sciences:

• Principle of Phase Transition Dynamics.

A couple of remarks are now in order.

Remark 2.4.Because various conservation laws can be deduced by PHD, PLDand sym-
metries based on the Noether theorem, these laws are not listed in the fundamental princi-
ples.

Remark 2.5.Mathematical models form the skeleton of theoretical physics, and most, if not
all, mathematical models (differential equations) in theoretic physics can be derived based on
the principles listed above. One of the ultimate goals of this book is to derive physical models
based on a first principles.

2.1.4 Symmetry

Symmetry plays an important role in physics. We start with intuitively examining how sym-
metry works, keeping in mind the relation between equationsand laws of physics in First
Principle2.1, which can be simply recast as

(2.1.12) Laws of Physics = Differential Equations.

The laws of Nature on the left hand side of (2.1.12) are often beyond words, and are
best expressed by differential equations. It is this characteristic, together with the Noether
theorem, that illustrates the importance of symmetry in physics, as illustrated below:

Invariance Covariance Symmetry

⇓ ⇓ ⇓

Form of Equation Space Structure Conservation Law

Namely, symmetry dictates and determines

1) the explicit form of differential equations governing the underlying physical system,

2) the space-time structure of the Universe, and the mechanism of fundamental interac-

tions of Nature, and

3) physical conservation laws.

We now give a simple example to demonstrate how symmetry determines the explicit
form of equations. We begin with two basic implications of a symmetry:

a) Fundamental laws of Nature are universal, and their validity is independent of the

space, time, and directions of experiments and observations, and