Mathematical Principles of Theoretical Physics

2.1. ESSENCE OF PHYSICS 39

b) By (2.1.12), the universality of laws of Nature implies that the differential equations

representing them are covariant. Equivalently the Lagrange actions are invariant under

the corresponding coordinate transformations.

As an example, consider a physical system obeying PLD2.3. For simplicity, we assume
that the Lagrange action of this fictitious system is finite-dimensional. Namely, let

F:Rn→R^1

be a finite dimensional function regarded as the action of thephysical system, and we need
to determine the explicit form of this function

(2.1.13) F=F(x) for x= (x 1 ,···,xn)∈Rn.

Consider the case where the system satisfies the following invariance principle, called the

2.3.2 Principle of equivalence

Principle 2.6(Rotational Invariance).The laws that the system (2.1.13) obeys are the same
under all orthogonal coordinate systems.

Based on the implication b) of symmetry above, Principle of Rotational Invariance2.6
says that the action (2.1.13) is invariant under orthogonal coordinate transformations:

(2.1.14) x ̃=Ax whereAis an arbitrary orthogonal matrix.

We know that the two forms as

|x|^2 =x^21 +···+x^2 n, x·y=x 1 y 1 +···+xnyn

are invariant, wherey∈Rnis given, and the invariant functionF(x)must be of the form:

F(x) =F(|x|,x·y)

In addition, by the simplicity principle in Essence of Physics2.2, the exponent ofFinxis
two. Thus the action (2.1.13) is defined in the form as

(2.1.15) F(x) =α|x|^2 +βx·y,

whereα,βare two constants, andyis a given vector.
Thus we deduce an explicit expression (2.1.15) for the functionF, which are invariant
under rotational symmetry, Principle2.6. The graph of (2.1.15) is a sphere, and the invariance
of (2.1.15) means that the sphere looks the same from different directions. This is the reason
why an invariance is called a symmetric principle.

2.1.5 Invariance and tensors

Symmetries are characterized by three ingredients:

spaces, transformation groups, and tensors,