Mathematical Principles of Theoretical Physics

2.5. PRINCIPLE OF LAGRANGIAN DYNAMICS (PLD) 89

2.5.5 Symmetries and conservation laws

The importance of PLD lies in the following three points:

1) Physics are established based on a few universal principles, which provides a solid

foundation for physics;

2) Based on PLD, many physical problems become simpler. In particular, by means of

invariance it is easier to find the Lagrange actions than to seek for the differential

equations; and

3) Lagrange actions contain more physical information thanthe differential equations. In

fact, a conservation law of a physical system can be derived from the invariance of the

Lagrange action under the associated symmetric transformation.

The correspondencebetween symmetries and conservation laws are revealed by the Noether
theorem, to be introduced below. For this purpose, we need tointroduce some related con-
cepts on group action and symmetry.
We begin with a simple example. A circle with radiusris described by

(2.5.43) x^2 +y^2 =r^2.

x

y

x′

y′

Figure 2.2

We can clearly see the symmetry of the circle shown in Figure2.2—the graph is the same
from whatever the direction we look at it. This phenomenon isexpressed in mathematics as
the invariance of the equation (2.5.43) under the following coordinate transformation

(2.5.44)

(

x′

y′

)

=

(

cosθ sinθ

−sinθ cosθ

)(

x

y

)

.

Namely, in the coordinate system(x′,y′), the equation describing the circle is invariant and
still takes form:

(2.5.45) x′^2 +y′^2 =r^2.