Mathematical Principles of Theoretical Physics

2.1. ESSENCE OF PHYSICS 37

Principle of Lagrangian Dynamics 2.3.For a physical motion system, there are functions

(2.1.9) u= (u 1 ,···,un),

which describe the states of this system, and there exists a functional of u, given by

(2.1.10) L(u) =

∫

Ω

L(u,Du,···,Dmu)dx,

whereΩis the domain of u, and Dku is the k-th derivative of u for any 0 ≤k≤m. Then the
state functions of this system are the extremum points of (2.1.10). Namely the state functions
satisfy the variational equation of (2.1.10):

(2.1.11) δL(u) = 0.

The functional L is called the Lagrange action, andLis called the Lagrange density.

By PLD, to derive dynamical equations for a physical system,it suffices to find the corre-
sponding Lagrange action. In the next subsection, we demonstrate that symmetries in physics
dictate the precise forms of the Lagrange actions.
A list of known important physical principles and laws of Nature in various subfields of
physics is given as follows.

1) Universal principles in all fields:

2.6 Principle of Hamiltonian Dynamics (PHD).

• Principle of Hamiltonian Dynamics (PHD),


• Principle of Interaction Dynamics (PID),


• Lorentz Invariance,


• Principle of General Relativity,

2.4.4 Principle of gauge invariance.

• Principle of Representation Invariance (PRI),

2.1.7 Principle of symmetry-breaking

2) Classical mechanics:

• Newtonian Second Law,


• Principle of Least Action,


• Principle of Galilean Invariance,


• Fick and Fourier Diffusion Laws.

3) Quantum physics:

6.5.2 Basic postulates forN-body quantum physics.

• Pauli Exclusion Principle.

4) Statistical physics: