This is the last of the three blog posts for the above paper. The first was on postulating the potentialdescending principle (PDP) as the first principle of statistical physics, and the second was on irreversibility and the problems in the Boltzmann equation.
This blog focuses on the general dynamic law for all physical motion systems, and on the new variational principle with constraintinfinitesimals.
Dynamical Law for isolated physical motion systems
1. For each isolated physical motion system, there are a set of state functions , describing the states of the system, and a potential functional , representing a certain form of energy intrinsic to the underlying physical system. An important basic ingredient of modeling the underlying physical system is to determine the state functions and the potential functional.
Then it is physically clear that the rate of change of the state functions should equal to the driving force derived from the potential functional .
More precisely, we postulate the following dynamical law of physical motion:
where is the coefficient matrix, and is the variation with constraint infinitesimals, representing the driving force of the physical motion system, and is a differential operator representing the infinitesimal constraint.
2. We show that proper constraints should be imposed on the infinitesimals (variation elements) for the variation of the potential functional . These constraints can be considered as generalized energy and/or momentum conservation laws for the infinitesimal variation elements. The variation under constraint infinitesimals is motivated in part by the recent work of the authors on the principle of interaction dynamics (PID) for the four fundamental interactions, which was required by the dark energy and dark matter phenomena, the Higgs fields, and the quark confinement. Basically, PID takes the variation of the Lagrangian actions for the four interactions, under energymomentum conservation constraints. We refer interested readers to an earlier blog on PID, and the following book for details:
3. The linear operator in the dynamic law (1) takes the form of a differential operator or its dual . The constraints can be imposed either on the kernel of the dual operator or on the range of the operator , given as follows:
Using the orthogonal decomposition theorem below, we show that the above variations with constraint infinitesimals take the following form:
for some function , which plays a similar role as the pressure in incompressible fluid flows. Here is the usual derivative operator.
4. As an example, we consider the compressible NavierStokes equations:
Let the constraint operator be the gradient operator, with dual operator .
Also, let the potential functional be
Then the above compressible NavierStokes equations are written as
which is in the form of (1) with coefficient matrix . Also, the pressure is given by
Here
5. There are two types of physical motion systems: the dissipative systems and the conservation systems. The coefficient matrix is symmetric and positive definite if and only if the system is a dissipative system, and is antisymmetry if and only if the system is a conservation system.
Dynamical Law for Coupled Physical Motion Systems
Symmetry plays a fundamental role in understanding Nature. In [MaWang, MPTP], we have demonstrated that for the four fundamental interactions, the Lagrangian actions are dictated by the general covariance (principle of general relativity), the gauge symmetry and the Lorentz symmetry; the field equations are then derived using PID as mentioned earlier.
For isolated motion systems, all energy functionals obey certain symmetries such as () symmetry. In searching for laws of Nature, one inevitably encounters a system consisting of a number of subsystems, each of which enjoys its own symmetry principle with its own symmetry group. To derive the basic law of the coupled system, we postulated in [MaWang, MPTP] the principle of symmetrybreaking (PSB), which is of fundamental importance for deriving physical laws for both fundamental interactions and motion dynamics: Physical systems in different levels obey different laws, which are dictated by their corresponding symmetries. For a system coupling different levels of physical laws, part of these symmetries must be broken.
In view of this principle, for a system coupling different subsystems, the motion equations become
where represents the symmetrybreaking.
OrthogonalDecomposition Theorem
To establish the needed mathematical foundation for the dynamical law of physical motion systems, we need to prove an orthogonal decomposition theorem, Theorem~6.1 in the paper. Basically, for a linear operator between two Hilbert spaces and , with dual operator , any then can be decomposed as
where and are orthogonal in .
Summary
The dynamical law given by (1) and (2) is essentially known for motion system in classical mechanics, quantum mechanics and astrophysics; see among others [MaWang, MPTP] and [Landau and Lifshitz, Course of theoretical physics, Vol. 2, The classical theory of fields].
Thanks to the variation with infinitesimal constraints, the law of fluid motion is now in the form of (1) and (2).
The potentialdescending principle (PDP) addressed in the previous blog shows that nonequilibrium thermodynamical systems are governed by the dynamical law (1) and (2), as well.
In a nutshell,
the dynamical law (1) and (2) are the law for all physical motions systems.
We end this blog by emphasizing that in deriving dynamical law and the basic laws for the four fundamental interactions [MaWang, MPTP], the following guiding principle of physics played a crucial role:

The heart of physics is to seek experimentally verifiable, fundamental laws and principles of Nature. In this process, physical concepts and theories are transformed into mathematical models:

the predictions derived from these models can be verified experimentally and conform to reality.
The true understanding of (3) is a subtle process, and is utterly important.
Tian Ma & Shouhong Wang