## Dynamical Law of Physical Motion Systems

Tian Ma & Shouhong Wang, Dynamical Law of Physical Motion and Potential-Descending Principle, The Institute for Scientific Computing and Applied Mathematics Preprint #1701, July 6, 2017

This is the last of the three blog posts for the above paper. The first  was on postulating the potential-descending 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 constraint-infinitesimals.

Dynamical Law for isolated physical motion systems

1. For each isolated physical motion system, there are a set of state functions ${u=(u_1, \cdots, u_N)}$, describing the states of the system, and a potential functional ${F(u)}$, 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 ${{du}/ dt}$ should equal to the driving force derived from the potential functional ${F}$.
More precisely, we postulate the following dynamical law of physical motion:

$\displaystyle \frac{du}{dt } =- A \delta_{\mathcal L} F(u), \ \ \ \ \ (1)$

where ${A}$ is the coefficient matrix, and ${- \delta_{\mathcal L} F(u)}$ is the variation with constraint infinitesimals, representing the driving force of the physical motion system, and ${\mathcal L}$ 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 ${F}$. 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 energy-momentum conservation constraints. We refer interested readers to an earlier blog on PID, and the following book for details:

[Ma-Wang, MPTP] Tian Ma & Shouhong Wang, Mathematical Principles of Theoretical Physics, 524pp., Science Press, 2015

3. The linear operator ${\mathcal L}$ in the dynamic law (1) takes the form of a differential operator ${L}$ or its dual ${L^\ast}$. The constraints can be imposed either on the kernel ${\mathcal N^\ast}$ of the dual operator ${L^\ast}$ or on the range of the operator ${L}$, given as follows:

$\displaystyle \langle \delta_{L^\ast}F(u),v\rangle_{H}=\frac{\mbox{d}}{\mbox{d}t}\bigg|_{t=0} F(u+tv), \qquad \forall\ L^\ast v=0,$

$\displaystyle \langle \delta_{L}F(u),\varphi\rangle_{H_1}=\frac{\mbox{d}}{\mbox{d}t}\bigg|_{t=0} F(u+tL\varphi), \qquad \forall\ \varphi\in H_1.$

Using the orthogonal decomposition theorem below, we show that the above variations with constraint infinitesimals take the following form:

$\displaystyle \delta_{L^\ast}F(u)=\delta F(u)+Lp,$

$\displaystyle \delta_{L}F(u)=L^\ast\delta F(u),$

for some function ${p}$, which plays a similar role as the pressure in incompressible fluid flows. Here ${\delta F(u)}$ is the usual derivative operator.

4. As an example, we consider the compressible Navier-Stokes equations:

$\displaystyle \frac{\partial u}{\partial t} + (u \cdot \nabla) u = \frac{1}{\rho} \left[ \mu \Delta u - \nabla p + f\right],$

$\displaystyle \frac{\partial \rho}{\partial t} = -\text{div} \left(\rho u \right).$

Let the constraint operator ${L=- \nabla}$ be the gradient operator, with dual operator ${L^\ast =div}$.
Also, let the potential functional be

$\displaystyle \Phi(u, \rho) = \int_{\Omega}\left[\frac{\mu}{2}|\nabla u|^2 - fu+ \frac12 \rho^2 \text{div}u\right]\text{d} x.$

Then the above compressible Navier-Stokes equations are written as

$\displaystyle \frac{\text{d} u}{\text{d} t} = - \frac{1}{\rho} \frac{\delta_{L^\ast} }{\delta u} \Phi (u, \rho),$

$\displaystyle \frac{d \rho}{d t} = - \frac{\delta}{\delta \rho} \Phi(u, \rho),$

which is in the form of (1) with coefficient matrix ${A=\text{diag}(1/\rho, 1)}$. Also, the pressure is given by

$\displaystyle p=p_0 - \lambda \text{div} u - \frac12 \rho^2.$

Here

$\displaystyle \frac{\text{d} u}{\text{d}t} = \frac{\partial u }{\partial t} + \frac{\partial u}{\partial x_i} \frac{\text{d} x_i}{\text{d} t } = \frac{\partial u }{\partial t} + (u \cdot \nabla ) u, \qquad \frac{d\rho}{dt}= \frac{\partial \rho}{\partial t} + u\cdot \nabla \rho.$

5. There are two types of physical motion systems: the dissipative systems and the conservation systems. The coefficient matrix ${A}$ is symmetric and positive definite if and only if the system is a dissipative system, and ${A}$ is anti-symmetry 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 [Ma-Wang, 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 ${F}$ obey certain symmetries such as ${SO(n)}$ (${n=2, 3}$) 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 [Ma-Wang, MPTP] the principle of symmetry-breaking (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

$\displaystyle \frac{\text{d} u}{\text{d}t} = -A\delta_{\mathcal L} F(u) + B(u), \ \ \ \ \ (2)$

where ${B(u)}$ represents the symmetry-breaking.

Orthogonal-Decomposition 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 ${L:H_1 \rightarrow H}$ between two Hilbert spaces ${H}$ and ${H_1}$, with dual operator ${L^\ast}$, any ${u \in H}$ then can be decomposed as

$\displaystyle u=L\varphi+v,\quad L^\ast v=0,$

where ${L\varphi}$ and ${v}$ are orthogonal in ${H}$.

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 [Ma-Wang, 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 potential-descending principle (PDP) addressed in the previous blog shows that non-equilibrium 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 [Ma-Wang, 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:

$\displaystyle \text{ physical laws } = \text{ mathematical equations} \ \ \ \ \ (3)$

• 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