To capture the dynamics of a system, constraints are put forward to form a closed set of equations:

  1. First the quantities describing system need to be consistent;
  2. Then the idea of conservation laws should hold by logic, or belief;
  3. Consistency and compatibility conditions may be necessary beyond conservation laws.
  4. The system of equations comes to a closure when we bridge the gap with constitutive relations, most of which appeared in natural phenomena are linear (approximations).

Conservation Laws and Continuity Equations

Continuity equation represents a conservation law: $\partial_\mu j^\mu=0$, where $j^\mu$ as a conserved current.

Table 1: Pairs of conserved currents and corresponding continuity equations

Conserved current Continuity equations
Mass flux ${\partial \rho \over \partial t} + \nabla \cdot (\rho \mathbf{u}) = 0$
Energy flux $\nabla \cdot \mathbf{q} + \frac{ \partial u}{\partial t} = 0$
Electric current $\nabla \cdot \mathbf{J} = - {\partial \rho \over \partial t}$
Probability current $\nabla \cdot \mathbf{j} + \frac{\partial \lvert \Psi \rvert^2}{\partial t} = 0$
Current of particles in phase space $\frac{\partial\rho}{\partial t}+\nabla \mathbf{J}=0$, where $J = (\rho\dot{q}^i,\rho\dot{p}_i)$ and $\rho = \rho(q^i, p_i)$

Justification of prevalence of conservation laws:

  1. 守恒律最基本,可以认为普遍存在;
  2. 这几个方程联系的变量是大部分问题的主要影响因素;
  3. 方程个数要和未知数个数匹配,当所研究问题的主要影响参量确定时,用于联系这几个参量的方程数就被限定了, 额外的方程(关系)只可能给已封闭系统引入新的变量,否则就出现矛盾了。

Equations of Motion and Equilibrium

By introducing force, we could write time derivative of (conserved) quantities as equations of motion.

Table 2: Equations of Motion and Equilibrium in Mechanics

Field of study Equations of motion/equilibrium
Classic mechanics $\frac{d p }{d t} = F$; $\frac{\mathrm{d}\mathbf{L}}{\mathrm{d}t} = \boldsymbol{\tau}$; $\text{d} T = \text{d} W_{ext} + \text{d} W_{int}$
Elasticity $\boldsymbol{\nabla}\cdot\boldsymbol{\sigma} + \mathbf{F} = \rho\ddot{\mathbf{u}}$
Fluid mechanics $\rho\frac{D\mathbf{v}}{D t} = \nabla\cdot\mathbb{P} + \rho\mathbf{f}$

Some dismiss the concept of force.

Because force is the time derivative of momentum, the concept of force is redundant and subordinate to the conservation of momentum, and is not used in fundamental theories.

While arguments also go against the perception of quantity conservation.

The discovery of the Second Law of Thermodynamics by Carnot in the 19th century showed that every physical quantity is not conserved over time. ... Hence, a "steady-state" worldview based solely on Newton's laws and the conservation laws does not take entropy into account.

Consistency and Compatibility Conditions

Reynolds Transport Theorem in continuum mechanics: (a generalization of the Leibniz integral rule, i.e. differentiation under the integral sign.)

$$\frac{\mathrm{d}}{\mathrm{d}t}\int_{\tau}{\varphi}~{\text{d}\tau} = \frac{\partial}{\partial t} \int_{\text{CV}}{ \varphi ~{\text{d}\tau} } + \int_{\text{CS}} \varphi ({\mathbf{v}}^{r}\cdot {\mathbf{n}}) ~{\text{d}A}$$

Consistency and compatibility conditions in elasticity:

  1. Geometric relations (strain-displacement equations): $\boldsymbol{\varepsilon} =\tfrac{1}{2} \left[ \boldsymbol{\nabla}\mathbf{u}+\mathbf{u} \boldsymbol{\nabla}\right]$
  2. Saint-Venant's compatibility condition: $\nabla \times \Gamma \times \nabla = 0$

Constitutive Relations


Polarization $P$ is the electric dipole moment per unit volume. Magnetization $M$ is the magnetic dipole moment per unit volume. The relations between electromagnetic fields and polarization or magnetization depend on how the dipoles respond to the applied fields. When the applied fields are weak, to first order approximation, the dipole moments will have a linear relation to the field.

Linear relations between polarizations and electromagnetic fields:

  1. Polarization and electric field: $\mathbf{P} = \varepsilon_0 \chi_e \mathbf{E}$
  2. Magnetization and magnetic field: $\mathbf{M} = \chi_m \mathbf{H}$

Here, $\varepsilon_0$ is the electric constant, aka vacuum permittivity; $\chi_e$ is electric susceptibility; $\chi_m$ is magnetic susceptibility; $\mathbf{H}$ is the auxiliary magnetic field, defined by $\mathbf{B}=\mu_0(\mathbf{H + M})$.

Transport Phenomena

Several laws describe the transport of matter. In each case they read, "flux (density) is proportional to a gradient, the constant of proportionality is the characteristic of the material."

Table 3: Linear Constitutive Relations of Transport Phenomena

Name Expression
Fick's law of diffusion $\mathbf{J} = -D\nabla \phi$
Newton's law of viscosity $\tau = \mu \frac{\partial u}{\partial y}$
Fourier's law of thermal conduction $\mathbf{q} = - k {\nabla} T$
Ohm's law of electric conduction $\mathbf{J} = \sigma \mathbf{E}$
Darcy's law for porous flow $q = -\frac{k}{\mu} \nabla P$

Continuum Mechanics

Field/Subject Constitutive Relations
fluid mechanics (Newtonian fluid) $\tau_{ij} = 2\mu (s_{ij} - \tfrac{1}{3} s_{kk} \delta_{ij})$
linear elasticity (tensor and isotropic form) $\boldsymbol{\sigma} = \mathsf{C}:\boldsymbol{\varepsilon}$; $\sigma_{ij} = \lambda \varepsilon_{kk} \delta_{ij} + 2\mu \varepsilon_{ij}$

Dynamic (shear) viscosity $\mu$ is equivalent to shear modulus $\mu$; in the sense of linear isotropic materials both of which are Lamé's second parameters. The elimination of second viscosity (bulk viscosity) is called Stokes assumption.

Plasticity. Several nonlinear constitutive relations are presented, either in functional form or differential form.

Governing Equations

Eventually, we can close a system by collecting all the above into a set of governing equations.

Navier–Stokes equations for fluid mechanics:

$$\rho \left({\frac {\partial {\mathbf {v}}}{\partial t}}+{\mathbf {v}}\cdot \nabla {\mathbf {v}}\right)=-\nabla p+\nabla \cdot {\boldsymbol {{\mathsf {T}}}}+{\mathbf {f}}$$

Schrödinger equation for quantum physics:

$$i\hbar\frac{\partial}{\partial t} \Psi(\mathbf{r},t) = \left [ \frac{-\hbar^2}{2m}\nabla^2 + V(\mathbf{r},t)\right ] \Psi(\mathbf{r},t)$$

Maxwell's equations for electrodynamics:

  1. Gauss' law: $\nabla \cdot \mathbf{E} = \frac {\rho} {\varepsilon_0}$
  2. Faraday's law: $\nabla \cdot \mathbf{B} = 0$
  3. Gauss's law for magnetism: $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$
  4. Ampere's law with Maxwell's correction: $\nabla \times \mathbf{B} = \mu_0\left(\mathbf{J} + \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t} \right)$

🏷 Category=Topics Category=Dynamical System