# 16.9: Summary and Implications


The goal of this chapter is to provide a glimpse into the classical mechanics of the continua which introduces the Lagrangian density and Hamiltonian density formulations of classical mechanics.

## Lagrangian density formulation

In three dimensional Lagrangian density $$\mathfrak{L}(\mathbf{q}, \frac{d\mathbf{q}}{ dt} ,\boldsymbol{\nabla} \cdot \mathbf{q}, x, y, z, t)$$ is related to the Lagrangian $$L$$ by taking the volume integral of the Lagrangian density.

$L = \int \mathfrak{L}(\mathbf{q}, \frac{d\mathbf{q} }{dt }, \boldsymbol{\nabla} \cdot \mathbf{q}, x, y, z, t)d\tau \label{16.21}$

Applying Hamilton’s Principle to the three-dimensional Lagrangian density leads to the following set of differential equations of motion

$\frac{\partial}{\partial t }\left(\frac{ \partial \mathfrak{L}}{\frac{ \partial \mathbf{q} }{\partial t}} \right) + \frac{\partial}{\partial x} \left(\frac{ \partial \mathfrak{L}}{\frac{ \partial \mathbf{q} }{\partial x}} \right) + \frac{\partial}{\partial y} \left(\frac{ \partial \mathfrak{L}}{\frac{ \partial \mathbf{q} }{\partial y}} \right) + \frac{\partial}{\partial z} \left(\frac{ \partial \mathfrak{L}}{\frac{ \partial \mathbf{q} }{\partial z}} \right) − \frac{\partial \mathfrak{L}} {\partial \mathbf{q}} = 0 \label{16.22}$

## Hamiltonian density formulation

In the limit that the coordinates $$q,p$$ are continuous, then the Hamiltonian density can be expressed in terms of a volume integral over the momentum density $$\pi$$ and the Lagrangian density $$\mathfrak{L}$$ where

$\boldsymbol{\pi} \equiv \frac{\partial \mathfrak{L}}{ \partial \mathbf{\dot{q}}} \label{16.27}$

Then the obvious definition of the Hamiltonian density $$\mathfrak{H}$$ is

$H = \int \mathfrak{H} dV = \int (\boldsymbol{\pi} \cdot \mathbf{\dot{q}} - \mathfrak{L}) d\tau \label{16.28}$

where the Hamiltonian density is given by

$\mathfrak{H} =\boldsymbol{\pi} \cdot \mathbf{\dot{q}} − \mathfrak{L} \label{16.29}$

These Lagrangian and Hamiltonian density formulations are of considerable importance to field theory and fluid mechanics.

## Linear elastic solids

The theory of continuous systems was applied to the case of linear elastic solids. The stress tensor $$\mathbf{T}$$ is a rank 2 tensor defined as the ratio of the force vector $$d\mathbf{F}$$ and the surface element vector $$d\mathbf{A}$$. That is, the force vector is given by the inner product of the stress tensor $$\mathbf{T}$$ and the surface element vector $$d\mathbf{A}$$.

$d\mathbf{F} = \mathbf{T}\cdot d\mathbf{A} \label{16.33}$

The strain tensor $$\boldsymbol{\sigma}$$ also is a rank 2 tensor defined as the ratio of the strain vector $$\boldsymbol{\xi}$$ and infinitessimal area $$d\mathbf{A}$$.

$d\boldsymbol{\xi} = \boldsymbol{\sigma}\cdot d\mathbf{A} \label{16.38}$

where the component form of the rank 2 strain tensor is

$\boldsymbol{\sigma} = \frac{1}{ 2 } \begin{vmatrix} \frac{d\xi_1 }{dx_1} & \frac{d\xi_1}{ dx_2} & \frac{d\xi_1 }{dx_3} \\ \frac{d\xi_2}{ dx_1} & \frac{d\xi_2 }{dx_2} & \frac{d\xi_2 }{dx_3} \\ \frac{d\xi_3}{ dx_1} & \frac{d\xi_3}{ dx_2 } & \frac{d\xi_3}{ dx_3} \end{vmatrix} \label{16.39}$

The modulus of elasticity is defined as the slope of the stress-strain curve. For linear, homogeneous, elastic matter, the potential energy density $$U$$ separates into diagonal and off-diagonal components of the strain tensor

$U = \frac{1}{2} \left[ \lambda \sum_i (\sigma_{ii})^2 + 2\mu \sum_{ik} (\sigma_{ik})^2 \right] \label{16.42}$

where the constants $$\lambda$$ and $$\mu$$ are Lamé’s moduli of elasticity which are positive. The stress tensor is related to the strain tensor by

$T_{ij} = \lambda \delta_{ij} \sum_k \frac{\partial\xi_k }{\partial x_k} + \mu \left( \frac{d\xi_i}{ dx_j } + \frac{d\xi_j}{ dx_i} \right) = \lambda \delta_{ij} \sum_k \sigma_{kk} + 2\mu \sigma_{ij} \label{16.43}$

## Electromagnetic field theory

The rank 2 Maxwell stress tensor $$\mathbf{T}$$ has components

$T_{ij} \equiv \epsilon_0 \left( E_iE_j − \frac{1}{2} \delta_{ij}E^2 \right) + \frac{1 }{\mu_0} \left( B_iB_j − \frac{1}{2} \delta_{ij}B^2 \right) \label{16.71}$

The divergence theorem allows the total electromagnetic force, acting of the volume $$\tau$$, to be written as

$\mathbf{F}= \int \left( \boldsymbol{\nabla} \cdot \mathbf{T} −\epsilon_0\mu_0 \frac{\partial \mathbf{S} }{\partial t} \right) d \tau = \oint \mathbf{T} \cdot d\mathbf{a}−\epsilon_0\mu_0 \frac{d}{dt} \int \mathbf{Sd}\boldsymbol{\tau} \label{16.74}$

The total momentum flux density is given by

$\frac{\partial}{ \partial t} (\boldsymbol{\pi}_{mech} + \boldsymbol{\pi}_{field}) = \boldsymbol{\nabla} \cdot \mathbf{T} \label{16.79}$

where the electromagnetic field momentum density is given by the Poynting vector $$\mathbf{S}$$ as $$\boldsymbol{\pi}_{field}=\epsilon_0 \mu_0 \mathbf{S}$$.

## Ideal fluid dynamics

Mass conservation leads to the continuity equation

$\frac{\partial \rho }{ \partial t} + \boldsymbol{\nabla}\cdot (\rho \mathbf{v})=0 \label{16.82}$

Euler’s hydrodynamic equation gives

$\frac{\partial \mathbf{v} }{\partial t} + (\mathbf{v} \cdot \boldsymbol{\nabla}) \mathbf{v} = −\frac{1}{ \rho} \boldsymbol{\nabla} (P + \rho V ) \label{16.90}$

where $$V$$ is the scalar gravitational potential. If the flow is irrotational and time independent then

$\left(\frac{1}{ 2} \rho v^2 + P + \rho V \right) = \text{ constant} \label{16.94}$

## Viscous fluid dynamics

For incompressible flow the stress tensor term simplifies to $$\boldsymbol{\nabla} \cdot \mathbf{T} =\mu \boldsymbol{\nabla}^2\mathbf{v}$$. Then the Navier-Stokes equation becomes

$\rho \left[ \frac{\partial \mathbf{v} }{\partial t} + \mathbf{v} \cdot \boldsymbol{\nabla}\mathbf{v} \right] = −\boldsymbol{\nabla}P + \mu \boldsymbol{\nabla}^2\mathbf{v}+ \mathbf{f} \label{16.98}$

where $$\mu \boldsymbol{\nabla}^2\mathbf{v}$$ is the viscosity drag term. The left-hand side of Equation \ref{16.98} represents the rate of change of momentum per unit volume while the right-hand side represents the summation of the forces per unit volume that are acting.

The Reynolds number is a dimensionless number that characterizes the ratio of inertial forces to viscous forces in a viscous medium. The evolution of flow from laminar flow to turbulent flow, with increase of Reynolds number, was discussed.

The classical mechanics of continuous fields encompasses a remarkably broad range of phenomena with important applications to laminar and turbulent fluid flow, gravitation, electromagnetism, relativity, and quantum fields.

This page titled 16.9: Summary and Implications is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Douglas Cline via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.