Notations
=========

.. csv-table::

    ":math:`\wedge` ", "vector product symbol."
    ":math:`\theta \wedge` ", "multiplication operator by the vector :math:`\theta` on the left."
    ":math:`q\text{'}` ", "derived from :math:`q` with respect to the curvilinear abscissa."
    ":math:`\dot{q}` ", "derived from :math:`q` with respect to time."
    ":math:`s` ", "curvilinear abscissa on the line of the centers of the sections."
    ":math:`\stackrel{ˆ}{u}` ", "associated 3rd order antisymmetric matrix of axial vector :math:`u`."
    ":math:`1` ", "3rd order unit matrix."
    ":math:`\text{Df}\text{.}\Delta x` ", "directional derivative of :math:`f` in the :math:`\Delta x` direction"
    ":math:`\left[\frac{d}{\mathrm{ds}}1\right]` ", ":math:`\text{DIAG}\left[\frac{d}{\text{ds}},\frac{d}{\text{ds}},\frac{d}{\text{ds}}\right]` diagonal matrix."


.. csv-table::

    ":math:`A,{I}_{\mathrm{1,2}\text{ou}3}` ", "area and moments of inertia with respect to the main axes 1, 2 or 3 of the right section."
    ":math:`B` ", "deformation matrix."
    ":math:`C` ", "behavior matrix."
    ":math:`E,G,\rho` ", "Young's modulus and shear stiffness, density."
    ":math:`{e}_{i\text{}i=\mathrm{1,3}}` ", "general coordinate axes."
    ":math:`{E}_{i}{(s)}_{i=\mathrm{1,3}}` ", "main inertia axes of the :math:`s` x-axis section in the reference position."
    ":math:`\overline{f}(s,t)` ", "linear external force exerted on the beam."
    ":math:`f(s,t)` ", "force in the beam at x-axis :math:`s` and at the instant :math:`t`."
    ":math:`F(s,t)` "," :math:`{R}^{T}(s,t)f(s,t)`."
    ":math:`{F}_{\text{ext}}` ", "external forces given to the nodes."
    ":math:`{F}_{\text{iner}},{F}_{\text{int}}` ", "inertial forces and forces within the nodes."
    ":math:`{I}_{\rho }` ", "inertia tensor of a unit length of a beam in a deformed position, expressed in the general axes."
    ":math:`{J}_{\rho }` ", "inertia tensor with a unit length of a beam in the reference position, expressed in the general axes."
    ":math:`\overline{m}(s,t)` ", "linear external moment exerted on the beam."
    ":math:`m(s,t)` ", "moment in the beam at the x-axis :math:`s` and at the moment :math:`t`."
    ":math:`M(s,t)` "," :math:`{R}^{T}(s,t)m(s,t)`."
    ":math:`{N}_{i}` ", "form function relating to node :math:`i`."
    ":math:`R(s,t)` ", "operator or matrix, in general axes, for rotating the right cross section of abscissa :math:`s`, from the reference configuration to that at time :math:`t`."
    ":math:`{R}_{o}(s)` ", "rotation passing from the general axes to the main inertia axes of the :math:`s` x-axis in the reference configuration."
    ":math:`{R}_{\text{tot}}(\text{s,t})` "," :math:`R(s,t){R}_{o}(s)`."
    ":math:`\text{SO}(3)` ", "group of rotation operators in 3-dimensional space."
    ":math:`{t}_{i}{(s,t)}_{i=\mathrm{1,3}}` ", "main inertia axes of the :math:`s` x-axis section at the time :math:`t`."
    ":math:`{x}_{o}(s,t)` ", "position, at time :math:`t`, of the center of the right section of the right cross section of abscissa :math:`s`."
    ":math:`\varepsilon` "," :math:`{x}_{o}^{\text{'}}-{t}_{1}`."
    ":math:`E` "," :math:`{R}^{T}\varepsilon`."
    ":math:`\Pi` "," :math:`(\begin{array}{cc}{R}_{\text{tot}}& 0\\ 0& {R}_{\text{tot}}\end{array})` position of the abscissa section :math:`s` the instant :math:`t`, :math:`d` finite by the position vector :math:`{x}_{0}` from the center and the rotation vector :math:`\Psi`"
    ":math:`\phi (s,t)` "," :math:`\{\begin{array}{c}{x}_{o}(s,t)\\ \Psi (s,t)\end{array}`"
    ":math:`\delta \phi (s)` "," :math:`\left\{\begin{array}{c}\Delta {x}_{o}(s)\\ \Delta \Psi (s)\end{array}\right\}`: virtual displacement at abscissa :math:`s`."
    ":math:`\Delta \phi (s)` "," :math:`\left\{\begin{array}{c}\Delta {x}_{o}(s)\\ \Delta \Psi (s)\end{array}\right\}`: correction of displacement to abscissa :math:`s`."
    ":math:`\chi (s,t)` ", "vector defining, at abscissa :math:`s` and at time :math:`t`, the variation in curvature with respect to the reference configuration."
    ":math:`X` "," :math:`{R}^{T}\chi`."
    ":math:`\Psi (s,t)` ", "vector rotation, at time :math:`t`, of the x-axis section :math:`s` in relation to its reference position."
    ":math:`{\Psi }_{i-\mathrm{1,}i}^{n}` ", "rotation vector between time :math:`i-1` and iteration :math:`n` of time :math:`i`."
    ":math:`{\omega }_{i}^{n}` ", "angular speed of a beam section calculated at iteration :math:`n` of time :math:`i`."
    ":math:`Q,{Q}^{-1}` ", "operator for passing a rotation vector to the associated quaternion and its inverse."