Browse & Filter

$\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}  
\newcommand{\dd}[2][]{\frac{d#1}{d#2}}$    $\newcommand{\mHighLight}[1]{\mbox{#1}}$

Whenever the generalized force \(Q_k\) \begin{equation}\label{AB01} Q_k=\sum_\alpha F^\text{(e)}_\alpha \pp[\vec{x}_\alpha]{q_k}. \end{equation} can be written in the form \begin{equation}\label{AB02} Q_k = \dd{t}\Big(\pp[U]{\dot{q}_k}\Big)-\pp[U]{q_k} . \end{equation} the equations of motion take the Euler Lagrange form with \(L=T-U\). It is demonstrated that the Lorentz force,the force on a charged particle in e.m. fields, has the form as in \eqref{AB02}. An expression for the generalized potential \(U\) is derived.

The equations of motion in Newtonian mechanics are always written in Cartesian coordinates. If there are constraints among the coordinates, these have to be taken into account separately. A special type of constraints, called holonomic constraints and the number of degrees of freedom is defined.

If the Lagrangian does not depend on time explicitly, the Hamiltonian \(H=\sum_{k=1}\frac {\partial{L}}{\partial{\dot q_k}}\dot q_k-L\) is a constant of motion. For conservative systems of many particles, the Hamiltonian coincides with the total energy. is conserved.

 Euler Lagrange equations are obtained using Newton's laws and D' Alembert's principle.
$\newcommand{\dd}[2][]{\frac{d#1}{d#2}}\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}$

The Lagrangian for conservative systems  is defined as \(L=T-V\) and the Euler Lagrange equations take the form \begin{equation*} \frac{d}{dt}\frac{\partial{L}}{\partial\dot{q_k}}-\frac{\partial{L}}{\partial{ q_k}}=0. \end{equation*}