Lagrangian

From Physics wiki

Building on d'Alembert's principle, Newton's second law can be recast in terms of generalized coordinates $q_i\,$ in the form of the Euler-Lagrange equations

$\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) - \frac{\partial L}{\partial q_i} = Q_i^{(nc)}\,$ ,

where $L(q_1, ..., q_n, \dot{q}_1, ..., \dot{q}_n, t)\,$ is the Lagrangian, which is sometimes called Lagrange's function, and is given by the difference in the kinetic and potential energies:

$L(q_i, \dot{q}_i, t) = T(q_i, \dot{q}_i, t) - V(q_i,\dot{q}_i, t)\,$ ,

and where $Q_i^{(nc)}\,$ represent the non-conservative forces, or forces that have otherwise not been included in constructing the potential $V\,$ . In terms of conventional coordinates, $T\,$ is given by