Editing Analytical mechanics (section)

==Lagrangian mechanics==
{{Main | Lagrangian mechanics | Euler–Lagrange equations}}

The introduction of generalized coordinates and the fundamental Lagrangian function:

:<math>L(\mathbf{q},\mathbf{\dot{q}},t) = T(\mathbf{q},\mathbf{\dot{q}},t) - V(\mathbf{q},\mathbf{\dot{q}},t)</math>

where ''T'' is the total [[kinetic energy]] and ''V'' is the total [[potential energy]] of the entire system, then either following the [[calculus of variations]] or using the above formula – lead to the [[Euler–Lagrange equations]];

:<math>\frac{d}{dt}\left(\frac{\partial L}{\partial \mathbf{\dot{q}}}\right) = \frac{\partial L}{\partial \mathbf{q}} \,,</math>

which are a set of ''N'' second-order [[ordinary differential equation]]s, one for each ''q<sub>i</sub>''(''t'').

This formulation identifies the actual path followed by the motion as a selection of the path over which the [[time integral]] of [[kinetic energy]] is least, assuming the total energy to be fixed, and imposing no conditions on the time of transit.

The Lagrangian formulation uses the '''[[Configuration space (physics)|configuration space]]''' of the system, the [[set (mathematics)|set]] of all possible generalized coordinates:

:<math>\mathcal{C} = \{ \mathbf{q} \in \mathbb{R}^N \}\,,</math>

where <math>\mathbb{R}^N</math> is ''N''-dimensional [[real number|real]] space (see also [[set-builder notation]]). The particular solution to the Euler–Lagrange equations is called a ''(configuration) path or trajectory'', i.e. one particular '''q'''(''t'') subject to the required [[initial conditions]]. The general solutions form a set of possible configurations as functions of time:

:<math>\{ \mathbf{q}(t) \in \mathbb{R}^N \,:\,t\ge 0,t\in \mathbb{R}\}\subseteq\mathcal{C}\,,</math>

The configuration space can be defined more generally, and indeed more deeply, in terms of [[topology|topological]] [[manifold]]s and the [[tangent bundle]].