Analytical Mechanics - Lagrangian Mechanics

Lagrangian Mechanics

Lagrangian and Euler-Lagrange equations

The introduction of generalized coordinates and the fundamental Lagrangian function:

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;

which are a set of N 2nd-order ordinary differential equations, one for each q_i(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.

Configuration space

The Lagrangian formulation uses the configuration space of the system, the set of all possible generalized coordinates:

where is N-dimensional 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:

The configuration space can be defined more generally, and indeed more deeply, in terms of topological manifolds and the tangent bundle.