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 qi(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.
Read more about this topic: Analytical Mechanics
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