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; (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 methods of analytical mechanics apply to discrete particles, each with a finite number of degrees of freedom.
They can be modified to describe continuous fields or fluids, which have infinite degrees of freedom.
Configuration space The Lagrangian formulation uses the configuration space of the system, the set of all possible generalized coordinates: 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.
Since Newtonian mechanics considers vector quantities of motion, particularly accelerations, momenta, forces, of the constituents of the system, an alternative name for the mechanics governed by Newton's laws and Euler's laws is vectorial mechanics.
By contrast, analytical mechanics uses scalar properties of motion representing the system as a whole—usually its total kinetic energy and potential energy—not Newton's vectorial forces of individual particles.
In fact the same principles and formalisms can be used in relativistic mechanics and general relativity, and with some modifications, quantum mechanics and quantum field theory.
Analytical mechanics is used widely, from fundamental physics to applied mathematics, particularly chaos theory.