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In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his presentation to the Turin Academy of Science in 1760 [ 1 ] culminating in his 1788 ...
Action principles are "integral" approaches rather than the "differential" approach of Newtonian mechanics.[2]: 162 The core ideas are based on energy, paths, an energy function called the Lagrangian along paths, and selection of a path according to the "action", a continuous sum or integral of the Lagrangian along the path.
The equations of motion in physical theories can often be derived from an object called the Lagrangian. In classical mechanics, this object is usually of the form, 'kinetic energy − potential energy'. In general, the Lagrangian is that function which when integrated over produces the Action functional.
Action is significant because it is an input to the principle of stationary action, an approach to classical mechanics that is simpler for multiple objects. [1] Action and the variational principle are used in Feynman's formulation of quantum mechanics [2] and in general relativity. [3]
Hamilton's principle states that the true evolution q(t) of a system described by N generalized coordinates q = (q 1, q 2, ..., q N) between two specified states q 1 = q(t 1) and q 2 = q(t 2) at two specified times t 1 and t 2 is a stationary point (a point where the variation is zero) of the action functional [] = ((), ˙ (),) where (, ˙,) is the Lagrangian function for the system.
The basic principle of Lagrangian mechanics, the principle of stationary action, is that an object subjected to outside influences will "choose" a path which makes a certain quantity, the action, an extremum. The action is a functional, a mathematical relationship which takes an entire path and produces a single number.
The Lagrangian for a scalar field moving in a potential () can be written as = = =! It is not at all an accident that the scalar theory resembles the undergraduate textbook Lagrangian = for the kinetic term of a free point particle written as = /. The scalar theory is the field-theory generalization of a particle moving in a potential.
The Einstein–Hilbert action in general relativity is the action that yields the Einstein field equations through the stationary-action principle.With the (− + + +) metric signature, the gravitational part of the action is given as [1]