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where the action
denoting the set of parameters of the system.The equations of motion obtained by means of the functional derivative are identical to the usual Euler-Lagrange equations. A dynamical system whose equations of motion are obtainable by means of an action principle on a suitably chosen Lagrangian are known as Lagrangian dynamical systems. Examples of Lagrangian dynamical systems range from the (classical version of the) Standard Model to Newton's equations to purely mathematical problems such as geodesic equations and Plateau's problem.
The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics known as Lagrangian mechanics. In this context, the Lagrangian is usually taken to be the kinetic energyKinetic energy (also called vis viva or living force is energy possessed by a body by virtue of its motion. The kinetic energy of a body is equal to the amount of work needed to establish its velocity and rotation, starting from rest. Equations Definition of a mechanical system minus its potential energyPotential energy U or E , a kind of scalar potential, is energy by virtue of matter being able to move to a lower-energy state, releasing energy in some form. For example a mass released above the Earth has energy resulting from the gravitational attracti.
Suppose we have a three dimensional space and the Lagrangian
Then, the Euler-Lagrange equation is where the time derivative is written conventionally as a dot above the quantity being differentiated.
Using the above result we can easily show that the Lagrangian approach is equivalent to the Newtonian one by writing the force in terms of the potential , then the resulting equation is , exactly the same equation in a Newtonian approach for a constant mass object, a very similar deduction gives us the expression which is Newton's Second Law in its general form.
Suppose we have a three-dimensional space in spherical coordinates, r, θ, φ with the Lagrangian
Then, the Euler-Lagrange equations are:
Here, the set of parameters is just the time , and the dynamical variables are the trajectories of the particle.
In field theory, occasionally a distinction is made between the Lagrangian , of which the action is the time integral
and the Lagrangian density , which one integrates over all space-time to get the action:
The Lagrangian is then the spatial integral of the Lagrangian density. However, is also frequently simply called the Lagrangian, especially in modern use; it is far more useful in relativistic theories since it is a locally defined, Lorentz scalar field. Both definitions of the Lagrangian can be seen as special cases of the general form, depending on whether the spatial variable is incorporated into the index or the parameters in . Quantum field theories in particle physics, such as quantum electrodynamics, are usually described in terms of , and the terms in this form of the Lagrangian translate quickly to the rules used in evaluating Feynman diagrams.
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