Lagrangian function, definition

The definition of a Lagrangian function can be generalized to a system with many particles and eventually also to a field that represents a continuously... 

We return to the definition of the Lagrangian function, Eq. (3.36.2) for a particle with mass m and electrical charge e subjected to a magnetic vector potential A and to a scalar potential 4> ... 

From the definition of the Lagrangian function (2.6) it can be shown that the time coordinate is both homogeneous and isotropic meaning that its properties are the same in both directions [52]. For, if t is replaced by —t, the Lagrangian is unchanged, and therefore so are the equations of motion. In this sense all motions which obey the laws of classical mechanics are reversible. 

There are three such equations for every particle, as before. These results are definitely restricted to Cartesian coordinates but by introducing a new function, the Lagrangian function L, defined for Newtonian systems as the difference of the kinetic and potential energy,... 

It can now be shown that the Lagrangian equations are equivalent to the more familiar Newton s second law of motion. If qi = r, the generaUzed coordinates are simply the Cartesian coordinates. Introducing this definition of the generalized coordinates and the corresponding Lagrangian function (2.6) into the Lagrangian equations of motion (2.14), we get ... 

The penalty term of an augmented Lagrangian method is designed to add positive curvature so that the Hessian of the augmented function is positive-definite. 

The definition of X incorporates the fact that the Lagrangian integral time scale is of the order of Zilw,. The statistical theory of turbulent diffusion outlined in the beginning of Section VIII,B can be used to estimate the functional dependence of g as... 

The subsystem or atomic Lagrangian integral is defined by the standard mode of integration used in the definition of the subsystem functional [ F, iJ] for a stationary state and for the definition of subsystem properties. 

It was emphasized in Chapter 6 that the definition of an atomic stationary state property is determined by the form of the atomic stationary state functional fl]. In precisely the same manner, the definition of an atomic property in the general time-dependent case is determined by the form of the atomic Lagrangian integral 2,t]. In both the stationary-state and... 

Worth noting that the variation in time is not explicitly taken in functional derivative of action functional since wile intervening in its integral definition respecting the Lagrangian is to be integrated out before explicit derivation however, using the derivative equivalence... 

It is understood that the density function of the second Piola-Kirchhoff stress = TjjEI 0 is the energy-conjugate or dual stress to E under the definition of the increment of internal energy dua = Lagrangian description (cf. Sect. 2.7). 

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