For example, even in classical mechanics we have equations (like Newton s) and the variational principle of the least action. If we introduced something similar to varying constants into the Lagrangian we will change the equations. Similar [Pg.239]

Following Hamilton s principle in classical mechanics, the required time dependence can be derived from a variational principle based on a seemingly artificial Lagrangian density, integrated over both space and time to define the functional [Pg.78]

The equations of motion of classical mechanics can be derived from variational principles such as Hamilton s principle of least action [74,75]. This principle states that a physical path connecting a given initial configuration with a given final configuration in time T makes the action [Pg.180]

The theory described above can be formulated in the form of a variational principle which is similar to the Lagrangian formulation of classical mechanics. The advantage of this formulation is that it is independent of the coordinate system, and allows a great flexibUity in choosing coordinates. [Pg.82]

Before undertaking the major subject of variational principles in quantum mechanics, the present chapter is intended as a brief introduction to the extension of variational theory to linear dynamical systems and to classical optimization methods. References given above and in the Bibliography will be of interest to the reader who wishes to pursue this subject in fields outside the context of contemporary theoretical physics and chemistry. The specialized subject of optimization of molecular geometries in theoretical chemistry is treated here in some detail. [Pg.25]

Hamiltonian mechanics refers to a mathematical formalism in classical mechanics invented by the Irish mathematician William Rowan Hamilton (1805-1865) during the early 1830 s arising from Lagrangian mechanics which was introduced about 50 years earlier by Joseph-Louis Lagrange (1736-1813). The Hamiltonian equations can however be formulated on the basis of a variational principle without recourse to Lagrangian mechanics [95] [2j. [Pg.201]

For a family of trajectories all starting at the value X(to) and at t=t all arriving at X(t), there is one trajectory that renders the action stationary. The classical mechanical trajectory of a given dynamical system is the one for which 5S=0, i.e. the action becomes stationary. The equation of motion is obtained from this variational principle [59], The corresponding Euler-Lagrange equations are obtained d(3L/3vk)/dt = 9L/dXk. In Cartesian coordinates these equations become Newton s equations of motion for each nucleus of mass Mk [Pg.290]

We will limit ourselves here to introducing the simplest of the quantum mechanics procedures the Modified Electron Gas (MEG) treatment of Gordon and Kim (1971). This procedure is on the borderline between the classical atomistic approach and quantum mechanics ab initio calculations that determine energy by applying the variational principle. A short introduction to MEG treatment should thus be of help in filling the conceptual gap between the two theories. [Pg.81]

From another point of view, the statistical interpretation of wave functions suggests how the radiation emitted by the atom may be calculated on wave-mechanical principles. In the classical theory this radiation is determined by the electric dipole moment p of the atom, or rather by its time-rate of variation. By the correspondence principle, this connexion must continue to subsist in the wave mechanics. Now the dipole moment is easily calculated by wave mechanics if we adhere to the analogy with classical atomic mechanics, it is given by [Pg.132]

This part is concerned with variational theory prior to modem quantum mechanics. The exception, saved for Chapter 10, is electromagnetic theory as formulated by Maxwell, which was relativistic before Einstein, and remains as fundamental as it was a century ago, the first example of a Lorentz and gauge covariant field theory. Chapter lisa brief survey of the history of variational principles, from Greek philosophers and a religious faith in God as the perfect engineer to a set of mathematical principles that could solve practical problems of optimization and rationalize the laws of dynamics. Chapter 2 traces these ideas in classical mechanics, while Chapter 3 discusses selected topics in applied mathematics concerned with optimization and stationary principles. [Pg.1]

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