Variation principle Hamilton

Assume now that two different external potentials (which may be from nuclei), Vext and Vgjjj, result in the same electron density, p. Two different potentials imply that the two Hamilton operators are different, H and H, and the corresponding lowest energy wave functions are different, and Taking as an approximate wave function for H and using the variational principle yields... 

The variational principle now states that the energy computed via equation (1-11) as the expectation value of the Hamilton operator H from any guessed xTtrial will be an upper bound to the true energy of the ground state, i. e.,... 

Since dt cannot be singled out for special treatment, the covariant generalization of Hamilton s variational principle for a single particle requires an invariant action integral... 

Any multicomponent system whose dynamical behavior is governed by coupled linear equations can be modelled by an effective Lagrangian, quadratic in the system variables. Hamilton s variational principle is postulated to determine the time behavior of the system. A dynamical model of some system of interest is valid if it satisfies the same system of coupled equations. This makes it possible, for example,... 

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... 

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. 

In many cases the system of equations (1) is equivalent to a variation principle, known as Hamilton s Principle, viz. ... 

Here L ig a certain function of the co-ordinates and velocities of all the particles, and, in certain circumstances, also an explicit function of the time, and equation (2) as an expression of Hamilton s Principle is to be interpreted as follows the configuration (co-ordinates) of the system of particles is given at the times l1 and t2 and the motion is sought (i.e. the co-ordinates as function of the time) which will take the system from the first configuration to the second in such a way that the integral will have a stationary value.2 The chief advantage of such a variation principle is its independence of the system of coordinates. 

The proof of this theorem is relatively simple but will not be reproduced here (the interested reader may consult ref. 9). It is based on the variational principle and uses that from the integral of the electron density one knows the total number of electrons and accordingly the kinetic-energy and the electron-electron-interaction parts of the total Hamilton operator. Only the external potential (which above was only the Coulomb potential of the nuclei, but which may contain other parts, too) is unspecified, but assuming that this can be written as a sum of identical single-particle terms, Hohenberg and Kohn proved that also this is uniquely determined within an additive constant. 

Extending Hamilton s variational principle to piezoelectric media gives an equivalent description of the above boundary value problem (BVP) ... 

The fact that an infinity of front velocities occurs for pulled fronts gives rise to the problem of velocity selection. In this section we present two methods to tackle this problem. The first method employs the Hamilton-Jacobi theory to analyze the dynamics of the front position. It is equivalent to the marginal stability analysis (MSA) [448] and applies only to pulled fronts propagating into unstable states. However, in contrast to the MSA method, the Hamilton-Jacobi approach can also deal with pulled fronts propagating in heterogeneous media, see Chap. 6. The second method is a variational principle that works both for pulled and pushed fronts propagating into unstable states as well as for those propagating into metastable states. This principle can deal with the problem of velocity selection, if it is possible to find the proper trial function. Otherwise, it provides only lower and upper bounds for the front velocity. 

Such a variation principle may be looked upon as the extension from one to three dimensions of Hamilton s Principle of Stationary Action in dynamics. We will show below that the analogous extension from one to three dimensions of Hamilton s less known Principle of Variable Action " (which regards the integral I in its dependence on both the boundary B of D and on the boundary values of ip) throws significant light on the structure of colloid theory. 

It has been shown [22] that the time-dependent non- linear Schrodinger Eq. (3.1) can be obtained from the Hamilton principle when a suitable QM Lagrangian density is defined (see Appendix A.l), and that this Lagrangian density implies an extension of the time-dependent Frenkel s variational principle [23]. 

The principle of virtual work is suitable for solving a wide range of problems. There are tasks however where different but related formulations might be more useful. Thus, two prominent variational principles will be extended here to take into account materials with electromechanical couplings. This novel approach to Dirichlet s principle of minimum potential energy will be employed later in Section 6.3.2. In comparison to the principle of virtual work, the extended general Hamilton s principle is considered to be equivalent and even more versatile, but only its derivation will be demonstrated here. 

In the first place, the laws formulated as variational principles are themselves important, irrespective of their mathematical equivalence to those expressed in a set of differential equations, as seen from the importance of Hamilton s and Maupertuis principles in theoretical mechanics. Physical insight into the behavior of rather complicated phenomena can be acquired much more easily from laws in this form than from those in the form of a complicated set of differential equations, and often more intuitive conceptions of the phenomena can be obtained. 

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... 

By using the calculus of variations, the integral form of Lagrange s method can be obtained. This is known as Hamilton s principle (Goldstein et al. 2002). It can be stated as follows ... 

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