Hamilton principle

El-Sayed, M.E.M., Marjadi, D. Sandgren, E. 1991. Force Method Formulations Based on Hamilton Principle. Computers Structures, 38(3) 301-316. 

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]. 

In fact, by introducing in the Euler-Lagrange equations associated to the Hamilton principle (A.20) we have ... 

Hamilton principle, 50, 51 Harmonic vibrational frequencies, 14 Hartree—Fock method (HF), 6, 8 Hellmann—Feynman theorem, 13, 20, 49... 

By applying the Hamilton principle of least action [16], it follows that the surface / satisfies the inhomogeneous biharmonic plate equation... 

In order to formulate the equations of a mechanical model (the U system) of an engineering system we start from the Hamilton principle... 

To formulate the equations of the U system we shall apply the Lagrangian approach [1,2]. Let us derive from the Hamilton principle (Eq. (1)) two systems of differential equations. 

Let us now derive from the Hamilton principle the second sy stem of the differential equations - the second order Lagrange equation system... 

An alternative procedure is the dynamic programming method of Bellman (1957) which is based on the principle of optimality and the imbedding approach. The principle of optimality yields the Hamilton-Jacobi partial differential equation, whose solution results in an optimal control policy. Euler-Lagrange and Pontrya-gin s equations are applicable to systems with non-linear, time-varying state equations and non-quadratic, time varying performance criteria. The Hamilton-Jacobi equation is usually solved for the important and special case of the linear time-invariant plant with quadratic performance criterion (called the performance index), which takes the form of the matrix Riccati (1724) equation. This produces an optimal control law as a linear function of the state vector components which is always stable, providing the system is controllable. 

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 Linear Algebraic Problem.—Familiarity with the basic theory of finite vectors and matrices—the notions of rank and linear dependence, the Cayley-Hamilton theorem, the Jordan normal form, orthogonality, and related principles—will be presupposed. In this section and the next, matrices will generally be represented by capital letters, column vectors by lower case English letters, scalars, except for indices and dimensions, by lower case Greek letters. The vectors a,b,x,y,..., will have elements au f it gt, r) . .. the matrices A, B,...,... 

N. Friis and A. E. Hamielec, Principles of Polymer Reactor Design, in Polymer Reaction Engineering Course Notes, McMaster University, Hamilton, Ontario, Canada, p.55. 

It is now shown how the abrupt changes in the eigenvalue distribution around the central critical point relate to changes in the classical mechanics, bearing in mind that the analog of quantization in classical mechanics is a transformation of the Hamiltonian from a representation in the variables pR, p, R, 0) to one in angle-action variables (/, /e, Qr, 0) such that the transformed Hamiltonian depends only on the actions 1r, /e) [37]. Hamilton s equations diR/dt = (0///00 j), etc.) then show that the actions are constants of the motion, which are related to the quantum numbers by the Bohr correspondence principle [23]. In the present case,... 

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.,... 

Evans and Baranyai [51, 52] have explored what they describe as a nonlinear generalization of Prigogine s principle of minimum entropy production. In their theory the rate of (first) entropy production is equated to the rate of phase space compression. Since phase space is incompressible under Hamilton s equations of motion, which all real systems obey, the compression of phase space that occurs in nonequilibrium molecular dynamics (NEMD) simulations is purely an artifact of the non-Hamiltonian equations of motion that arise in implementing the Evans-Hoover thermostat [53, 54]. (See Section VIIIC for a critical discussion of the NEMD method.) While the NEMD method is a valid simulation approach in the linear regime, the phase space compression induced by the thermostat awaits physical interpretation even if it does turn out to be related to the rate of first entropy production, then the hurdle posed by Question (3) remains to be surmounted. 

Hamilton Standard of the U.S. EPA has reported several coil coating plants wastewater treatment case histories.8,9 A full-scale wastewater treatment plant system has performed well for treatment of the wastewater generated from coil coating steel subcategory operations. The process principles and operational data of the full-scale treatment of a steel subcategory wastewater are summarized herein for the convenience of readers ... 

Since the proof-of-principle test, considerable improvements to the BWA monitoring method hardware and software were made by Hamilton Sund-strand Sensor Systems, and they are now being incorporated into LRIP units for the final phase of BWA method work. The BWA monitoring method and... 

Russell WMS, Burch RL (1959) The principles of humane experimental technique. Methuen, London. Reprinted by UFAW, 1992 8 Hamilton Close, South Mimms, Potters Bar, Herts EN6 3QD England. ISBN 0 900767 78... 

It applies for both formulations above that the expansion in principle contains an infinite number of terms. The convergence to a few lowest order terms relies on the ability to orderly separate influences of the dominant rf irradiation terms (through a suitable interaction frame) from the less dominant internal terms of the Hamiltonian. In principle, this may be overcome using the spectral theorem (or the Caley-Hamilton theorem [57]) providing a closed (i.e., exact) solution to the Baker-Campbell-Hausdorf problem with all dependencies included in n terms where n designates the dimension of the Hilbert-space matrix representation (e.g., 2 for a single spin-1/2, 4 for a two-spin-1/2 system) [58, 59]. 

To develop a system of mechanics from here without the introduction of any other concepts, apart from energy, some general principle that predicts the course of a mechanical change is required. This could be like the Maupertuis principle of least action or Fermat s principle of least time. It means that the actual path of the change will have an extreme value e.g. minimum) of either action or time, compared to all other possible paths. Based on considerations like these Hamilton formulated the principle that the action integral... 

Hamilton s principle is equivalent to the statement that for the actual motion the average kinetic energy approaches the average potential energy as closely as possible. If the potential energy is a function of position only the equivalent mathematical statement is... 

The principal function S is the same quantity as the one defined by Hamilton s principle as in equation (2). It has the advantage of dealing with the scalars T and V only, whereas transformations of force components using Newton s laws deal with more complicated vector quantities. 

The Hamiltonian for a charged particle in an electromagnetic field can be obtained from Hamilton s principle and Lagrange s equations of motion (Section 3.3) ... 

By analogy with Hamilton s principle of least action, the simplest proposition that could solve the thermodynamic problem is that equilibrium also depends on an extremum principle. In other words, the extensive parameters in the equilibrium state either maximize or minimize some function. 

The equations of motion for the nuclei are obtained from Hamilton s least action principle. The nuclei total kinetic energy, K, is given by the sum of individual nucleus kinetic energy, (l/2)Mk(dXk/dt)2. The time integral of the Lagrangian L(X,dX /dt,t) = K-V is the action S of the system. For different paths (X=X(t)) the action has different numerical values. 

In principle, with given initial conditions, the time evolution of the coordinates and momenta [rjt p ) of each particle is given by Hamilton s equations 15... 

Exotic atomic nuclei may be described as structures than do not occur in nature, but are produced in collisions. These nuclei have abundances of neurons and protons that are quite different from the natural nuclei. In 1949, M.G, Mayer (Argonne National Laboratory) and J.H.D. Jensen (University of Heidelberg) introduced a sphencal-shell model of die nucleus. The model, however, did not meet the requirements and restrains imposed by quantum mechanics and the Pauli exclusion principle, Hamilton (Vanderbilt University) and Maruhn (University of Frankfurt) reported on additional research of exotic atomic nuclei in a paper published in mid-1986 (see reference listedi. In addition to the aforementioned spherical model, there are several other fundamental shapes, including other geometric shapes with three mutually peipendicular axes—prolate spheroid (football shape), oblate spheroid (discus shape), and triaxial nucleus (all axes unequal). 

Ismail, A.A., Nicodemo, A., Sedman, J., van de Voort, F.R., and Holzbauer, I.E. 1998. Infrared spectroscopy of lipids Principles and applications. In Spectral Properties of Lipids (R.J. Hamilton and J. Cast, eds.) pp. 235-269. Sheffield Academic Press/CRC Press, Boca Raton, Fla. 

Quantitative Formulations. Computer simulations (213) have been used to put the Gurney-Mott mechanism on a more quantitative basis. Hamilton s recent formulation (214) uses "a more analytical approach. ., that gives a maximum insight into the concepts involved." The method is based on the principle that when there is a branch in a sequence of events allowing two or more possible pathways, a particular event j will be selected with probability p given by... 

Following Atkins [68], the propagation of particles follows a path dictated by Newton s laws, equivalent to Hamilton s principle, that particles select paths between two points such that the action associated with the path is a minimum. Therefore, Fermat s principle for light propagation is Hamilton s principle for particles. The formal definition of action is an integral identical in structure with the phase length in physical optics. Therefore, particles are associated with wave motion, the wave-particle dualism. Hamilton s principle of least... 

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