Lagrangian time-dependent

Keywords coupled cluster, CCSD, CC3, response theory, quasi-energy Lagrangian, time-dependent... 

W.F. Noh, CEL A Time-Dependent, Two-Space-Dimensional, Coupled Eulerian-Lagrangian Code, in Methods in Computational Physics, Volume 3 (edited by B. Alder, S. Fernbach and M. Rotenberg), Academic Press, New York, 1964. 

The definition of the Lagrangian in Eq. (2.9) in terms of the paths x(r) needs to be modified when time-dependent densities p t) are considered as the... 

Lagrangian as a functional -(v,v). Note however that unlike functionals used in the Time-dependent Hartree Fock approximation (14), this Lagrangian is not complex analytic in the variables (v,v) separately. 

The Cauchy moments have been derived in Ref. [4] for CC wavefunctions, using the time-dependent quasi-energy Lagrangian technique [I]. In Section 2.1 we recapitulate the important points of that derivation and use it in Section 2.2 to derive the CC3-specific formulas. 

Figure 24.2 Calculation of the time-dependent concentration along the river axis x of a reactive compound. The effect of the reaction is calculated in the Lagrangian framework the effect of advection is accounted for by the relation between flow time t and distance x, x(t).

S q(t)] has the form of an action functional for an auxiliary dynamical system with time-dependent Lagrangian L(q, q- 4>) ... 

By Noether s theorem, invariance of the Lagrangian under an infinitesimal time displacement implies conservation of energy. This is consistent with the direct proof of energy conservation given above, when L and by implication H have no explicit time dependence. Define a continuous time displacement by the transformation t = t + oi(t ) whereat/(,) = a(t ) = 0. subject to a —0. Time intervals on the original and displaced trajectories are related by dt = (1 + a )dt or dt = (1 — a )dt. The transformed Lagrangian is... 

Since time here is an ignorable variable, it can be eliminated from the dynamics by subtracting ptt from the modified Lagrangian and by solving H = E for t as a function of the spatial coordinates and momenta. This produces Jacobi s version of the principle of least action as a dynamical theory of trajectories, from which time dependence has been removed. The modified Lagrangian is... 

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... 

The time-dependent (hyperbolic) Lagrangian framework should also generalize to three dimensions as well as resolve reactive interfaces dynamically. 

Not explicitly time dependent systems axe called autonomous. For autonomous systems dH/dt = 0 and we have H — E = const, i.e. the total energy of the system is conserved. Clearly the system of equations (3.1.21) is more symmetric than the set (3.1.6) of second order dilferential equations obtained from the Lagrangian formalism. 

The application of the chemical schemes to atmospheric phenomena requires a diffusion formulation that reflects time-dependence and spatial variability of meteorological conditions. An attempt has been made to keep the mathematical description near the level of detail and precision of the observational data. This has resulted in a Lagrangian air parcel formulation with finite-rate vertical diffusion. The approach avoids the artificial numerical diffusion because it uses natural (or intrinsic) coordinates that are aligned with fluid motion. This allows us simultaneously to include upward dispersion and chemical change. Figure 1 schematically illustrates the main features of the formulation. Highspeed memory requirements are limited by allowing sequential point-by-point output of the history of the air parcel. 

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... 

Equation (8.144) is an alternative form of the expression given in eqn (8.125) for the total system. The principle of stationary action for a subsystem can be expressed for an infinitesimal time interval in terms of a variation of the Lagrangian integral, similar to that given in eqn (8.127) for the total system. For the atomic Lagrangian, assuming F to have no explicit time dependence, this statement is... 

In the case of the coaxial mixer, the rotation kinematics is much more complex since the two sets of agitators counter-rotate at different speeds. For the sake of simplicity, we decided to simulate the flow using the frame of reference of the anchor. In this Lagrangian viewpoint, the anchor is fixed but the vessel wall rotates at —Qanchor and the turbine rotates at anchor + turbine- such a situation. Contrary to the simple propeller problem, the resolution of the flow equations is time-dependent as the position of the central agitator changes with time. 

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