Lagrangian density mechanics

Another possibility to represent the quantum mechanical Lagrangian density is using the logarithm of the amplitude X = Ina, a = e. In that particular representation, the Lagrangean density takes the following symmetrical fomi... 

This approach to defining the Lagrangian density with the aid of both forward and backward Euler densities ip and ip uses the neat construct that ip ip is time-invariant. This is as true in the quantum mechanical analogy. 

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 quantum mechanical Lagrangian density for a system of many particles interacting via a many-particle potential energy operator V is... 

The beauty of the above topological definition of the atom in a molecule lies in the fact that it coincides with the rigorous quantum mechanical definition of an open subsystem [27, 33, 34]. In particular, the atomic action integral, which is defined through the atomic one-particle Lagrangian density, is zero within the atomic volume ... 

It has been demonstrated already that local gauge transformation on this Lagrangian leads to Eq. (153), which contains new charge current density terms due to the Higgs mechanism. For our present purposes, however, it is clearer to use the locally invariant Lagrangian obtained from Eq. (325), specifically... 

The use of functionals and their derivatives is not limited to density-functional theory, or even to quantum mechanics. In classical mechanics, e.g., one expresses the Lagrangian C in terms of of generalized coordinates q(x,t) and their temporal derivatives q(x,t), and obtains the equations of motion from extremizing the action functional 4[g] = J C q, q t)dt. The resulting equations of motion are the well-known Euler-Lagrange equations 0 = = fy — > which are a special case of Eq. (14). 

The fundamental link between the particle (Lagrangian) and wave-mechanical (Eulerian) pictures is defined by the following expression for the Eulerian density ... 

The large maxima of the electron density are expected and are found at the nuclear positions Ra. These points are m-limits for the trajectories of Vp(r), in this sense they are attractors of the gradient field although they are not critical points for the exact density because the nuclear cusp condition makes Vp(Ra) not defined. The stable manifold of the nuclear attractors are the atomic basins. The non-nuclear attractors occur in metal clusters [59-62], bulk metals [63] and between homonu-clear groups at intemuclear distances far away from the equilibrium geometry [64]. In the Quantum Theory of Atoms in Molecules (QTAIM) an atom is defined as the union of a nucleus and of the electron density of its atomic basin. It is an open quantum system for which a Lagrangian formulation of quantum mechanics [65-70] enables the derivation of many theorems such as the virial and hypervirial theorems [71]. As the QTAIM atoms are not overlapping, they cannot share electron pairs and therefore the Lewis s model is not consistent with the description of the matter provided by QTAIM. 

With known bond lengths (or density) and vibration frequencies, Lagrangian mechanics maps H-bond asymmetric, local, short-range potentials Water and ice prefer statistically the fluctuated, tetrahedrally coordinated structure with supersolid skin despite the fluctuations in 0 H length and <0 H-... 

---