Least-action path

For JT problems of higher dimensions such as that for the T (e t2) problem, the adiabatic potential V is complicated and cannot be written down in an analytical form. However, in such problems, the least action path can be approximated by the minimum energy path (or path of steepest descent) on the adiabatic potential surface. It is the path for which the tangent to it is parallel to the gradient of the APES. 

Fig. 1. The path and the contour plot of the lowest potential energy surface in the T t2 system. By symmetry considerations, the 3D problem is reduced to a calculation in 2D (X, Y) coordinate space. The five paths have been computed with different initial conditions and the one (in the middle) that reaches the zero-point energy (the innermost ring) is the least action path.

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... 

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. 

Euler s proof of the least action principle for a single particle (mass point in motion) was extended by Lagrange (c. 1760) to the general case of mutually interacting particles, appropriate to celestial mechanics. In Lagrange s derivation [436], action along a system path from initial coordinates P to final coordinates Q is defined by... 

SA = 0 subject to the energy constraint restates the principle of least action. When the external potential function is constant, the definition of ds as a path element implies that the system trajectory is a geodesic in the Riemann space defined by the mass tensor m . This anticipates the profound geometrization of dynamics introduced by Einstein in the general theory of relativity. 

Then, the principle of least action states that the path a mechanical system takes from (qi,ti) to (921 2) is such as to minimize the action... 

The path of mechanical systems has been described by extremal principles. We emphasize the principles of Fermat Hamilton. The principle of least action is named after Maupertui but this concept is also associated with Leibnitz, Euler, and Jacobi For details, cf. any textbook of theoretical physics, e.g., the book of Lindsay [16, p. 129]. Further, it is interesting to note that the importance of minimal principles has been pointed out in the field of molecular evolution by Davis [17]. So, in his words... 

For intermediate reaction-path curvature, one may use either the SCSA or LC3 approximation, but even more accurate results are obtained by a least-action (LA) method.In the LA method, the tunneling paths are linear interpolations between the MEP and the LC3 paths. Thus this method does not require knowing the potential over a wider swath than is necessary for the LC3 method. 

The least-action principle leads to the Euler-Lagrange equation determining motion paths. Suppose that a mechanical system is located at two different points with coordinates, qi and q2, at different times, t = t and t = t2, respectively. Then, the system transfers between these points under the condition that the action... 

This situation is illustrated in Figure 2.3. As in most cases of practical interest the actual path minimizes the action, the Hamiltonian principle defined by Eq. (2.48) is sometimes also denoted the principle of least action. It may be considered as the most fundamental law of Nature, valid in all areas of physics, and all other fundamental laws or equations can be deduced from this principle. 

It is a path of least action At any point along the path, the gradient of the potential has no component perpendicular to the path. 

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... 

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