Variational principles least action

The most celebrated textual embodiment of the science of energy was Thomson and Tait s Treatise on Natural Philosophy (1867). Originally intending to treat all branches of natural philosophy, Thomson and Tait in fact produced only the first volume of the Treatise. Taking statics to be derivative from dynamics, they reinterpreted Newton s third law (action-reaction) as conservation of energy, with action viewed as rate of working. Fundamental to the new energy physics was the move to make extremum (maximum or minimum) conditions, rather than point forces, the theoretical foundation of dynamics. The tendency of an entire system to move from one place to another in the most economical way would determine the forces and motions of the various parts of the system. Variational principles (especially least action) thus played a central role in the new dynamics. 

Variational principles for classical mechanics originated in modem times with the principle of least action, formulated first imprecisely by Maupertuis and then as an example of the new calculus of variations by Euler (1744) [436], Although not stated explicitly by either Maupertuis or Euler, stationary action is valid only for motion in which energy is conserved. With this proviso, in modem notation for generalized coordinates,... 

The necessary condition for S to have a minimum is that the variation of the integral is zero. Hence, the variational principle of least action is written in the form ... 

The principle of least action, 8A = 0 for a particular choice of system state vector li/r), yields the equations that describe the system dynamics. In the case of a state vector that can explore the entire Hilbert space, the stationarity of the action yields the time-dependent Schrodinger equation. For any approximate family of state vectors this procedure yields an equation that approximates the time-dependent Schrodinger equation in a manner that is variationally optimal for the particular choice of state vector form. 

For example, even in classical mechanics we have equations (like Newton s) and the variational principle of the least action. If we introduced something similar to varying constants into the Lagrangian we will change the equations. Similar... 

The variational calculus approach to classical mechanics is based on minimizing the action Al over the class G of parameterized curves. This is normally referred to as the principle of least action . It is difficult to provide a physical motivation for this concept, but it is normally taken as a foundation stone for classical mechanics. 

Using a variational formulation, it can be shown that Hamilton s principle of least action leads to the following Lagrange s equations of motion ... 

Lagrange Joseph Louis (1736-1813) Fr. math., showed mechanics could be found on the principle of least action, studied perturbations, hydrodynamics, developed calculus of variations, partial differential equations... 

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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