Maupertuis principle

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... 

FVom the lemma just proved there follows the Maupertuis principle. The mapping x TTU T U carries the trajectories of the geodesic flow of the metric g, ... 

Proof By the Maupertuis principle, the function / ox on (7 is an analytic first integral of restriction of the geodesic flow... 

In the first place, the laws formulated as variational principles are themselves important, irrespective of their mathematical equivalence to those expressed in a set of differential equations, as seen from the importance of Hamilton s and Maupertuis principles in theoretical mechanics. Physical insight into the behavior of rather complicated phenomena can be acquired much more easily from laws in this form than from those in the form of a complicated set of differential equations, and often more intuitive conceptions of the phenomena can be obtained. 

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,... 

Now that we have introduced coordinates and velocities, the next question is how to predict the time evolution of a mechanical system. This is accomplished by solving a set of ordinary differential equations, the equations of motion, which can be derived from the principle of least action. It was discovered by Maupertuis and was further developed by Euler, Lagrange and Hamilton (d Abro (1951)). 

Mendel s 1866 paper on plant hybridization proposed a numerical, integral pattern ( N) for the distribution of hereditary qualities, a view contrasting strongly with Darwin s non-quantitative, infinitesimal variations. Mendel, in his paper, did not explore the nature of possible particle units that might account for his laws. Others before him, Maupertuis, Buffon, Diderot, and Herbert Spencer in his Principles of Biology of 1864 had propounded ideas of particulate inheritance. 

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... 

Before moving on to the Schrodinger equation, let us briefly review the relevant analytical mechanics. The most significant aspects of analytical mechanics are the least-action principle and the conservation laws based on it. In 1753, L. Euler arranged P.-L.M. de Maupertuis s thoughts in his paper entitled On the least-action principle and proved that the kinetics of mechanical systems obey the least-action principle, to apply this principle to general problems (Ekeland 2009). 

Maupertuis Pierre Louis Moreau (1698-1759) Fr. math., known for the principle of least action ... 

---