Thus, by applying a single idea : the principle of least action, to a single quantity : the scalar curvature on the bundle ( as a whole ), one obtains simultaneously all of the needed field equations, for both the space-time and the gauge field.
42.
The corresponding dynamics problem is that of a rock thrown upwards; the path the rock follows is a critical point ( extremum ) of the Lagrangian; as before, this critical point is typically a minimum, so this is called the " principle of least action ":
43.
The second volume contains a long paper embodying the results of several papers in the first volume on the theory and notation of the calculus of variations; and he illustrates its use by deducing the principle of least action, and by solutions of various problems in dynamics.
44.
He also analyzes the movement of a photon near a mass with the principle of least action : the photon tries to make as big steps ( oscillations with low frequency ); therefore passing a zone where the second slows down and the unit of length increases.
45.
Although it is based on the calculus of variations and may be said to belong to the general class of problems included under the principle of least action which had been studied earlier by Arnold, have extended Hamilton's work, thereby expanding our knowledge of mechanics and differential equations.
46.
Newton's 2nd law explains conservation of energy and momentum and can be used to show equivalency with least action when forces are properly conserved, e . g . when forces are summed to zero in accordance with Newton's 1st and 3rd laws and when accounting for heat generated by friction.
47.
The Principle of least action was already a well known principle that explained Snell's law when Newton was investigating optics, so why did he come to the conclusion that light went faster when it entered glass rather than slower ? Preceding contribs ) 21 : 42, 4 August 2008 ( UTC)
48.
In the latter case, the geodesic equations are seen to arise from the principle of least action : they describe the motion of a " free particle " ( a particle feeling no forces ) that is confined to move on the manifold, but otherwise moves freely, with constant momentum, within the manifold.
49.
Comparison of the Ricci tensor with the Ricci tensor calculated from the principle of least action, Theoretical motivation for general relativity # Principle of least action in general relativity identifying the stress energy tensor with the Hilbert stress-energy, and remembering that A + B = 1 removes the ambiguity in A, B, and C.
50.
Two years later, Maupertuis cites Euler's 1744 work as a " beautiful application of my principle to the motion of the planets " and goes on to apply the principle of least action to the lever problem in mechanical equilibrium and to perfectly elastic and perfectly inelastic collisions ( see the 1746 publication below ).
