The Euler-Lagrange mechanical paradigm
There is another way of discussing mechanics, which we can illustrate by the game of football discussed above. As the striker approaches the opposition’s goal mouth, their action in kicking the ball into the goal is not done meaninglessly. The striker has a great deal of visual information about the dispositions of all the other players in the match and how they are running. The striker can imagine where those players might be if the ball came their way, so the kick has to be such that the trajectory of the ball is clear to the back of the net. In some currently not understood way, the striker will analyse, in their brain and before the kick, potential future trajectories of the ball. Then they will decide almost spontaneously which potential trajectory is best or optimal in some sense, and then arrange to kick the ball so as to make it follow that trajectory. A good player will often succeed. Pele usually succeeded.
According to this way of looking at it, football could be regarded as an exercise in teleology, the performance of certain actions so as to ensure a desired final result.
In the Euler-Lagrange approach to CM, an SUO is analysed non-locally in time via the Calculus of Variations. At any time between initial and final times, the instantaneous state of a classical SUO is fully specified by the current values of a number n of configuration degrees of freedom q = (q1, q2,. . ., qn) and their time derivatives (velocities) q = (q1, q2,. . ., qn). These are fed into the Lagrangian L = L(q, q, t), a chosen function of the coordinates, their velocities, and time, specified over some trajectory Г running from initial configuration at initial time t, to final configuration at final time f. The action integral А[Г] is then the integral of the Lagrangian over the time interval of interest, that is,
The Calculus of Variations explores variants of the trajectory Г and selects an optimum one, according to some principle [Goldstein et al., 2002]. For fixed end points, the chosen principle is Hamilton’s principle, whilst for variable end points we use the Weiss Action Principle [Weiss, 1936; Sudarshan & Mukunda, 1983]. The resulting equations of motion are known as the Euler-Lagrange equations of motion,
One of the great mysteries of science crops up here. No one knows why all useful Lagrangians involve only the coordinates qi and their velocities q and no higher time derivatives. The fact is, canonical Lagrangians, those of the form
L = L(q, q, t), are used in all known fundamental theories, including those of elementary particle physics and general relativity. This has the following consequence: the Euler-Lagrange equations (14.4) give second-order in time differential equations. This impacts directly on all theories of time, because the only viable way we have of investigating time is through observation, and all observations are based on the laws of physics consistent with canonical Lagrangians.
Paul Dirac [1902-84] found third-order in time differential equations of motion for extended classical electric charges [Dirac, 1933]. He took into account the possibility of the dissipation of electromagnetic field energy whenever an electric charge accelerates. In consequence, his equations of motion have a built-in temporal asymmetry, in that they are not time-reversal invariant. Time reversal is discussed in Chapter 26.
-  It is possible to use Lagrangians containing higher time derivatives of the coordinates [Tapia,1988], but quantization becomes problematical.