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The Euler-Lagrange equation is about the functional

$$ \int_{t_1}^{t_2} L(q, \dot{q}, t ) dt . $$

From a mathematical point of view, a simpler functional might be

$$ \int_{t_1}^{t_2} L(q, t ) dt , $$

right? So, why is this functional not so common in physics? Can anyone give an example where this functional is relevant?

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    $\begingroup$ The last subquestion (v2) seems like a list question. $\endgroup$ – Qmechanic Jun 12 '16 at 7:15
  • $\begingroup$ The simple reason is that physics must be compatible with Newton's 2nd law. $\endgroup$ – auxsvr Jun 12 '16 at 12:47
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The latter functional would not be very useful in physics, because without the $\dot{q}$ dependence there would be no way to capture the particle's kinetic energy in your Lagrangian. There would be no dynamics, as the functional would be extremized simply by setting $\partial{L}/\partial{q} = 0$. For a typical Lagrangian of the form $L = T - U$, this would just translate to saying that the particle sits at a stationary point of the potential. This result can be found much more simply from elementary considerations without requiring the action principle.

This in turn answers your second question: the latter functional is simply extremized when $\partial{L}/\partial{q} = 0$, so it would usually be easier to just start directly from that requirement without introducing a functional at all.

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Here is the short version: Physics is often about calculating the time evolution of a dynamical system. There the kinetic term $T(\dot{q})$ plays an important role. In contrast, in static problems, the kinetic term $T(\dot{q})$ is absent or can be neglected, and we minimize just the potential energy $V(q)$.

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Well, if approaching the issue from general principles, there is a reason why the equation of evolution of mechanical dynamical systems is of second (or higher) order: all inertial reference frame are equivalent to formulate the laws of physics (not only the laws regarding mechanics). Since these reference frames have arbitrary relative constant velocity, there is no way to fix an absolute velocity for every given dynamical system and every inertial reference frame is admissible in principle. If the evolution laws -- viewed as systems of differential equations -- were of the first order, it would mean that the initial velocity is given when assigning the initial conditions to get the solution of the problem of motion. In other words a preferred reference frame among the class of inertial reference frames would exist. Euler-Lagrange equations produces differential equations of first order if $L$ is function of $t$ and $q$ only. This is a reason why $\dot{q}$ is also necessary in the Lagrangian of a mechanical system.

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