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Principle of Least Action

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The principle of least action, which asserts that the motion of mechanical systems may be described thusly:

There exists a function $L(\mathbf{q},\mathbf{\dot{q}},t)$ , called the Lagrangian, whose integral over each sufficiently small interval of time, $t ε[t 1, t 2]$ , takes the least possible value. This integral is called the action . This is described mathematically thusly

For each mechanical system, $\exists L: \Re^{2n + 1} \rightarrow \Re, L(\mathbf{q},\mathbf{\dot{q}},t)$ , such that

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

is at minimum, for some selection of intervals $[t 1, i, t 2, i]$ which completely cover the region of interest.

This does not actually impose terribly many restrictions on $L$ . First, clearly, it must be integrable . One notes that if a Lagrangian $L$ describes a system, so does $L' = L + \frac{d}{dt} F$ , where F is any function.

The main point is that, for nearly every system of interest (a proof here, or more insightful description of what actually is described by Lagrangian physics, would actually be very interesting. It should be noted that for every Quantum System, the classical variational principle follows from the use of Feynman Propagator for the time evolution of the system. Are potentially non-quantum justifications for this principle? Are there other systems entirely), mechanical systems may be described by the principle of least action. In other words, this isn't voodoo . The toolset for dealing with mechanical systems, once it is in this form, is very advanced, and has computational and practical advantages over dealing with forces directly -- many of the details are taken care of. (Note: It may be interesting to prove some upper/lower bounds for this).

Now, the requirement that the action $S$ be minimized automatically places some restrictions on the form of the solution of $\mathbf{q},\mathbf{\dot{q}}$ (that's right, after finding a Lagrangian, somehow, after examining systems carefully, we work backwards to impose to find the mechanical path of the system. Luckily, nature isn't terribly malevolent, and the Lagrangians we find are governed by fairly simple laws which we can use outside of the context where we discovered them. It feels backwards, even if we don't think about it much.)