Feynman problem on action

Question

It is very weird for me that a newbie can ask a new (may be silly, sorry...) question but must have 50 reputation to comment. When I see a good question like this but have no answer what I have to do? So I know that the integral $$S = \frac{m}{2}\int\limits_{t_a}^{t_b}{\left(\frac{d{x}}{d{t}}\right)^2 \mathbb dt}$$ occurs in the famous Feynman Lectures on Physics. This is so called action for the free moving particle, i.e the particle that have only kinetic energy (KE). So the question keep in force: how to calculate the action $S$. Btw, such a question is on Mathematica.SE too.

Answer 1

(comments converted in an answer)

For a definite path $\;t\mapsto x(t)\;$ you must integrate the square of its derivative $\,(\dot{x}(t))^2\,$ but the point is not to calculate this integral but to find the path (i.e. a function $x$) such that the action $S$ is minimal (more generally an extrema).

You don't really need to suppose that the function $x$ is derivable everywhere and could consider a regular sampling like in this picture :

_{(source: fsu.edu)}

(the derivative would be constant in every short integral and the sum of the squares rather large => this graph would clearly not return the "least action"!).

Between all the possible paths $\;t\mapsto x(t)\;$ we have to choose the one such that $S$ is minimal.
For the very simple Lagrangian $\;L=\frac 12\,m\,\dot{x}^2\,$ the function defined by the straight line from $\,(x_0,t_0)\to (x',t')\,$ in the picture will return the minimal $S$ (shortly because every larger derivative would contribute "too much").

In the actual solution we are thus simply integrating the square of the constant speed $\;v:=\dfrac{x'-x_0}{t'-t_0}\;$ so that the action will be : $$S_{real}=\dfrac m2 \left(\dfrac{x'-x_0}{t'-t_0}\right)^2(t'-t_0)=\dfrac m2\dfrac{(x'-x_0)^2}{t'-t_0}$$

What to do next? Well read the excellent Feynman lectures to the end to discover (while studying non-relativistic quantum mechanics) that the path $x$ such that the action is minimal is not the only one of interest : all paths play a role and contribute!

But before that you may learn calculus of variation while studying classical mechanics (Lagrangian and Hamiltonian mechanics) for example in one of these excellent references :

1. Lev D. Landau and E. Lifshitz Mechanics : handwaving, deep insight and practical computations
2. Goldstein Classical Mechanics : with references to other works
3. Vladimir Arnold Mathematical Methods of Classical Mechanics (first pages only) : beautiful mathematical presentation by a master of geometry (more advanced at the end)

For more about the quantum mechanics method (all paths considered) and once you know some basis about QM say from the first chapter of his Vol. 3 you may read Feynman's Nobel conference and other references to Feynman path integrals (extending an idea of Paul Dirac) :

4. Feynman & Hibbs "Quantum Mechanics and Path Integrals"
5. Taylor & all "Teaching Feynman's sum-over-paths quantum theory"