Tell me more ×  Here is a question that's been bothering me
since I was a sophomore in university, and should have probably
asked before graduating: In analytic (Lagrangian) mechanics, the
derivation of the Euler-Lagrange equations from the principle of
least action assumes that the start and end coordinates at the
initial and final times are known. As a consequence, any
variation on the physical path must vanish at its boundaries.
This conveniently cancels out the contributions of the boundary
terms after integration by parts, and setting the requirement
for minimal action, we obtain the E.L. equations.  This is all
nice and dandy, but our intention is finding the location of a
particle at a time in the future, which we do not know a priori;
after we derive any equations of motion for a system, we solve
them by applying initial values instead of boundary conditions.
How are these two approaches consistent?  share|improve this
question Most treatments contain a footnote or remark that the
method can be exten