# Initial in time conditions and Lagrangian approach in classical mechanics

When we derive Euler-Lagrange equations in classical mechanics following the Lagrangian approach we introduce Boundary conditions at the starting- and end-points of the path in the configuration space. Usually,though not necessarily, one requires that $$q(t_{initial})=q(t_{final})=0$$. But the standard problem in classical mechanics is to instead assume Initial (not Boundary) conditions, in practice it is assumed that the initial position $$q(t_{initial})$$ and initial velocity $$q'(t_{initial})$$ are known.

Is there any way to modify the Lagrangian such that the extremisation of the Action would yield the Euler-Lagrange equations together with the above mentioned Initial conditions?

• The Euler Lagrange equations don't depend on initial conditions? Do you mean $\delta q(t_{initial}=\delta q(t_{final})=0$? The variation of $q$ has to be zero because any varied path should still match the boundary conditions but there are no restrictions on the B.C. of $q$. – AccidentalTaylorExpansion Nov 21 '19 at 17:03
• Related: physics.stackexchange.com/q/38348/2451 and links therein. – Qmechanic Nov 21 '19 at 17:22

$$x=Asin(\omega t)$$
But you don't know what $$A$$ and $$\omega$$ are without substituting certain conditions.