In Lagrangian formalism, given two points $(x_1,t_1)$ and $(x_2,t_2)$, we ask the question which paths $x(t)$ make the action $S=\displaystyle \int_{t_1}^{t_2}L\ \mathrm dt$ stationary and satisfy the boundary condition $x(t_1)=x_1,\ x(t_2)=x_2$. This question is equivalent to solving the Euler-Lagrange equation
$$\frac{\mathrm d}{\mathrm dt}\frac{\partial L}{\partial \dot{x}}=\frac{\partial L}{\partial q}$$
with boundary condtion $x(t_1)=x_1,\ x(t_2)=x_2$.
My question is why we are authorized to use the Euler-Lagrange equation to solve the initial condition problem $x(t_1)=x_1,\ \dot{x}(t_1)=v_1$.
It seems that these are two different problems. One problem is to find a path satisfying the boundary condition $x(t_1)=x_1,\ x(t_2)=x_2$ and make the action stationary.
The other problem is to find a path with initial condition $x(t_1)=x_1,\ \dot{x}(t_1)=v_1$ and I even don't know how to put other requirements such that its equation of motion is the Euler-Lagrange equation.
How can you prove these two problems are equivalent if you can make the second problem clear? Or maybe it is an axiom that we require the initial condition problem is solved by Euler-Lagrange equation. I'm confused about the logic of Lagrangian formalism.
Best Answer
Indeed the problem with boundary conditions, generally speaking, is not well-posed.
There are boundary conditions admitting no curves or admitting many curves, satisfying both these conditions and Euler-Lagrange equations.
Examples.
(1) Think of a particle constrained to stay on a smooth sphere where it can freely move. If you assign the North and the South pole of the sphere as boundary conditions, you get infinitely many solutions as the motion always describes a geodesic.
(2) Similarly, if you remove an open ball in $\mathbb R^3$ you do not have solutions when assigning boundary conditions on the opposite sides of the ball for a free particle in $\mathbb R^3$.
If $L$ is quadratic with respect to the $\dot{q}$ variables and this quadratic form is strictly positively defined, as is the case for systems of classical particles (also satisfying ideal holonomic constraints), the problem with initial conditions is always well-posed provided $L$ is sufficiently regular. There is exactly one maximal solution satisfying both Euler-Lagrange equations and initial conditions.
With these hypotheses also the problem with boundary conditions is well-posed with the additional condition that the two boundary conditions are sufficiently close to each other (this is evident from the two examples I presented above).
For these reasons a safer (mathematically minded) viewpoint is assuming that the variational principle determines the equation of motion, but not the solutions themselves.