Question

How does it make sense to vary the position and the velocity independently?

Edit:

Velocity is the derivative of position, so how can you treat them as independent variables? Doesn't every physics student ask this question when he learns calculus of variations? Does anybody ever answer this question? Ever? If so, please educate me.

Answer 1

Here is my answer, which is basically an expanded version of Greg Graviton's answer.

The question of why one can treat position and velocity as independent variables arises in the definition of the Lagrangian L itself, before one thinks about varying the action , and has therefore nothing to do with calculus of variation.

I) On one hand, let us first consider the role of the Lagrangian. Let there be given an arbitrary but fixed instant of time . The (instantaneous) Lagrangian is a function of both the instantaneous position and the instantaneous velocity at the instant t0. Here and are independent variables. Note that the (instantaneous) Lagrangian does not depend on the past nor the future . (One may object that the velocity profile   is the derivative of the position profile so how can and be truly independent variables? The point is that since the equation of motion is of 2nd order, one is still entitled to make 2 independent choices of initial conditions: 1 initial position and 1 initial velocity.) We can repeat this argument for any other instant

II) On the other hand, let us consider calculus of variation. The action functional depends on the whole (perhaps virtual) path. Here the time derivative does depend on the function . Extremizing the action functional

with appropriate boundary conditions leads to Euler-Lagrange equation,

III) Note that

is a total time derivative, not an explicit time derivative , so that the Euler-Lagrange equation (2) is really a 2nd-order ordinary differential equation (ODE),

To solve for the path , one should specify two initial conditions, e.g., and