Why do we use the Lagrangian and Hamiltonian instead of other related functions?

Question

There are 4 main functions in mechanics. $L(q,\dot{q},t)$, $H(p,q,t)$, $K(\dot{p},\dot{q},t)$, $G(p,\dot{p},t)$. First two are Lagrangian and Hamiltonian. Second two are some kind of analogical to first two, but we don't use them. Every where they write that using them can cause problems. But why?

Answer 1

Here is one argument:

1. Starting from Newton's 2nd law, the Lagrangian $L(q,v,t)$ is just one step away.

2. A Legendre transformation $v\leftrightarrow p$ to the Hamiltonian $H(q,p,t)$ is well-defined for a wide class of systems because there is typically a bijective relation between velocity $v$ and momentum $p$.

3. On the other hand, there is seldomly a bijective relation between position $q$ and force $f$ (although Hooke's law is a notable exception). Therefore the Legendre transforms $K(f,v,t)$ and $G(f,p,t)$ are often ill-defined.

Answer 2

To describe the motion of bodies, it is enough to know two variables — the coordinates and velocitys, or the coordinates and momentums, then the initial conditions are determined. In the case of setting other variables, it is necessary to use boundary conditions, which is more complicated. It is necessary to set the initial coordinates and speeds to satisfy the initial values of other variables. Only the initial coordinates and speed determine the movement of bodies. The value of other variables is a function of coordinates and velocity, and the inverse function does not always exist. When the inverse function does not exist or is not unique, the motion is undefined. To get the initial value of variables $\dot p,\dot q$, it is necessary to calculate the solution with initial variables - coordinate, speed, go back and get the initial value of variables $\dot p,\dot q$. In addition, by describing the motion in variables $\dot p,\dot q$, you will not achieve a complete description; instead of derivatives $\dot p,\dot q$, knowledge of variables $p,q$ is necessary. To find coordinates and impulses by their derivatives, it is necessary to know their initial values. The circle is closed.