Short question:

What is the physics interpretation of max & inflection points in the principle of least action?

Long question:

If $$L(q_1,q_2;t)=K-V$$

then let $$S = \int^{t_1}_{t_2} L(q_1,q_2;t)dt$$

Taking $$\delta S=0$$ is what usually called the least action of principle

The path taken by the system between times $t_1$ and $t_2$ is the one for which the action is stationary (no change) to first order. -wiki

eventually it gives us Euler–Lagrange equation.

"Given there is a minimum point", that physically means assuming the system tends to be steady.

However, logically, those points can be maximum or inflection points.

I wonder what maximum points and inflection points mean physically? Will we observe things different if those points turn out to be saddle points?

I found this in wiki:

In 1842, Carl Gustav Jacobi tackled the problem of whether the variational principle always found minima as opposed to other stationary points (maxima or stationary saddle points); most of his work focused on geodesics on two-dimensional surfaces. The first clear general statements were given by Marston Morse in the 1920s and 1930s, leading to what is now known as Morse theory. For example, Morse showed that the number of conjugate points in a trajectory equalled the number of negative eigenvalues in the second variation of the Lagrangian.

Would anyone explain the work by Jacobi and Morse to me?