Euler-lagrange, positivity of second order term

[jostpuur]

For twice differentiable path $x:[t_A,t_B]\to\mathbb{R}^N$ the action is defined as

$$S(x) = \int\limits_{t_A}^{t_B} L\big(t,x(t),\dot{x}(t)\big) dt$$

For a small real parameter $\delta$ and some path $\eta:[t_A,t_B]\to\mathbb{R}^N$ such that $\eta(t_A)=0$ and $\eta(t_B)=0$ the action for $x+\delta\eta$ can be approximated as follows:

$$S(x+\delta\eta) = \int\limits_{t_A}^{t_B}\Big( L\big(t,x(t),\dot{x}(t)\big)$$
$$\quad+ \delta\sum_{n=1}^N \Big(\eta_n(t) \frac{\partial L(t,x(t),\dot{x}(t))}{\partial x_n} + \dot{\eta}_n(t) \frac{\partial L(t,x(t),\dot{x}(t))}{\partial \dot{x}_n}\Big)$$
$$\quad+\frac{1}{2}\delta^2 \sum_{n,n'=1}^N\Big(\eta_n(t)\eta_{n'}(t) \frac{\partial^2 L(t,x(t),\dot{x}(t))}{\partial x_n\partial x_{n'}} + 2\eta_n(t)\dot{\eta}_n(t) \frac{\partial^2 L(t,x(t),\dot{x}(t))}{\partial x_n\partial \dot{x}_{n'}}$$
$$\quad\quad + \dot{\eta}_n(t)\dot{\eta}_{n'}(t)\frac{\partial^2 L(t,x(t),\dot{x}(t))}{\partial\dot{x}_n\partial \dot{x}_{n'}}\Big) + O(\delta^3)\Big)dt$$

The Euler-Lagrange equations deal with condition that the first order term is zero. Has the positivity of the second order term been studied at all? It is quite common, that in classical mechanics it is believed that the action is minimized, but it is not proven in any ordinary books. What if the zero of the gradient was a saddle point in some case? Would that be a surprise? Saddle points can exist in infinite dimensional spaces too.

 

[Matterwave]

I don't think it's true that in general the second order term must be positive for any physical Lagrangian. For example, the path taken by a particle in free fall in relativistic mechanics is the path of maximum proper time, not minimum proper time.

The Lagrangian in that case is basically a constant with the proper time as the integration variable. And the action is therefore maximized not minimized.

 

[UltrafastPED]

This is not a physics question, but rather a question about the variational calculus.

The entire field has been studied extensively by mathematicians. You should be able to find discussions which satisfy your curiosity in texts like Forsyth's "Calculus of Variations", first published in 1927:
https://archive.org/details/CalculusOfVariations

A reprint is available from Dover; here is a review:

"Forsyth's Calculus of Variations was published in 1927, and is a marvelous example of solid early twentieth century mathematics. It looks at how to find a FUNCTION that will minimize a given integral. The book looks at half-a-dozen different types of problems (dealing with different numbers of independent and dependent variables). It looks at weak and strong variations. This book covers several times more material than many modern books on Calculus of Variations. The down-side, of course, is that the proofs move quickly (it can take the reader a few hours to fill in the missing steps in order to verify Forsyth's calculations in a proof) and do not worry about truly bizarre behavior (such as encountered in nonlinear dynamics). But the proofs are complete (given the 1927 understanding of derivatives of functions) and quite solid (again, by 1927 standards).

I reccomend this book to anyone who wishes to explore the wild, wild world of Calculus of Variations. Yes, there are easier books on the subject, but this one is a gem."