Euler-lagrange, positivity of second order term

In summary, the action for a twice differentiable path is defined as the integral of the Lagrangian over the given time interval. For a small real parameter, the action for a slightly altered path can be approximated. The Euler-Lagrange equations deal with the condition that the first order term is zero. The positivity of the second order term has been studied extensively in the field of variational calculus. A classic text on the subject is Forsyth's "Calculus of Variations", which covers a wide range of problems and variations. While the proofs may be challenging, the book offers a solid understanding of the subject.
  • #1
jostpuur
2,116
19
For twice differentiable path [itex]x:[t_A,t_B]\to\mathbb{R}^N[/itex] the action is defined as

[tex]
S(x) = \int\limits_{t_A}^{t_B} L\big(t,x(t),\dot{x}(t)\big) dt
[/tex]

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

[tex]
S(x+\delta\eta) = \int\limits_{t_A}^{t_B}\Big( L\big(t,x(t),\dot{x}(t)\big)
[/tex]
[tex]
\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)
[/tex]
[tex]
\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'}}
[/tex]
[tex]
\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
[/tex]

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.
 
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  • #2
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.
 
  • #3
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."
 

FAQ: Euler-lagrange, positivity of second order term

1. What is the Euler-Lagrange equation?

The Euler-Lagrange equation is a mathematical expression used to determine the stationary points of a functional. It is commonly used in the calculus of variations, a branch of mathematics that deals with finding the path or function that minimizes or maximizes a given functional.

2. How is the Euler-Lagrange equation derived?

The Euler-Lagrange equation is derived by setting the functional's derivative equal to zero and solving for the unknown function. This results in a differential equation that represents the stationary points of the functional.

3. What is the significance of the positivity of the second order term in the Euler-Lagrange equation?

The positivity of the second order term in the Euler-Lagrange equation is important because it ensures the functional has a minimum or maximum value at the stationary point. If the second order term is negative, the functional may have a saddle point instead of a minimum or maximum.

4. How does the positivity of the second order term relate to the stability of a system?

The positivity of the second order term in the Euler-Lagrange equation is directly related to the stability of a system. A positive second order term indicates that the system is stable, while a negative second order term indicates instability. This is because a positive second order term ensures the system will tend towards its minimum or maximum value.

5. Are there any practical applications of the Euler-Lagrange equation and the positivity of the second order term?

Yes, the Euler-Lagrange equation and the positivity of the second order term have many practical applications in fields such as physics, engineering, and economics. They can be used to optimize systems, find the most efficient paths, and solve various optimization problems.

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