Help a novice with EL equation derivation

In summary: This expansion can be done explicitly, or using the Laplacian. Doing it explicitly gives a way to calculate the variation as a function of the differentials, which can be useful for understanding the minimum principle.
  • #1
Alex Cros
28
1
Hello everyone,

Reading Landau and Lifshitz Course of Theoretical Physics Volume 1: Mechanics (page 3) I got suck in the following step (and I cite in italics):

The change in S when q is replaced by q+δq is
[tex]\int_{t_1}^{t_2} L(q+δq, \dot q +δ\dot q, t)dt - \int_{t_1}^{t_2} L(q, \dot q, t)dt[/tex]

(So far so good)

When this difference is expanded in powers of δq and [itex]δ\dot q[/itex] in the integrand, the leading terms are of the first order.

How do you expand that? Could anyone show me how explicitly if you don't know the explicit form of the Lagrangian?

The necessary condition for S (where S is the action) to have a minimum is that these terms (called the first variation, or simply the variation, of the integral) should be zero. Thus the principle of least action may be written in the form
[tex]δS = δ \int_{t_1}^{t_2} L(q, \dot q, t)dt = 0[/tex]


(Which I'm fine with the above expression)

Or, effecting the variation,
[tex]\int_{t_1}^{t_2} (
\frac{\partial L}{\partial q}δq+
\frac{\partial L}{\partial \dot q}δ\dot q)
dt = 0[/tex]

Now my guess would have included [itex]\frac{\partial L}{\partial t}[/itex] like:
[tex]\int_{t_1}^{t_2} (
\frac{\partial L}{\partial q}δq+
\frac{\partial L}{\partial \dot q}δ\dot q + \frac{\partial L}{\partial t}dt)
dt = 0[/tex]

To perform the total differential of all variables.
Explain me like if I'm five why this guess is wrong.

Thanks so much in advance! And sorry for my lack of elemental knowledge.
-Alex
 
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  • #2
Hi,

##t## plays a different role here: it is the integration variable. We want to minimize ##S## by finding a path ##\Bigl ( q(t), \dot q(t) \Bigr )## that achieves this ##\delta S=0## condition. There is no value of ##t## to find; it just runs from ##t_1## to ##t_2## and nothing can be done with ##\partial L \over \partial t##. The time dependence of ##L## is present in the integration, though.
 
  • #3
Thanks BvU!
Any ideas on how to do this and why is it relevant?
When this difference is expanded in powers of δq and δ[itex]\dot q[/itex] in the integrand, the leading terms are of the first order.
 
  • #4
Alex Cros said:
Thanks BvU!
Any ideas on how to do this and why is it relevant?

The expansion is basically a Taylor expansion about δq and δ{itex}\dot q{\itex}t keeping the first order powers in in the differentials. For a deeper understanding, read on functional derivatives and or the calculus of variations.
 

1. What is the EL equation and why is it important in science?

The EL equation, also known as the Euler-Lagrange equation, is a fundamental tool used in mathematical physics to describe the motion of particles or systems. It is derived from the principle of least action, which states that a physical system will follow a path that minimizes the action (a measure of the integral of the kinetic and potential energies) between two points. The EL equation is important because it allows us to mathematically describe the behavior of physical systems and make predictions about their motion.

2. How is the EL equation derived?

The EL equation is derived by applying the calculus of variations to the principle of least action. This involves taking the partial derivative of the action with respect to the position or velocity of the system, setting it equal to zero, and solving for the equations of motion. The result is a set of differential equations that describe the system's behavior and can be solved to determine its trajectory.

3. Can the EL equation be applied to any physical system?

Yes, the EL equation can be applied to any physical system, as long as the system can be described using a Lagrangian function. This function takes into account the kinetic and potential energies of the system and any external forces acting on it. By using the appropriate Lagrangian, the EL equation can be derived and applied to a wide range of systems, from simple particles to complex systems of particles.

4. How is the EL equation used in scientific research?

The EL equation is used in many areas of scientific research, including classical mechanics, quantum mechanics, and field theory. It allows scientists to analyze the behavior of physical systems and make predictions about their motion. The EL equation is also used in the development of mathematical models and simulations to study complex systems and phenomena.

5. Are there any limitations or assumptions when using the EL equation?

Like any mathematical model, the EL equation has its limitations and assumptions. It assumes that the system is conservative (i.e. energy is conserved) and that the Lagrangian is a well-defined function. It also does not take into account relativistic effects, so it is not applicable to systems moving at speeds close to the speed of light. Additionally, the EL equation may not always have a unique solution, and numerical methods may be required for complex systems.

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