Obtaining Lagrange-Euler's equations for a system

[Elwin.Martin]

I've been trying to learn the basics of the Lagrangian formulation of Mechanics and I skipped over something that I'd like to ask about (I basically just took it for granted and did problems assuming it to be true).

Most of the books I've looked at introduce action as the integral of the Lagrangian for a given interval. I've read that q and $\dot{q}$ are enough to define a system's Lagrangian. Is there a proof for the least action principle or am I just missing something really obvious?

I have another question too, this one is math related though. Most of the resources I've looked at state that the condition for an extremum is that the first variation should be zero. This leads to an integral of the Lagrangian that simplifies to [see attached image] and this leads to Lagrange's equations (something along the lines of

Wiki says that the last step is a result of the "Fundamental Lemma of Calculus of Variations" but I'm just missing something that is supposed to be obvious here, again.

Thank you for your time, any and all help is greatly appreciated.

Elwin

 

[Fightfish]

Hamilton's Principle is often mistakenly referred to as the "principle of least action"; to be more accurate it is actually the "principle of stationary action", for the value of the action need not be a minimum.

The Euler-Lagrange equations can actually be derived from D'Alembert's Principle:
$$\sum_{i} (F_{i} - m_{i} a_{i})\cdot \delta r_{i} = 0$$

Well, we can easily show that Euler-Lagrange equations (& consequently the principle of stationary action) are consistent with Newton's Second Law. If you want to go deeper though, the origin of the principle of stationary action can be traced back to quantum mechanics. To put it simply, it turns out that the particle actually takes all possible paths from one point to another. Each path is characterised by a phase. Destructive interference occurs when the action is non-stationary while constructive interference happens when the action is at a stationary value, hence that is the path that we observe.

The fundamental lemma of the calculus of variations states that for a functional
$$J[f(x)] = \int_{a}^{b} f(x) g(x) dx = 0$$
if g (a) = g (b) = 0 ie. g (x) = 0 at the endpoints, then it follows that f (x) must be equal to 0 over the interval [a,b]. The proof is available in most texts and online sources.

 

[Elwin.Martin]

Hamilton's Principle is often mistakenly referred to as the "principle of least action"; to be more accurate it is actually the "principle of stationary action", for the value of the action need not be a minimum.

The Euler-Lagrange equations can actually be derived from D'Alembert's Principle:
$$\sum_{i} (F_{i} - m_{i} a_{i})\cdot \delta r_{i} = 0$$

Well, we can easily show that Euler-Lagrange equations (& consequently the principle of stationary action) are consistent with Newton's Second Law. If you want to go deeper though, the origin of the principle of stationary action can be traced back to quantum mechanics. To put it simply, it turns out that the particle actually takes all possible paths from one point to another. Each path is characterised by a phase. Destructive interference occurs when the action is non-stationary while constructive interference happens when the action is at a stationary value, hence that is the path that we observe.

The fundamental lemma of the calculus of variations states that for a functional
$$J[f(x)] = \int_{a}^{b} f(x) g(x) dx = 0$$
if g (a) = g (b) = 0 ie. g (x) = 0 at the endpoints, then it follows that f (x) must be equal to 0 over the interval [a,b]. The proof is available in most texts and online sources.

I'm aware of the distinction, but I understand that in most cases this is a minimum, hence the name (I've read the sections in Goldstein, Landau and Marion).

I'm less concerned about how to derive the equations (I have a functional understanding [no pun intended] I've solved the problems in Landau and in Marion), and more concerned with understanding why using q and q-dot specifies the system entirely (I'm also familiar with the basics of path integral formulation, so I know where the equations come from in this sense too [via A. Zee's text]).

Is this like knowing the momentum and position of a particle and knowing it's evolution from then on? (This is something else that I've been told but I haven't actually seen, I just know it to be true). I am just having some sort of basic problem here.

If I understand correctly, the goal of most mechanics problems is to determine the state of a system at a given time based on some initial conditions and the equations of motion that we derive from the information given. In a basic one dimensional case, this could be discovering that springs oscillate harmonically (or in a more complicated setting showing the the path of the end of the string with some sort of dampening). But the equation comes down to
something in the form of
x(t)= ___ or A(x,t)=___ some actual description of the systems position for a given time (or time and x coordinate)
Do you see what I'm asking about? I'm just not sure what is meant by the q and q-dot representing the system. (I understand these are generalized coordinates, too)

I guess I'll just Google a proof for the fundamental lemma, then.

Edited for clarity**

 

[Fredrik]

more concerned with understanding why using q and q-dot specifies the system entirely

The key part of that is that the E-L equations, just like Newton's second law, is a differential equation of the form $$x''(t)=f(x(t),x'(t),t)$$ where f is a nice enough function to ensure that we can use the existence and uniqueness theorem that says that such an equation has exactly one solution for each initial condition of the form $x(t_0)=x_0,\ x'(t_0)=v_0$.

 

[PJC01]

I can understand your frustration and confusion with the Lagrangian formulation of Mechanics. It is a complex and abstract concept that can be difficult to grasp at first. However, with patience and practice, it can become a powerful tool for understanding and solving problems in physics.

To answer your first question, the proof for the least action principle lies in the fundamental principle of stationary action, which states that the true path of a system is the one that minimizes the action integral. This can be derived using the Euler-Lagrange equations, which are a direct consequence of the least action principle. These equations are essentially a set of mathematical conditions that describe the path of a system that minimizes the action.

As for your second question, the Fundamental Lemma of Calculus of Variations is a mathematical theorem that states that if a function has a stationary point (where the first variation is zero), then it is also a local minimum or maximum. In the case of the Lagrangian, this means that the path of the system that satisfies the Euler-Lagrange equations is a minimum of the action integral. This is why we use the first variation to find the path that minimizes the action.

I hope this helps clarify some of your concerns. Keep practicing and seeking out resources to deepen your understanding of the Lagrangian formulation. It may seem daunting at first, but with perseverance, you will be able to master it and use it to solve complex problems in physics.