Intuition behind Hamilton's Variational Principle

Question

Background: I am an upper level undergraduate physics student who just completed a course in classical mechanics, concluding with Lagrangian Mechanics and Hamilton's Variational Principle.

My professor gave a lecture on the material, and his explanation struck me as a truism.

Essentially, he argued that the difference between the Lagrangian evaluated along the parameters describing the true path and the Lagrangian evaluated along parameters corresponding to a mild perturbation of the parameters by a function an(x), where a is a scale factor, is zero.

Where exactly is the profundity in this statement? I understood it as "If we deviate the parameters away from the parameters that minimize the integral, and then take the limit as that deviation vanishes, the difference between the path described by these two sets of parameters is zero and the path must be the true path." Well of course this is true. What am I missing?

Alternatively are there any decent texts that outline this principle at an undergraduate level?

Answer 1

1) The key is, to FIRST order. If a function attains extremum at $x_0$, then $f(x_0+\epsilon)-f(x_0)\approx 0$ to first order in $\epsilon$. This is often written as $df|_{x_0}=0$. This is no longer true if you include terms proportional to $\epsilon^2$; write down the taylor series about the maximum for instance.

2) It is important to realise what these 'perturbations' are. These 'virtual displacements', i.e. you are looking at infinitesimally seperated alternate trajectories than the one you are considering. Call them $x(t)$ and $x(t)+\epsilon(t)=x(t)+a\eta(t)$. If the one you are considering, $x(t)$, is some kind of extremum, then, by definition,; to FIRST order in $\epsilon$ i.e the scale parameter $a$, you have that the difference vanishes.

3) Finally, why does such an extremum correspond to the true path amounts to a postulate. You have hints in other situations, such as Fermat's principle in optics, but why should such a principle extend has a priori little justification.

4) In Quantum mechanics, however, such a formalism a little more natural. When constructing the path integral, you only assume the existence of an operator that generates time translations, and your path integral then contains something you call the action(after you assume a specific form for the Hamiltonian). You then see that the trajectory that extremises the action contributes the most to the path integral, and you then identify it with the classical trajectory. However, this is rather circular and is meant to provide a plausibility argument, not a proof of the principle.

Answer 2

Short answer: Already "true path" is an ugly choice of wording to say the "path that minimizes the action", taken from Hamilton's principle of least action, intuitively: Mechanical systems favor paths along which the difference between the kinetic and potential energy is as small as possible. More formally Hamilton's principle says:

Given the action $S[\gamma]$, a functional of the path $\gamma$, the motion of classical systems coincides with extremals (see about functional derivatives) of this functional: $$S[\gamma ]= \int_{t_1}^{t_2}L dt$$

Where $L=T-U$, kinetic minus potential energy. What this principle doesn't necessarily imply is whether the extremal of $S$ corresponds always to a minimum or not. (Deeper insight: the Euler-Lagrange's equation corresponds only to a differential $F$ that depends only linearly on $\delta \gamma$, i.e. deviations from $\gamma.$ Higher order variations are omitted ($\mathcal{O}(\delta \gamma^2)$).) This you can also realize by noticing that we only have first order derivatives in the Euler-Lagrange equation.

In chaos theory for example, one often deals with points called conjugate points, which actually correspond to a maximum of $S[\gamma]$, and are to be considered as forbidden classically.

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As for the literature you asked about: I personally like a lot (V.I.)Arnold's Mathematical Methods of Classical Mechanics, which although is designed for a graduate level, you can still learn (very readable) a lot from the first 3 chapters of it (I'm sure you can borrow it from your univ's library). Alternatively, Landau & Lifshitz volume on Mechanics is among the best ones, you should definitely give it a try. If you like their style of writing, you will learn a lot from, if you don't there are tons of other classical books written differently.