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Given a particle in space, the EL equations give us a differential equation for determining how the partilce will move as time evolves.

I am comfortable deriving a least action principle which recovers the EL equations, with one caveat. I cannot follow the reasoning of requiring a fixed endpoint in the variation.

My derivation follows below: $$ \int dLdt = \int \left(\frac{dL}{dx}dx +\frac{dL}{d\dot{x}} d\dot{x}\right)dt $$ $$ =\int \left(\frac{dL}{dx}dx - \left(\frac{d}{dt} \frac{dL}{d\dot{x}}dx \right) +\frac{d}{dt}\left( \frac{dL}{d\dot{x}}dx \right)\right)dt. $$ At this point, I can make the final term vanish by assuming the start and end points of the integral are fixed, so $dx=0$.

However, I do not follow the reasoning of this.

Perhaps to outline my confusion, and provide some steps to my reasoning, I have considered the following:

  1. A separate question on a similar vein prompted discussion of a wave version of the least action principle, which does not require a fixed endpoint. This is the "Hamilton-Jacobi" formulation, linked here. My thinking here is, while the initial point is fixed, from there, the path spreads out in all directions until reaching all possible spatial locations at the end point. Contrasting this with the least action principle required to arrive at the EL equations, I think thus. The requirement of a fixed endpoint is not that we are stating a specific fixed endpoint, but that there exists 'a' fixed endpoint of the motion. This being in contrast to the wave version of the principle.

  2. My qualm with the above reasoning, is to ask as to the intermediate points of the motion. Upon solving the EL equations, the intermediate points of the motion are fixed, in addition to the endpoints. If (1) above is correct, how are the intermediate points distinguished in the EL derivation?

If my confusion is unclear, please prompt elaboration in a comment, and I can amend as required.

A note on fixed intermediate points

In the above, my reference to 'fixed intermediate points', describes the following. A particle in motion follows a path with a unique value at each point, $x(t)$. It is in this sense I refer to a particles path being 'fixed' at each point, in contrast to a wave, where there are a collection of points for every time $t$.

It is something about the notion of 'fixed points' at the heart of this question. Hopefully the above addendum is sufficient to warrant re-opening.

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  • $\begingroup$ Comment to this post (v3): Consider for clarity to elaborate and/or provide references to the the following mentioned claims: 1. A wave version of the least action principle, which does not require a fixed endpoint. 2. That intermediate points of the motion are fixed. $\endgroup$ Commented Nov 3 at 12:24

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  1. In the principle of least/extremal/stationary action/Hamilton's principle [which is a variational boundary value problem (BVP)], one should impose one of the following boundary conditions (BC):

    • Essential/Dirichlet BC: $\quad q(t_i)~=~q_i\quad\text{and}\quad q(t_f)~=~q_f,$

    • Natural BC: $\quad p(t_i)~=~0\quad\text{and}\quad p(t_f)~=~0,$

    • Combinations thereof,

    cf. e.g. my Phys.SE answer here.

  2. Boundary value problems (BVP) and initial value problems (IVP) should not be conflated, cf. e.g. my Phys.SE answers here & here.

  3. Hamilton's principal function $S(q,\alpha,t)$ is a solution to a non-linear 1st-order PDE [the Hamilton-Jacobi (HJ) equation]. This is not a variational boundary value problem per se.

    The Hamilton's principal function $S(q,\alpha,t)$ and the action functional $S[q]$ should not be conflated, cf. e.g. my Phys.SE answer here.

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At this point, I can make the final term vanish by assuming the start and end points of the integral are fixed, so $dx=0$.

However, I do not follow the reasoning of this.

If you are interested in a derivation of Hamilton's principle from the d'Alembert principle, it can be found in Section 51 of the textbook by Lanczos ("The Variation Principles of Mechanics").

The upshot is that we must require that the variation vanish at the endpoints. It is only with this requirement that d'Alembert's principle can be reformulated as a stationary action principle (Hamilton's principle).

As a brief recap of what is written in Lanczos, the key insight was to recognize that we can actually simply the problem by integrating over time the virtual work due to external and inertial forces: $$ \int_{t_1}^{t_2} dt \delta W_v $$ $$ =-\left[\sum_i m_i \vec v_i\cdot\delta \vec r_i\right]_{t_1}^{t_2} + \delta \int_{t_1}^{t_2} dt L\;,\tag{1} $$ where that equality elides a lot of steps (see Lanczos Eqs (51.1) through (51.8)).

Now, if we assert that no variations are allowed at the endpoints, the first term on the RHS of Eq. (1) is zero and the time integral of the virtual work becomes the same as the variation of the integral that we call the "action" $S$: $$ S=\int_{t_1}^{t_2}dt L \;. $$

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