9

I prefer to think like this. Inertial observer ($X$) is at rest in his/hers reference frame. The world-line of $X$ is the longest possible route between any two events. This follows since, in its frame, $X$ is moving fully along the temporal axis, therefore $ds=cd\tau$ (displacement in time; $c$ is the speed of light), and $d\mathbf{r}=\mathbf{0}$, and one ...


6

For a single particle, it does not matter what prefactor you use, the equations of motion and everything else stays the same. The factors only start to matter when you couple different systems to each other. For example, consider a charged particle in an electromagnetic field described by a vector potential $A_\mu$. The right action describing its movement ...


4

The equation you mention is the action of a single point particle. $$S =-mc^2\int d\tau$$ The unit of action is energy multiplied by time, in the present case the rest energy which corresponds to the mass of the particle multiplied by the proper time of the particle. This equation refers to the action from the point of view of the reference frame of the ...


3

Extremising the action just means finding the classical solution by the principle of stationary action, which eventually boils down to solving the Euler-Lagrange equations $$ \partial_\mu \frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)} = \frac{\partial \mathcal{L}}{\partial \phi}, $$ in this case $$ \partial_\mu \partial^\mu \phi = V'(\phi) = m^2 \...


2

Let us for simplicity consider Minkowski space although the generalization to curved spacetime is straightforward. Lorentz invariance suggests that the Lagrangian one-form for a massive point particle should be $$\mathbb{L}~=~ f(\dot{x}^2)\mathrm{d}\lambda, \qquad \dot{x}^2~:=~\eta_{\mu\nu} \dot{x}^{\mu}\dot{x}^{\nu}~>~0, \qquad \dot{x}^{\mu}~:=~\frac{dx^{...


2

This is because you work in first order in $\varepsilon$. So you need to consider only the linear terms in $\varepsilon$: $$ \delta g^{\mu\nu}(x)=\bar g^{\mu\nu}(x)-g^{\mu\nu}(x)=-\frac{\partial{ \bar{g}^{\mu\nu}}}{\partial x^{\alpha}}\varepsilon^{\alpha}+ \frac{\partial \varepsilon^{\mu}}{\partial x_{\nu}}+\frac{\partial \varepsilon^{\nu}}{\partial x_{\mu}}...


1

Well, it is possible to work directly with the Lagrangian in Lagrange's equations, cf. e.g this Phys.SE post. However, if one wants to have a variational principle that leads to Euler-Lagrange (EL) equations, it is necessary to introduce the action functional $S=\int_{t_i}^{t_f}\! dt~L$. This leads to the principle of stationary action/Hamilton's principle. ...


1

Since action has the units of angular momentum, the proportionality constant needs the units of energy by dimensional analysis. It must also be Lorentz-invariant, so is $mc^2$ times some real number. This number's modulus can be varied without changing the resulting equations of motion, but @Cryo's answer shows a modulus of $1$ recovers the usual $\int(T-V)...


1

Note that the use of Hamilton's principle (a.k.a. the principle of stationary action) for systems with semi-holonomic constraints in Ref. 1 is inconsistent with Newton's laws, and has been retracted on the errata homepage for Ref. 1. See Ref. 2 for details. See also this & this related Phys.SE posts. For starters, Ref. 1 provides a wrong (or at best an ...


1

The quantum equivalent to the optimal control of densities is a topic of interest in the areas of large population control, optimal mass transport and mean field games (try https://arxiv.org/abs/1810.06064) in the control theory. The idea in the related literature is to transport a density given the individual micro-dynamics. The question of the minimum time ...


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