action Definition and 16 Threads

In Dirac's "General Theory of Relativity", he develops the "comprehensive action principle" in chapter 30. Simply put, he writes a combined action for the gravitational field and all other matter-energy fields ##I=I_g+I'##. Varying this: $$\delta(I_g+I')=\int ( p^{\mu\nu}\delta g_{\mu\nu} +...

The action for a free particle is $$I=-m\int{ds} = \int \left(-m\frac{ds}{dt}\right) dt \quad\quad\quad(*)$$ hence the Lagrangian is $$L=-m\frac{ds}{dt}=-\frac{m}{\gamma}=-m\sqrt{1-v^2} .$$ Dirac ("General Theory of Relativity", p. 52) infers this by checking that it gives the correct spatial...

I understand the principle of stationary action and why it is very powerful. However I'm trying to (at least qualitatively) understand why this principle works. I believe that, unlike theorems, principles are taken as true, and thus we just accept it. To me, this seems like a cop-out. If we...

The force two static electric charges exert on each other fulfills (velocity of light set unity) \begin{equation} F r^2 = N_1 N_2 \alpha \hbar \;, \end{equation} where ##F## is the force, ##r## is the mutual distance, ##\alpha## is the dimensionless fine structure constant, ##\hbar## is Planck's...

I've a doubt regarding the application of the principle of minimum action to real cases. Pick an inertial frame with a potential ##V## defined on it. The principle (aka Hamilton's principle) claims that the actual path taken from a body gives rise to a "stationary" action when calculated from a...

When I do a press-up I consciously push down on the ground, my muscles tense, I breathe harder, I sweat, etc. When the Earth pushes back up on me (Newton's 3rd Law) is it 'doing anything', or does it push back up on me merely by virtue of it being there?

Normally action reaction forces do not move things. In this problem they move. I wish to discuss in what way these so called constrain forces contributeto motion or kinetic energy L. Do they do wirk. Yes then why no then why not?

In matrix representation, the special unitary group is distinguished from the more general unitary group by the sign of the matrix determinant. However, this presupposes that the special unitary group is formulated in matrix representation. For a unitary group action NOT formulated in matrix...

I just need a hint to get started, and then I reckon the rest will follow... We consider a theory where matter is a covector field ##\omega_a## which is described by a diffeomorphism-invariant action ##S_m##. Define:$$E^{a} = \frac{1}{\sqrt{-g}} \frac{\delta S_m}{\delta \omega_a}$$Also, ##T^{ab}...

Dirac derives Einstein's field equations from the action principle ##\delta I=0## where $$I=\int R\sqrt{-g} \, d^4x$$ (##R## is the Ricci scalar). Using partial integration, he shows that $$I=\int L\sqrt{-g} \, d^4x$$ where ##L## involves only ##g_{\mu\nu}## and its first derivatives, unlike...

Under the RELATIVISTIC definition of Action, is the Action for a beam of light always zero? Thanks in advance.

Here is an action for a theory which couples gravity to a field in this way:$$S = \int d^4 x \ \sqrt{-g} e^{\Phi} (R + g^{ab} \Phi_{;a} \Phi_{;b})$$I determine\begin{align*} \frac{\partial L}{\partial \phi} &= \sqrt{-g} e^{\Phi} (R + g^{ab} \Phi_{;a} \Phi_{;b}) \\ \nabla_a \frac{\partial...

I am stuck at the final part where one is supposed to show that the derivative of the second term of the action gives the mass term in the Majorana equation. For $$\chi^T\sigma^2\chi = -(\chi^\dagger\sigma^2\chi^*)^*$$ we get $$\frac{\delta}{\delta\chi^\dagger}(\chi^\dagger\sigma^2\chi^*)^*$$...

How and/or why does the existence of the quantum of action (Planck's constant) cause indeterminism?

Here is the action: ##S = \frac{1}{16\pi} \int d^4 x \sqrt{-g} (R\phi - \frac{\omega}{\phi} g^{ab} \phi_{,a} \phi_{,b} + 16\pi L_m)## the ordinary matter is included via ##L_m##. Zeroing the variation ##\delta/\delta g^{\mu \nu}## in the usual way gives ##\frac{\delta}{\delta g^{\mu \nu}}[R\phi...