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Derivation of gravitational dynamics using Lagrangian?

First, is it essential to use two different Lagrangians? What is the physical basis and meaning of having two Lagrangians? Is it a rigorous approach? No, it is not essential to use two different ...
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3 votes

Derivation of gravitational dynamics using Lagrangian?

It is important to mention that in this model the point particle masses $m_i$ serve as sources $$ \rho({\bf x},t)=\sum_i m_i \delta^3({\bf x}-{\bf x}_i(t)) $$ for the gravitational potential field $\...
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Derivation of gravitational dynamics using Lagrangian?

The first Lagrangian is the Lagrangian for the motion of a point particle in a Newtonian gravitational potential (or field), whereas the second is the Lagrangian for the gravitational potential (or ...
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Variation of Pontryagin density $*R^{abcd}R_{abcd}$ with inverse metric $g^{ab}$

EDIT: I'm not quite so sure about any usefulness of this answer right now. I was not really familiar with the Pontryagin classes and all that stuff, so I attempted to vary the functional by hand. ...
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Variation of Pontryagin density $*R^{abcd}R_{abcd}$ with inverse metric $g^{ab}$

The Pontryagin Lagrangian: $$\int *R^{abcd}R_{abcd} \sqrt{-g}\,d^4x$$ is invariant against variation of metric. That is exactly why it's called a topological term. As you can easily see, the $$\sqrt{-...
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Fierz-Pauli action as an effective action from Einstein-Hilbert acition?

I've been looking for similar info as I'm interested in how to construct Lagrangians for higher spin fields. I came across this link. Hopefully you'll find it helpful too. http://www.ugr.es/~bjanssen/...
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How is proper time extremized?

This is a case where if we generalize from special relativity to general relativity, the possible options become fewer: We want to find the trajectory of a massive point particles that moves between 2 ...
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How is proper time extremized?

First off, an answer to the question "why do timelike geodesics extremize proper time". A geodesic is by definition the path with the shortest distance between two points on a manifold i.e. ...
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How is proper time extremized?

In flat spacetime in Cartesian coordinates, it is not hard to prove that given two events $\mathbf p_1 = (ct_1,x_1,y_1,z_1)$ and $\mathbf p_2\equiv (ct_2,x_2,y_2,z_2)$ which are timelike-separated (i....
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How is proper time extremized?

How is proper time extremized? "How" is easy. The proper time interval is: $$ S = \int d\tau $$ If the particle has a trajectory $x(t)$, going from $x_1$ at $t_1$ to $x_2$ at $t_2$, then ...
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Scalar field equation of motion in FRW metric

You made a mistake when you varied the action. Explicitly, the Lagrangian density is: $$ \mathcal L = (-\frac{1}{2}g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi-V(\phi))\sqrt{-g} $$ so the Euler-Lagrange ...
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How does nature know Hamilton's principle?

It might be a circular reasoning, but, if the path that extremizes the action satisfies the Euler-Lagrange equations, there is at every time a local condition that the path must satisfy without ...
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How does nature know Hamilton's principle?

This is a rather deep question. In classical physics, you have to be pragmatic: the theory gives results that line up with reality, so we're happy. From a quantum point of view, I'd suggest you have ...
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What is the intuitive implication behind $L' = L + \frac{df}{dt}$ not affecting equations of motion?

Well, the intuitive explanation is clear and can be generalized to field theory: The functional derivative $$\frac{\delta S}{\delta\phi^{\alpha}(x)}\tag{1}$$ (at an interior point $x\in M$ of ...
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2 votes
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What is the intuitive implication behind $L' = L + \frac{df}{dt}$ not affecting equations of motion?

There is the physics SE feature of offering (in the column on the right hand side of the page) suggestions, under the heading 'Related' In this particular case the column 'Related' offers a link to a ...
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Is there more than one GR action (if we include boundary terms)?

The action you call $S_2$ was discovered first! (Or if not first, then at around the same time. Remember we had the EFE before the EH action). This action is known by a few names, the Einstein action, ...
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What is the intuitive implication behind $L' = L + \frac{df}{dt}$ not affecting equations of motion?

I guess the question boils down to what the Lagrangian is in relation to the equations of motion. Yes, exactly. The clue is that the equations of motions for $q(t)$ are equivalent to the principle of ...
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Tight-Binding method and orthogonality of Bloch functions

Your second equation should be $$ \psi_n(\vec{r}-\vec{R}) = \sum_{n'= 1}^N C_{nn'} \phi_{n'}(\vec{r}-\vec{R})\;\;\; n = 1, ..., N, $$ and while the $\phi_{n'}$ and $\phi_{n}$ at the same site are ...
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