# Tagged Questions

The action is the integral of the Lagrangian over time, or the integral of the Lagrangian Density over both time and space.

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### Is there a Maupertuis principle for General Relativity?

The motion of a point particle in classical mechanics is given by Newton's equation, $\mathbf{F}=m\mathbf{a}$. Suppose all forces considered are conservative and we have a constant total energy $h$. ...
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### How to derive the true spatial paths (orbits) from the Jacobi-Maupertuis condition

How can differential equations describing a physical object's true spatial paths (orbits) be derived from the time-independent Jacobi-Maupertuis principle of least action? According to this, it is ...
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### Why do we sum relativistic intervals in relativistic action of a massive point-particle, and not a function from it?

Relativistic action as follows (which should explain relativistic motion of a classical particle): $$S = C \Delta s=C\int ds$$ Where $C$ is some constant and $\Delta s$ is relativistic interval. ...
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### Is boundary well defined if variation of metric don't vanish on the boundary?

Suppose that you want to calculate the variation $\delta S$ of an action induced by some arbitrary variation $\delta g_{\mu \nu}$ of the spacetime metric : S = \int_{\Omega} \mathscr{...
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### Origin of Hamilton's variational principle [duplicate]

My question is what is the theoretical origin of Hamilton's principle. I mean is there any rigorous mathematical proof of this principle from some more basic principles?
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### What are the boundary conditions associated to this lagrangian?

Suppose that $L(q^i, \dot{q}^i)$ is a standard and well behaved lagrangian associated to some Dirichlet boundary conditions : $q^i(t_1) = q_1^i$ and $q^i(t_2) = q_2^i$. Now I have this new lagrangian ...
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### Boundary conditions of fields from the stationary action principle

First principle of stationary action Consider a real Klein-Gordon scalar field $\phi$ living in a $D$ dimensional flat spacetime. The field is considered off shell (the on shell condition is defined ...
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### On the surface, is the law of maximum entropy production the same as principle of least action?

I just have read about the law of maximum entropy production. Someone has idolized it enough to make an whole website just for it: http://www.lawofmaximumentropyproduction.com/ A system will ...
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### Why can't we fix the metric and its derivatives at boundary, with the variational method?

In general relativity and for its Einstein-Hilbert action, we usually ask that the metric variations $\delta g_{\mu \nu}$ cancel on the boundary $\partial \, \Omega$ of some region $\Omega$ of the ...
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### Variation of the Einstein-Hilbert action in D dimensions without the Gibbons-Hawking-York term

Consider the standard Einstein-Hilbert action in $D \ne 2$ dimensions spacetimes : $$S_{EH} = \frac{1}{2 \kappa} \int_{\Omega} R \; \sqrt{- g} \; d^D x,$$ where $\Omega$ is ...
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### Action for solution of general nth order differential equation [duplicate]

Suppose I want to find solution to a general nth order differential equation. (If I am right about the logic then) one might say that the solution $y\equiv y(x)$ is that function for which the ...
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### Proof of Hamilton's principle [duplicate]

Is there a anything like a proof of Hamilton's principle? Where would I find it?
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### Action functional of Born-Infeld model

I have a Born-Infeld action functional like this $$I[A,\phi]~=~\int b^2(\sqrt{1+(|\bigtriangledown\times A|^2)/b^2}-1)+|D_A\phi|^2 + b^2(1-\sqrt{(1-|\phi|^2)^2/b^2} ).$$ Have any books or notes talk ...
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### What does it mean for an action to be defined “on-shell”?

Some actions like 11D supergravity are defined "on-shell". What does this mean exactly? Can you give me an example? Say for example the Klein-Gordon action. Can this be defined on-shell too?
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### Behaviour of action with respect to time

I was wondering if it was possible to say something general on the behaviour of the action : $$S[x(\tau)]=\int_0^T L(x,\frac{dx}{d\tau},t) dt$$ (where $x(\tau)$ defines a trajectory, with certain ...
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### Why does overall action need to have an extremum?

Quoting from Landau's and Lifshitz' Mechanics : The integral ${\int\limits_{t_1}^{t_2}}L(q, \dot{q},t)\,dt$ for the entire path must have an extremum, but not necessarily a minimum. This, ...
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### Deriving Expression for variation of Action (Lagrangian Mechanics)

I am studying Lagrangian mechanics and I have come across something that I do not understand. Basically the text I am reading skipped steps and I do not know how to get from point A to point B. I ...
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### Why are the Euler-Lagrange equations invariant if we add a surface term to the action?

In the lecture on Noether's theorem and the Lagrange formulation of classical field theories, my professor wrote A symmetry is a field variation that maps solutions to solutions, which is true if ...
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### Deriving velocity after elastic and inelastic collision via the Principle of Least Action

I am reading on the Principle of Least Action from a historical perspective. I am also trying to make sense of it from a contemporary point of view -- though my training in contemporary physics is ...
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### Why should physical theories always have a Lagrangian formalism? [duplicate]

I've often heard that every physical theory has some kind of Lagrangian formalism, or a formalism in terms of a principle of stationary action. The Standard Model has one, General Relativity has one, ...
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### Help me calculate the Euclidean action of a gravitating system!

I recently read Gibbons and Hawking's paper Action integrals and partition functions in quantum gravity, Phys. Rev. D 15 (1977) 2752. I am interested in repeating their calculations. It is fairly ...