any of several principles that find the physical trajectory of a system by minimizing or maximizing some value computed over the proposed path (for instance geometric optics can be reproduced by insisting on a minimum time principle).

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44 views

How do I derive geodesic equation using variational principle? [duplicate]

I am trying to derive the geodesic equation using variational principle. My Lagrangian is $$ L = \sqrt{g_{jk}(x(t)) \frac{dx^j}{dt} \frac{dx^k}{dt}}$$ Using the Euler-Lagrange equation, I have got ...
3
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2answers
61 views

shape formed by a stiff string with ends pinched together [on hold]

Suppose I have a string of length $L$ with a bending energy given by $$E=\frac{1}{2}\epsilon \int_0^L ds\, (\mathbf{R}''(s))^2 $$ Let's say I form a bight with it by pinching the ends together, ...
2
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1answer
43 views

Action with self-dual field strength

It is said that writing down an action in presence of a self-dual field strength is subtle and not known till date. The familiar example people give is that of type IIB super-gravity which has a ...
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1answer
53 views

Equivalence between principle of least action and minimum potential energy

Are the principle of least action and the principle of minimum potential energy equivalent? How does one show that? Also, are Newton's laws of motion equivalent to the principle of least action? How ...
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0answers
44 views

Principle of Most Action? [duplicate]

In Landau-Lifshitz - Vol 1. Mechanics, right after the introduction of the principle of leas action, there is the following comment: It should be mentioned that this formulation ($S = ...
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1answer
62 views

Conceptual problem with action considered as function of endpoints

I am having some trouble with understanding why it makes sense to consider action in classical mechanics as function of endpoints $q_{initial}, \ q_{final}$ and endtimes $t_{initial}, \ t_{final}$. ...
3
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34 views

“Simple” Variation of the gravity action with boundary

I'm concerned with the derivation of the quasi-local stress tensor (getting from eqn 2.4 to eqn 2.6 in this paper: http://arxiv.org/abs/hep-th/0508218). As is the case with all the references I have ...
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0answers
101 views

Canonical second quantization vs canonical quantization with multisymplectic form in AQFT

First of all, I'm a mathematician that knows less than the basics of QFT, so forgive me if this question is trivial. Please, keep in my mind that my background in physics is very poor. 1) The usual ...
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2answers
98 views

Principle of least action: $\frac{d S_{cl}}{dt_b} = \frac{\partial S_{cl}}{\partial t_b} + \frac{\partial S_{cl}}{\partial x_b}\dot{x}_b$

Question I cannot see how I can obtain the yellow highlighted section on the RHS from that of the LHS. The following equation can be found in both my lecture notes(*1) (page 9, equation 2.7) and is ...
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0answers
38 views

Confusion with Fermat’s principle of least time [closed]

Here is an illustration used in my book to prove format’s principle of least time. My book says that ACB will be the shortest path. It is obvious that light reaches E first, and then X. If the ...
3
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1answer
70 views

To derive the relation between work function and potential energy

I'm reading "The variational principles of mechanics- Lanczos", The author mentions a relation between Work-Function $U(q_1,q_2,\cdots,q_n,\dot q_1,\dot q_2,\cdots,\dot q_n)$ and the potential ...
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2answers
65 views

Are the generalized coordinates in Lagrangian mechanics really independent?

In Goldstein's Classical Mechanics, Chapter 2.3: Derivation of Lagrange's Equations From Hamilton's Principle part of the derivation involves each of the generalized coordinates being independent. $$ ...
3
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1answer
58 views

Variational proof of the Hellmann-Feynman theorem

I use the following notation and definition for the (first) variation of some functional $E[\psi]$: \begin{equation} \delta E[\psi][\delta\psi] := \lim_{\varepsilon \rightarrow 0} \frac{E[\psi + ...
6
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1answer
148 views

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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1answer
40 views

Why is the potential independent of the generalized velocity?

In Goldstein, Classical Mechanics, Chap. 1.4 we derive Lagrange's equations from D'Alembert's Principle. My question is regarding the last part of the derivation, specifically the part where he ...
0
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1answer
85 views

How to proceed (Tough Problem) [closed]

The problem that I am considering is to find the shortest path (or geodesic) on a surface with the equation $z=f(x,y)$. The path is parameterized by $s$ so that the path goes from ...
1
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1answer
45 views

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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2answers
75 views

Principle of least action and greedy algorithm

Is the principle of least action sort of a greedy algorithm that all mechanical systems follow?, sometimes to minimise and sometimes to maximise the quantity we call action, at each individual step.
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0answers
60 views

Is it better to use Lagrangian mechanics for all cases? [duplicate]

I am in high school and I need to solve lots of problems which involve spring-block systems, pendulums, damped oscillations and so on. I am learning Lagrangian mechanics and I am already quite ...
1
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1answer
63 views

Classical mechanics principle of least action

I don't understand here what does the book mean by expanding in terms of $\delta{q}$ and $\delta{\dot{q}}$ can someone explain that part.
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0answers
44 views

Derivative condition in the Brachistochrone problem

I know that in general, to find the minimum of some line integral with given end points I need to solve E-L equations. what disturbs me is that I have seen in my class the famous Brachistochrone ...
2
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0answers
49 views

How does GR determine the topology of spacetime? [duplicate]

The crux of GR is the action $$ S=\int _\mathcal M d^n x \sqrt{|g|}\,R $$ Varying this and setting $\delta S=0$ gives you the Einstein field equations. However, that only determines the metric, not ...
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28 views

Equation of string as hamiltonian field equations

I am using Hamiltonian field theory for the first time and I struggle with some final steps. The task is to derive the equation of vibrating string using Hamilton's field equations. Here is what I ...
4
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1answer
68 views

Energy in dynamical variational principle

In quantum mechanics we use variational principle in order to find approximate expression for the ground state. Lets assume our probe wavefunction $|\Psi\rangle$ can be expanded in orthonormal basis ...
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33 views

Gibbons-Hawking Variation

I know there already exist some questions about this and some very good answers. However, I am still having trouble understanding one part of the calculation. The GHY term is given by ...
2
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5answers
221 views

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 : \begin{equation} S = \int_{\Omega} ...
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35 views

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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4answers
223 views

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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1answer
106 views

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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32 views

Cyclicity of trace with fermionic arguments

I think this is a non-question, but it has me considerably worried. Consider the piece of a Lagrangian density given by, $$\mathcal{L} = ...
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0answers
56 views

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 ...
2
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0answers
108 views

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 ...
4
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185 views

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 : \begin{equation} S_{EH} = \frac{1}{2 \kappa} \int_{\Omega} R \; \sqrt{- g} \; d^D x, \end{equation} where $\Omega$ is ...
7
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0answers
71 views

Spinning a rope when hanging, what is the curve? [duplicate]

Holding a rope from one end and spinning it. As shown in the picture, what will be the curve of it?
0
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1answer
146 views

Acoustical wave equation from Hamilton's principle

It is common to show the features and power of the Hamilton's principle by deriving the equation of vibrating string, membrane etc. using this principle. But I have never seen that used for deriving ...
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0answers
40 views

Variation of a tensor quantity

How does one go about in finding a variation of a tensor quantity say, with respect to the variation in the metric tensor? I've gone through calculus of variations but I can't figure out how one ...
0
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1answer
80 views

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, ...
0
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1answer
52 views

Strong interaction under $SO(3)$ isospin transformation

I'm given the following strong interaction: $$S = \int d^{4}x [\frac{1}{2} \partial_{\mu} \phi^{a} \partial^{\mu} \phi^{a} - \frac{m^2}{2} \phi^{a} \phi^{a}] ,\qquad a = 1,2,3 \text{.}$$ It is stated ...
2
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0answers
69 views

Derivative of the action integral [closed]

I need to find the partial derivative of the action $S$ with respect to the generalized coordinate $q(t_f)$ and according to my textbook, it should equal the generalized momentum $p(t_f)$. I ...
2
votes
2answers
67 views

Action max, min, or saddle?

It is well known that $\delta S = 0$ lays the foundation for variational mechanics. But I am confused as to whether or not this S is a minimum, a maximum, or a saddle point. Some books address this ...
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2answers
86 views

How do you find potential by Lagrangian formalism?

Suppose a ball is falling towards earth and hence by Lagrange equation we can find $T$ and $V$ where $V$ is $mgh$. But we know $V$ only because we know $F = mg$. Now since Lagrange equation doesn't ...
1
vote
1answer
94 views

Help on understanding a concept in Noether's first theorem

Given a Lie group $G$, whose most general transform depends on $\rho$ parameters, under the action of which an integral $I$ is invariant, there are $\rho$ linearly independent combinations of the ...
4
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2answers
321 views

Goldstein's derivation of the 'principle of least action'

I want make an punctual question ands it's about The derivation of the expression $$ \Delta\int_{t_1}^{t_2} Ldt=L(t_2)\Delta t_2-L(t_1)\Delta t_1 + \int_{t_1}^{t_2} \delta L dt. \tag{8.74}$$ You can ...
2
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2answers
95 views

qualitative explanation of Principle of Least Action (vertical movement)

Consider the following situation I want to understand what the PLA means here from an intuitive and qualitative point of view. I understand the mathematical approach. Combining ...
0
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1answer
127 views

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 ...
2
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1answer
58 views

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 ...
12
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5answers
638 views

When/why does the principle of least action plus boundary conditions not uniquely specify a path?

A few months ago I was telling high school students about Fermat's principle. You can use it to show that light reflects off a surface at equal angles. To set it up, you put in boundary conditions, ...
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0answers
79 views

In Fermat's Principle of Least Time, how do we know that light is able to reach the end point?

From my understanding of Fermat's Principle, you decide a start point and an end point for a light ray to travel between, and the light 'chooses' whichever path takes the least time (or technically ...
6
votes
2answers
210 views

What canonical momenta are the “right” ones?

I'm doing some classical field theory exercises with the Lagrangian $$\mathscr{L} = -\frac{1}{4}F_{\mu \nu}F^{\mu \nu}$$ where $F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$. To find the ...
0
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2answers
67 views

Independent Variables of a Lagrangian

Let us consider a particle in one spatial dimension $x$ and one temporal dimension $t$. Its Lagrangian $L$ is given by \begin{eqnarray*} L &=& T- V \\ &=& \frac{1}{2} m\dot{x}^2 - ...