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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Formulating the Lagrangian in terms of invariant quantities

Consider a closed system consisting of $N$ point particles, whose Lagrangian is given in the standard way, by the total kinetic energy minus the potential energy: $\mathcal{L}(\dot{q},q):= T(\dot{q}) ...
7
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2answers
212 views

Deriving Newton's first law from the principle of least action

Newton's first law states that if the net force on an object is zero, then this object moves with constant velocity. I'm interested in the derivation of this law from the principle of least action. ...
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1answer
35 views

Elementary question about distributive property of variation operator on an exterior product

I am trying to work out the equations of motion of a 11-dimensional supergravity action $$S = \frac{1}{2\kappa^2}\left(\gamma\int d^{11}x\sqrt{|g|}\mathcal{R} - \frac{\alpha}{2}\int G \wedge \star G ...
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0answers
36 views

A question on elementary Lagrangian mechanics(perhaps a bit of maths) [duplicate]

I am stuck with this question Consider the action, from $t=0$ to $t=1$, of a ball dropped from rest. From the Euler-Lagrange equation, we know that $y(t)={-gt^2 \over 2}$ yields a stationary value ...
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3answers
82 views

The Nambu-Goto action how do we know the Hamilton's principle applies?

I am reading 'A first course in string theory' by Barton Zwiebach (2ed) on page 112 he comes up (after a small derivation) the action formula: $$S=-\frac{T_0}{c} \int d\tau d \sigma \sqrt{-\gamma}.$$ ...
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2answers
73 views

Do we have a deeper understanding of Fermat's Principle?

Fermat's principle says that light travels between two points along the path that requires least time as compared to other nearby paths. But why this is so? Why can't light follow other paths? How ...
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48 views

How to write the Lagrangian for a body that exhibits gravitation?

Can anyone tell me how one actually goes about writing the Lagrangian for a mass that exhibits gravity? If I wanted to write the Lagrangian for, say, a spherical mass in space that curves the ...
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1answer
35 views

Reversing time for a closed system of particles

For a closed system of particles, the lagrangian in classical mechanics is $$L=\sum \frac{1}{2}mv_a^2 - U(\mathbf{r_1},\mathbf{r_2}, \cdots)$$ For an arbitrary position function $x(t)$, to see the ...
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37 views

Variation of Gibbons-Hawking-York term. General boundary condition and total derivatives

It is actually a comment and question to the answer of Robert McNees in the following post: Explicit Variation of Gibbons-Hawking-York Boundary Term In deriving the variation of the extrinsic ...
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3answers
66 views

Two questions about Variational Method of quantum mechanics

I have two question about variational method of quantum mechanics. Why we always find the ground state energy by this approach. Why not the other excited states? When we find the ground state energy ...
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2answers
74 views

Classical trajectories that are not a minimum of the action [duplicate]

Are there physically realizable dynamical systems where the true trajectory is not a minumum action trajectory? Formally, Lagrangian mechanics only requires that the trajectory be an extremum (or ...
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2answers
119 views

Detailed conditions for symmetries of Lagrangian

Edit: To clarify the question, I am asking why we are justified in calling a continuous symmetry a symmetry of a system when it changes the Lagrangian by a total derivative of a function of $t, q(t)$ ...
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1answer
53 views

Equations of motion for Polyakov action

In Polchinski 2.1.10 we have the action in terms of complex coordinates $$S = \frac{1}{2\pi \alpha'} \int d^{2}z \partial X^{\mu}\bar{\partial}X_{\mu}\tag{2.1.10}$$ This should be a rather trivial ...
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2answers
28 views

Deriving the partition function in MaxEnt

I'm trying to understand this paper on Maximum Entropy by Jaynes, and am stuck on something which should be rather simple. We're attempting to maximize the entropy $-\sum_i p_i \ln(p_i)$ subject to ...
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1answer
54 views

Derivation of an ordinary, Lagrangian/Hamiltonian and action formulation

I am confused as to how the different formulations in physics are derived. In many fields of physics, we usually begin with an ordinary formulation (e.g Newton's Laws in classical mechanics), and ...
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1answer
86 views

Trivial conserved Noether's current with second derivatives

I'm considering a symmetry transformation on a Lagrangian $$ \delta A = \int L(q +\delta q, \dot{q} + \delta \dot{q} , \ddot{q} + \delta \ddot{q}) dt $$ the general variation takes the form $$ ...
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3answers
72 views

Hamilton's Principle - achieving Hamilton equations

Consider the action function: $$\mathcal{S}(t)=\int_{t_1}^{t_2}\mathcal{L}(q_i,\dot{q_i},t) dt$$ where $\mathcal{L}$ is the Lagrangian of the system. The Hamiltonian is defined by the following ...
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2answers
63 views

First fundamental form in the Gibbons-Hawking-York boundary term

Let me expose my problem, I am trying to perform the explicit variation of the Gibbons-Hawking-York boundary term, $$S_{GH}=\int_{\partial M} d^{n-1}x\sqrt{\left|h\right|}K$$ The problem I have is ...
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2answers
139 views

Minimizing the Lagrangian action of an impossible problem

I'm working my way through Structure and Interpretation of Classical Mechanics (SICM), and am stuck on an exercise in Section 1.4: Exercise 1.6. Minimizing action: Suppose we try to obtain a ...
2
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1answer
65 views

If time-like paths are geodesics, what physical principle applies to space-like intervals?

If I have a number of particles interacting with one another locally, then the center of mass of the system moves along a geodesic. Taking this further with the particles interacting via an EM field, ...
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35 views

Does the principle of stationary action always work? [duplicate]

Give some Lagrangian we use the principle of stationary action to find the desired euqations of motion for something (e.g. a field). A lot of modern physics seems to be based on the principle of ...
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2answers
77 views

“Shortest” path in general relativity

My professor in mechanics course sneakily teach us some basic idea of general relativity. Which one of the basic assumption is particle walks in shortest world line. I understand shortest path in ...
2
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1answer
59 views

Proving independence of the lagrangian on position of a free particle using the euler-lagrange equation

I asked a similar question some time back but am trying to work this from another angle. In deriving the lagrangian of a free particle, we use the homogeneity of space to conclude that the lagrangian ...
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1answer
62 views

Equation of motion of an auxiliary field

I'm a newbie in the field of QFT and SUSY, so I'm warning you: this might be a stupid question. I'm working with auxiliary fields to describe supersymmetric models and I understand that upon ...
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1answer
114 views

Carroll's derivation of the geodesic equations [duplicate]

In Carroll's derivation of the geodesic equations (page 69, http://preposterousuniverse.com/grnotes/grnotes-three.pdf), he starts with ...
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1answer
124 views

Why two different Lagrangians to derive geodesic equations?

I'm trying (very early stages) to understand the derivation of the geodesic equation ...
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1answer
53 views

Deriving lagrangian of a free particle - How do you arrive at Lagrangian independency conclusions

I guess this question has been asked before, but I'm looking at a slightly different aspect. I'm reading Landau's book on classical mechanics. In deriving the lagrangian for a free particle, I ...
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0answers
53 views

Fastest path of light [duplicate]

Fermat's principle of least time says that light always takes the fastest path to any point. So how can light know which is the fastest path without taking all the paths first?
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1answer
76 views

Calculating Christoffel symbols from Lagrangian

I was given the following metric for a sphere $$g_{\mu\nu} = diag(1, r^2, r^2\sin^2\theta)$$ and tasked to calculate the Christoffel symbols. There are 2 ways that I know of to calculate them. One ...
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0answers
58 views

Decoupling of generalized coordinates in lagrangian

Say you have a lagrangian $L$ for a system of 2 degrees of freedom. The action, S is: $S[y,z] = \int_{t_1}^{t_2} L(t,y,y',z,z')\,dt \tag{1}$ If $y$ and $z$ are associated with two parts of the ...
2
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1answer
138 views

Null geodesic equation

For a null geodesic curve $X^i$, $$0=g_{ij}V^iV^j.$$ When we derive the geodesic equation from E-L equations, will this affine parametrization cause it to blow up? How is it justified to use the ...
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1answer
154 views

Why a timelike geodesic maximizes path length?

I'm studying some GR and my book says that in Pseudo-Riemannian manifolds geodesics may even maximize the path locally. That's what happen to the timelike geodesics, for example. My first question: Is ...
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2answers
89 views

Deriving the geodesic equation [closed]

I having been reading a general relativity book, but when in comes to the geodesic equation, it is not derived. How does one go about doing this?
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1answer
81 views

What is incorrect about the original statement of Fermat's principle?

Here are some statements about Fermat's Principle taken from Eugene Hecht's Optics book. The original statement of Fermat's Principle : "The actual path between two points taken by a ray of light is ...
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1answer
44 views

second variation of the action for 1-d lagrangian

i know that the first variation of the action integral yields to the euler lagrange equation by setting $ \delta S [y(x)]=0 $ however given a Lagrangian in the form $$ \frac{1}{2}mv^ {2}-V(x)$$ how ...
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2answers
129 views

An inconsistency in Hamiltonian formulation for non-local Lagrangian: what am I doing wrong?

This question is based on a previous question I asked, Q. [1] In this question, I proposed an example of a non-local Lagrangian (functional), I'm revisiting it here: $$\mathbb{L}=\frac{1}{2}\int^t_0 ...
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1answer
143 views

Legendre transform for non-local Lagrangians, or Hamiltonian of non-local Lagrangian and their properties

This is sort of a multi-part question, mostly dealing with how to treat non-local Hamiltonians and how the corresponding properties of Hamiltonians work in a non-local framework. I proposed an example ...
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1answer
73 views

Geometry of Hamilton-Jacobi Equation

I'm trying to understand the geometry of the Hamilton-Jacobi equation (working from Gelfand + Fomin), but I'm stuck. I know that: If we define the function $S(t,y;t_0, y_0)$ as: $$S(t,y;t_0,y_0) = ...
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0answers
40 views

Explaining why planets are round [duplicate]

is it possible to prove that planets (and/or stars) are always round (elliptical if you consider the spin)? Is there a set of equation that demonstrate that fluids (after all, molten rocks "floating" ...
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0answers
65 views

The principle of least action [duplicate]

I have read about the principle of least action. This principle suggests that nature would allow a particle to travel in a path along which the integral of the difference between kinetic energy and ...
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0answers
47 views

Terminal conditions and boundary terms in Lagrangian formulations: what do different choices mean?

For the sake of having compact expressions: $$ \left\langle f,g\right\rangle=\int^T_0 f(t)g(t)\,\text{d}t $$ Given some functional: $$ F=\frac{1}{2}m\!\left\langle ...
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1answer
90 views

Hamilton-Jacobi theory and initial value problem?

Having read through some recent posts regarding the Lagrangian formulation being interpreted into an initial value problem rather than the familiar boundary condition problem we are familiar with, I ...
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1answer
63 views

Euler-Lagrange equation with torsion, question on derivatives

Consider a mechanical system, the Lagrangian of which is: $$-L(u,\dot u)=\int\left(\dfrac{\partial^2 u}{\partial x^2}\right)^2\mathrm{d}x$$ This would correspond to a system in torsion, for example. ...
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1answer
67 views

Definition of the Lagrangian for a relativistic point particle in curved space

I have read that the Lagrangian in GR is defined as $L=\frac{\mathrm{d}s}{\mathrm{d}u}$, where $\mathrm{d}s = g_{ab}\mathrm{d}x^a\mathrm{d}x^b$ is the line element with the metric tensor $g_ab$ and ...
3
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1answer
58 views

Is there a sensible fully-discretized Hamilton's principle?

In computational physics it is common to formulate Hamilton's principle in a semi-discrete way, where space is continuous but time is discrete: in other words the Lagrangian $$L(q, \dot q, t): ...
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132 views

Non-conservative Derivation of Lagrangian [closed]

I was previously led to a recent paper by a SE member that did an alternative derivation of the Lagrangian as an initial value problem with two paths rather than the traditional boundary value method. ...
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1answer
99 views

Independence of position and velocity in Lagrangian from the point of view of physics?

I would like to continue discussion from my previous post on time dependence of lagrangian Time dependence of the Lagrangian of a free particle?. I have also read this old post Why does calculus of ...
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3answers
104 views

Time dependence of the Lagrangian of a free particle?

I am working through Landau's book on Classical Mechanics. I understand the logic and physics of isotropy and homogeneity of space-time behind the derivation of the Lagrangian for a free particle, but ...
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0answers
36 views

What justification is necessary for convolutional variational principles to be considered legitimate?

I recently asked a related question and was interested in why/or why we cannot use convolutional variational principles in practice or in theory. Summarizing the points I made in the earlier post: ...
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2answers
132 views

Examples in which the light maximizes the optical path length

I posted a similar question about geodesics on Math.SE. Many sources (Wikibooks for instance) claim that the light could maximize the optical path length in some cases. But I don't think it's actually ...