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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1answer
76 views

Polyakov From Nambu-Goto Directly, for Strings?

The following derivation, for a classical relativistic point particle, of the 'Polyakov' form of the action from the 'Nambu-Goto' form of the action, without any tricks - no equations of motion or ...
3
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0answers
56 views
+150

Lower limit for atomic number in case of imaginary ion with $Z < 1$

In my programming project I calculate the minimal energy of an atom with 2 electrons in the $L=0, S=0$ state, using a Hylleraas wave function. The values I find for $Z=2$ (He) and $Z=1$ (H$^-$) are ...
6
votes
4answers
955 views

Lagrangian for relativistic massless point particle

For relativistic massive particle, the action is $$S ~=~ -m_0 \int ds ~=~ -m_0 \int d\lambda ~\sqrt{ g_{\mu\nu} \dot{x}^{\mu}\dot{x}^{\nu}} ~=~ \int d\lambda \ L,$$ where $ds$ is the proper time of ...
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2answers
72 views

Derivation of Euler-Lagrange equation from principle of least action

When deriving the Euler-Lagrange equation for a field $\phi$ the term $$ \int\textrm{d}x^{\mu}~\partial_{\mu}\left( \dfrac{\partial \mathcal{L} }{\partial(\partial_{\mu}\phi)}\right)\delta\phi $$ is ...
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2answers
926 views

Question about the apparent loophole in principle of least action

In Lagrangian formalism, given two points $(x_1,t_1)$ and $(x_2,t_2)$, we ask the question which paths $x(t)$ make the action $S=\displaystyle \int_{t_1}^{t_2}L\ \mathrm dt$ stationary and satisfy the ...
0
votes
1answer
178 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 ...
15
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2answers
559 views

Lagrangian Mechanics - Commutativity Rule $\frac{d}{dt}\delta q=\delta \frac{dq}{dt} $

I am reading about Lagrangian mechanics. At some point the difference between the temporal derivative of a variation and variation of the temporal derivative is discussed. The fact that the two are ...
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52 views

Physical motivation for Lagrangian formalism

This is more of a request for clarification of understanding and intuition rather than a question, but I hope people can help me with it. I have learned calculus of variations and have subsequently ...
4
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1answer
291 views

Equations of motion of displacement field

We have an action: $$S[\boldsymbol{u}] = \frac{1}{2} \int dt \int d^3x \left\{ \mu (\frac{\partial u_{i}}{\partial t})^{2} - \nu (u_{ii})^{2} - \rho(u_{ij})^{2}\right\} $$ Where $u_{ij} = ...
2
votes
1answer
417 views

Variation of the metric with respect to the metric

For a variation of the metric $g^{\mu\nu}$ with respect to $g^{\alpha\beta}$ you might expect the result (at least I did): \begin{equation} \frac{\delta g^{\mu\nu}}{\delta g^{\alpha\beta}}= ...
4
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2answers
90 views

Higher than Lagrangian/action?

When you begin learning physics, you start with equations of motion applied to various physics systems. In classical mechanics course you learn, that exists Lagrangian/action of a system, which gives ...
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0answers
54 views

Simple real life applications of Euler-Lagrange equations of motion

If you read some introductory mechanics text like David Morin's Introduction to Classical Mechanics about Euler Lagrange Equations you get a large amount of simple examples like the "moving plane" ...
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0answers
26 views

Must there exist a Lagrangian for any 2nd order ordinary derivative equation?

We know if there exist a Lagrangian of some ODE, then it must exist many equivalent Lagrangian. My question: Then must there exist a Lagrangian for any 2nd order ODE? If not, do we have some ...
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0answers
32 views

Is energy always a diminishing quantity in imaginary time?

If we write Schrodinger equation in imaginary time $\tau = it$, then one can easily show that the energy $E(\tau) = \langle \psi(\tau)| \hat{H} |\psi(\tau)\rangle$ is a diminishing quantity, i.e. ...
3
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1answer
108 views

Is there an action for every physical law?

Given an action, I can get the differential equation governing the evolution of the system by applying the principle of least action. Does it work the other way around? Given any differential ...
0
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1answer
38 views

Understanding Derivation of Euler-Lagrange

I am trying to understand the derivation of the Euler-Lagrange equation. I drew a graph below. So, according to the graph, $$ \int_{t_1}^{t_2} L(x+\delta{x},\dot{x}+\delta\dot{x}\,t) dt - ...
3
votes
1answer
176 views

Determine path of point mass using the Hamilton's principle

I am very new in this field but I try to solve a problem by using the Hamilton's principle and afterwards I want to compare the solution by solving the same problem using conservation laws. What I ...
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2answers
469 views

Time-dependent Schrodinger equation from variational principle

In the paper, "Density-functional theory for time-dependent systems" Physical Review Letters 52 (12): 997 the authors mentioned that the action $$ A= \int_{t_0}^{t_1} \mathrm dt \langle \Phi(t) | i ...
3
votes
1answer
180 views

Hamilton's equations from the action with boundary conditions involving position and momentum

Generally, when you are given the action $$ S=\int_{t_1}^{t_2}\mathrm dt (p\dot q - \mathcal H )$$ the boundary conditions are $q(t_1)=q_1$ and $q(t_2)=q_2$. This is useful because to calculate ...
3
votes
1answer
78 views

Why can you make $V$ stationary with respect to a parameter of the field in Derrick's theorem?

I'm going over Coleman's derivation of Derrick's theorem for real scalar fields in the chapter Classical lumps and their quantum descendants from Aspects of Symmetry (page 194). Theorem: Let ...
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2answers
392 views

Derrick’s theorem

Consider a theory in $D$ spatial dimensions involving one or more scalar fields $\phi_a$, with a Lagrangian density of the form $$L= \frac{1}{2} G_{ab}(\phi) \partial_\mu \phi_a \partial^\mu \phi_b- ...
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0answers
48 views

An interesting problem on potentials [closed]

I am thinking of an interesting problem on potentials. Consider a potential field $U(\mathbf R)$. Furthermore, consider a rocket with exhaust speed from 0 to $v$. I am wondering whether there is a way ...
2
votes
0answers
80 views

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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1answer
46 views

Principle of Stationary Action and Euler-Lagrange Equation

Principle of Stationary Action: Given a mechanical system, there exists an action $S$ such that it is extremitized, or $\delta S=0$, for the actual motion of the system. $$S = ...
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0answers
29 views

How to calculate Lagrangian density function in classical field theory

In Lagrangian mechanics observing the possible degrees of freedom we first write down our Lagrangian. Then we use E-L equation to determine equation of motion and using sufficient boundary condition ...
0
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1answer
141 views

“Principle of least action” and “Principle of conservation of energy”: Which one is fundamental and which one is derived? [closed]

Suppose I throw a ball upwards. First it will rise under gravity and then fall under gravity. During the rising part the kinetic energy gradually decreases and the potential energy increases until ...
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0answers
19 views

Max & inflection point in the principle of least action [duplicate]

Short question: What is the physics interpretation of max & inflection points in the principle of least action? Long question: If $$L(q_1,q_2;t)=K-V$$ then let $$S = \int^{t_1}_{t_2} ...
3
votes
2answers
113 views

Finding geodesics: Lagrangian vs Hamiltonian

I have a question referring to how to compute geodesics of a given spacetime (say, Kerr). I know that the direct way is via the geodesic equation ...
0
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1answer
38 views

Simplify calculation of geodesics from action principle

I don't understand a step with the calculation of geodesics equations from action principle on this link : demo geodesics equations My issue is the following step : $$\int ...
5
votes
3answers
1k views

Hanging chain in a planet's gravitational field

The curve for a chain hanging between two poles in a uniform gravitational field is known as the catenary. Is there known an expression for the curve of a hanging chain on a planet of mass $M$ which ...
4
votes
1answer
432 views

Least Action Principle (Classical and Quantum Theory)

I) My first question would be "why should classical systems obey the principle of least action ?" When we find out the propagator in quantum physics, we find the amplitude to be equal to the sum over ...
2
votes
1answer
94 views

Geometric derivation of quantum mechanics from Lagrangian mechanics

I have used classical Lagrangian mechanics for quite a while, and what I like about it is that everything can be derived from a very small number of geometric principles. There are just three things ...
0
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1answer
56 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
votes
2answers
64 views

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

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, ...
0
votes
1answer
197 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 ...
2
votes
1answer
57 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 ...
42
votes
6answers
5k views

Why does calculus of variations work?

How does it make sense to vary the position and the velocity independently? Edit: Velocity is the derivative of position, so how can you treat them as independent variables? Doesn't every physics ...
14
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4answers
11k views

Derivation of Maxwell's equations from field tensor lagrangian

I've started reading Peskin and Schroeder on my own time, and I'm a bit confused about how to obtain Maxwell's equations from the (source-free) lagrangian density $L = ...
2
votes
1answer
64 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 ...
2
votes
0answers
45 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 = ...
2
votes
1answer
72 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
votes
0answers
42 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 ...
6
votes
0answers
110 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 ...
3
votes
2answers
111 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
46 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
votes
1answer
76 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 ...
1
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2answers
80 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
votes
1answer
80 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 + ...
7
votes
2answers
730 views

Finding 3-Sphere Christoffel connection coefficients using variational calculus, Sean Carrol problem

I have A 3-Sphere with coordinates $x^{\mu} = (\psi,\theta,\phi)$ and the following metric: \begin{equation} ds^2 = d\psi^2 + \text{sin}^2\psi(d\theta^2 + \text{sin}^2\theta d\phi^2) \end{equation} ...
11
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3answers
3k views

Noether's current expression in Peskin and Schroeder

In the second chapter of Peskin and Schroeder, An Introduction to Quantum Field Theory, it is said that the action is invariant if the Lagrangian density changes by a four-divergence. But if we ...