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3
votes
3answers
75 views

Field equations of a given action

Provided an action: $$S[A_\nu] = \int\left(\frac{1}{4\mu_0}(A_{\gamma,\mu}-A_{\mu,\gamma})(A_{\zeta,\alpha}-A_{\alpha,\zeta})\eta^{\gamma\zeta}\eta^{\mu\alpha}+\frac{1}{2}\nu^2A_\mu A_\gamma -\beta ...
1
vote
0answers
41 views

How do I model the motion of a particle changing acceleration vector (2D)?

I want to model a particle with an arbitrary initial velocity, and estimate the time it takes to reach a final point given a constant magnitude of acceleration. It should take the quickest path to the ...
1
vote
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 ...
2
votes
2answers
55 views

General form for functional derivatives

Working on the hamiltonian formalism applied to canonical field theory, how do I deduce the general form for the functional derivatives $\frac{\delta}{\delta \pi}$ and $\frac{\delta}{\delta \phi}$ ...
0
votes
0answers
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 ...
1
vote
0answers
48 views

How to calculate the second functional derivative of the action of a one-particle system?

Given the Lagrangian $$L(q,\dot{q})=m\dot{q}^2/2-V(q)$$ and the corresponding action $$S[q]\equiv\int_0^t dt' (m\dot{q}^2/2-V(q)),$$ I need to be able to evaluate the second functional derivative ...
0
votes
1answer
48 views

Electromagnetism theory and complex scalar field

I've got the following problem for classical field theory lecture: Find equations of motion (equations of field?), canonical and symmetrical tensor of energy-momentum in electromagnetic field ...
1
vote
0answers
71 views

Euler-Lagrange equations in General Relativity

When obtaining the Euler Lagrange equations for a scalar field with higher order derivatives in curved space is it the same to use $$ -\partial_\nu\partial_\mu\frac{\partial \sqrt{-g} ...
0
votes
0answers
33 views

Question on basic tensorial calculus on field theory

Working on the Maxwell field as a gauge theory, at some point the following derivative comes up: $\frac{\partial(\partial_iA_0)}{\partial A_0}=0$ which must be, accordingly to the theory, zero. My ...
0
votes
1answer
54 views

Derivatives involving four vectors [closed]

The Schrödinger lagrangian for complex fields is $$L=\frac{1}{2m}(D_i \psi)^* Di \psi - \frac{i}{2} \left[\psi ^* D_0 \psi - (D_o \psi)^* \right] - \frac{1}{4}F^{\mu \nu}F_{\mu \nu}$$ Where ...
1
vote
1answer
23 views

Deriving Electromagnetism energy-stress tensor in GR [closed]

Please find the mistake in the following calculations. We have $L=-F^{\mu\nu}F_{\mu\nu}$, and try to derive the energy-stress tensor using $\delta(-g)^{1/2}=\frac{1}{2}(-g)^{1/2}g^{\mu\nu}\delta ...
0
votes
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 - ...
1
vote
0answers
20 views

Can you help me solve this using the current value Hamiltonian? [closed]

Okay, so I am getting a little stuck on this question, I will post it and then tell you how far I get. $$ max - \int_0^2 (x^2 + u^2)e^{-0.03t}dt\, $$ $$ x' = x-2u $$ $$ x(0) = 3 $$ $$ x(2)free $$ ...
0
votes
0answers
21 views

Action and equation of motion for codim-2 “cosmic” brane in Einstein gravity, in 3d

Consider the following action, $$ S = S_{EH} + S_{B} = -\frac{1}{8\pi G_N}\int d^3 X \sqrt{G}R + T \int dy \sqrt{g} , $$ where, $G_{\mu \nu}$ is the bulk metric, and $g$ is the induced metric on the ...
2
votes
0answers
58 views

Einstein equations from the Palatini action [closed]

I am trying to obtain the usual form of vacuum Einstein's equations $$ R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} + \Lambda g_{\mu \nu} = 0 $$ from the first-order (Palatini) tetradic action $$ ...
2
votes
1answer
68 views

Shape of water on top of a thin sheet of stretched plastic

Consider a thin sheet of plastic (a square sheet for simplicity) that is stretched taught in a plane parallel to the ground. If a volume of water is then placed on top of the thin plastic sheet, then ...
1
vote
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 ...
3
votes
1answer
62 views

Derivation of the Cartan Field equation

Please help me understand how, in this introduction to spacetime and fields, the Einstein Cartan equation: $$C^k_{\hspace{2mm} [ji]}-\delta_{[i}^{k}C^l_{\hspace{2mm} ...
6
votes
1answer
84 views

Use partial or covariant derivatives when deriving equations of a field theory?

I feel like this question has been asked before but I can't find it. would the Euler Lagrange equation for, say, the standard model Lagrangian be $$\frac{\partial L}{\partial \phi}=\partial_\mu ...
0
votes
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 ...
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, ...
4
votes
1answer
83 views

Why can the bra and ket be varied independently?

Given a functional which depends on a function (ket), and its complex conjugate (bra), e.g. $$F[\varphi] = \langle \varphi|\hat{F}|\varphi\rangle = \int \varphi^{*}(\mathbf{r}) \hat{F} ...
1
vote
0answers
48 views

What is the path taken by a “cable car”?

A well known result in variational calculus & Lagrangian Mechanics is the solution to the "brachistochrone" problem, where it is found the path connecting two points, A & B such that the time ...
0
votes
0answers
20 views

Is it possible to derive the shape of the bending plates by use calculus of variations?

Of course the main idea to solve this problem is find the physical quantity which is have smallest or largest value. I’ve tried some, such as area of surfaces, But I think it can’t be a solution. Does ...
1
vote
2answers
68 views

Variation of a Lagrange density Symmetries

So I am reading Goenner's Spezielle Relativitästheorie and I am currently in chapter §4.9.1 Variation under Inclusion of Coordinates p. 129. So basically we have: $$\delta W_\zeta=\int d^4x' ...
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 ...
0
votes
0answers
32 views

How to obtain calculus of variation of Einstein summation?

I have the Lagrange density for Maxwell field, which is $\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$, where $F_{\mu\nu}=\partial_{\mu}A_\nu-\partial_{\nu}A_{\mu}$. How can I obtain ...
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 + ...
0
votes
1answer
56 views

Euler-Lagrange for simple scalar field (Peskin & Shroeder)

I'm reading Peskin & Schroeder and they give as a simple example the Lagrangian $$\mathcal{L} = \frac{1}{2} (\partial_\mu \phi)^2$$ First of all, I'm guessing that $(\partial_\mu \phi)^2$ is ...
0
votes
0answers
43 views

Deriving Maxwell's equation from the Lagrangian of an electromagnetic field with a charge density $\rho$

I want to derive Maxwell's equation from the Lagrangian of an electromagnetic field with a charge density of $\rho$ The Lagrangian is given by ...
1
vote
1answer
54 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 ...
0
votes
1answer
61 views

Calculus of Variations - Virtual displacements

I am currently reading "The Variational Principles of Mechanics - Cornelius Lanczos", in which the author talks about the variation of a function $F(q_1, q_2, \dots q_n)$ where $q_1, q_2, \dots q_n$ ...
1
vote
1answer
64 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.
0
votes
0answers
50 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 ...
0
votes
0answers
36 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
votes
5answers
229 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} ...
1
vote
4answers
226 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 ...
1
vote
1answer
121 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 ...
1
vote
0answers
28 views

Commutativity Variation and derivative

When in general is true that $\nabla_{\mu}\delta=\delta\nabla_{\mu}$ where $\nabla_{\mu}$ is covariant derivative? I was thinking this when one has to find the equations of motion for example in ...
2
votes
0answers
110 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
votes
0answers
217 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 ...
1
vote
0answers
42 views

Proof of Hamilton's principle [duplicate]

Is there a anything like a proof of Hamilton's principle? Where would I find it?
1
vote
1answer
26 views

Function for which action is the minimum

On page 2 of "Mechanics" Landau & Lifshitz say that $q=q(t)$ is a function for which action is a minimum. Before this they say that at times $t_1$ and $t_2$, the system occupies coordinates ...
0
votes
2answers
107 views

Derivation of Euler-Lagrange equations in Landau's and Lifshitz's “Mechanics”

There's an integral ${\int\limits_{t_1}^{t_2}}(\frac{\partial{L}}{\partial{q}}{\delta}q+\frac{\partial{L}}{\partial{v}}{\delta}v)dt=0$. [1.] $ {\delta}v={\frac{d{\delta}q}{dt}}$ [2.] I should get $ ...
0
votes
1answer
47 views

Optimal Firing Rate to Maximize Vehicle Velocity

Suppose that you are in a vehicle in space with a power source generating a constant power of $P_0$ that you can use to fire iron pellets. The iron pellets compose 99% of the ship's mass, $m$. The ...
1
vote
1answer
127 views

Mathematics of the Virtual Displacement

So I'm pretty certain this question has been asked to death here, but I still can't find a good explanation of a very particular aspect of the virtual displacements in physics. Background For ...
3
votes
2answers
123 views

Equivalence of functional and partial derivatives

I am trying to derive Newton's second law from the principle of least action, that is, setting the functional derivative $\frac{\delta S}{\delta x(t)}$ equal to 0. $$S = \int dt' \left[ \frac{m}{2} ...
2
votes
3answers
105 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 ...
0
votes
0answers
163 views

What is the meaning of symbols $\delta f$ and $\delta^2f$?

Professor was using these symbols to derive the continuity equation. He defined the infinitesimal mass as $\delta^2m=\rho \delta V$ and the mass that leaves some closed boundary $\partial V$ as ...
3
votes
1answer
99 views

Noether's theorem in field theory: Jacobian factor

Following my earlier question in this Phys.SE post I have another question regarding the derivation I am struggling through! Considering the variation in the Lagrange density for $x'=x+\delta x$ and ...