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6
votes
2answers
90 views

Actions that are not integrals

So far every action I've seen in physics has been an integral of a Lagrangian, be it a point particle: $$S = \int dt\ L$$ or fields (relativistic or not): $$S = \int d^4x\ \mathcal{L}$$ and so ...
1
vote
0answers
50 views

Extrinsic Curvature variation

I have seen the post Explicit Variation of Gibbons-Hawking-York Boundary Term on variation of Gibbons-Hawking term, that was really helpful, however, I have problem evaluating $\delta K$ and getting ...
3
votes
2answers
113 views

Are the partial derivatives of Lagrangian in the varied action functional derivatives?

In particle mechanics Lagrangian $L$ depends upon position, velocity (and may be explicitly on time), whereas in field theory the Lagrangian density ${\cal L}$ similarly (or analogously) depends upon ...
1
vote
3answers
178 views

Why we do calculus of variation instead of finding maxima or miniama of function? [closed]

Why we do calculus of variation instead of finding maxima or minima of function? What is the difference between finding maxima or mimima i.e. critical point of a function and calculus of variation?
0
votes
0answers
50 views

Functional Differetiation of a complex functional

Suppose I have a simple functional $$F=\int{dx\;\phi^{*}(x)\phi(x)}\tag{1}.$$ Assuming $\phi(x)$ and $\phi^{*}(x)$ are independent and I take a functional differential with respect to $\phi(x)$ and $\...
4
votes
1answer
64 views

Vary action with respect to velocity

Variation of the action $S$ corresponding to a Lagrangian e.g. $L(x(t),\dot{x}(t))$ gives the Euler-Lagrange equations: $$ \frac{\delta S}{\delta x(t)} = 0 \\ \int du \left ( \frac{\delta L}{\delta x(...
3
votes
1answer
54 views

Problem obtaining string equations from Polyakov action [closed]

I am trying to obtain the string equations of motion from the Polyakov action in the conformal gauge, i.e.: $$ S=T\int{d\tau d\sigma (\dot{x}^2-x^{'2})}\equiv\int{d\tau d\sigma \mathcal{L}} $$ where ...
0
votes
0answers
36 views

What expression of Ricci tensor should we choose in order to obatin a correct field equations?

I have doubt regarding the choice of the Ricci tensor $R_{ij}$. I have seen many books and papers use the expression $R_{ij}=\Gamma^i_{jp,i}-\Gamma^i_{ji,p}+\Gamma^i_{in}\Gamma^n_{jp}-\Gamma^i_{pn}\...
0
votes
0answers
35 views

Variation with respect to the metric and other tensors

When varying an action with respect to tensors and the metric, I'm afraid I get confused as how to one organizes the Lagrangian and then performs the variation. Take for example, the following example ...
0
votes
0answers
25 views

How to obtain the Klein Gordon equation for DBI action?

The action for DBI field is given by $$S=d^{4}x\,\sqrt{-g}\left[- V(\phi)\sqrt{1-g^{ij}\partial_{i}\phi\partial_{j}\phi}\right]$$ And the required Klein Gordon is given by $$\square \phi+\frac{\...
0
votes
0answers
32 views

Variables in the Dirac Equation Lagrangian [duplicate]

(Warning: I'm a student of mathematics with no training in physics.) In derivations of the Dirac equation from an action principle, one encounters the action $$S= \displaystyle\int\,d^4x \,\bar\psi(x)...
3
votes
1answer
111 views

Lagrangian Equations of Motion, Conservative Forces

I'm new to this topic so please bear with me. Here on wikipedia we have the Lagrangian equations of motion: $$ \frac{d}{dt}\left(\frac{\partial T}{\partial \dot{q}}\right) - \frac{\partial T}{\...
3
votes
2answers
46 views

Variation of Lagrangian density $\mathcal{L}$ w.r.t $x_\mu$

If a function $f(x(t),y(t))$ has no explicit dependence on the variable $t$, then $\frac{\partial f}{\partial t}=0$. In quantum field theory, the Lagrangian density $\mathcal{L}(\phi,\partial_\mu\phi)...
8
votes
1answer
142 views

When is numerical value of Lagrangian evaluated on-shell a full differential?

I noticed recently that for many field equations, Lagrangian evaluated on-shell (i.e. using equations of motions) is a full derivative- a divergence or something, or in other words a boundary term. ...
3
votes
3answers
96 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 A_\...
1
vote
2answers
80 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 ...
1
vote
0answers
62 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 ...
2
votes
2answers
72 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
1answer
68 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 ...
17
votes
2answers
576 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 ...
0
votes
0answers
67 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
53 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 $\...
6
votes
1answer
120 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 \frac{\...
2
votes
1answer
429 views

Euler-Lagrange Equation with logarithmic potential

A particle moving towards the origin has initial conditions $x(t=0) = 1$ and $\dot{x}(t=0)=0$. If the Lagrangian is $$L:=\frac{m}{2}\dot{x}^2 -\frac{m}{2}\ln|x|$$ This should satisfy Euler ...
1
vote
0answers
78 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} \mathcal{L}}{\...
35
votes
3answers
3k views

Why treat complex scalar field and its complex conjugate as two different fields?

I am new to QFT, so I may have some of the terminology incorrect. Many QFT books provide an example of deriving equations of motion for various free theories. One example is for a complex scalar ...
0
votes
0answers
35 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 ...
4
votes
1answer
437 views

$SU(2)$ Yang-Mills EOM

I'm having trouble with some indices on my yang-mills lagrangian. I have a gauge group $SU(2)$ and a field strength tensor $$ F_{ab}^{i}=\partial_{a}A^{i}_{b}-\partial_{b}A^{i}_{a}+\epsilon^{i}_{\,\,...
0
votes
1answer
56 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 $D_\...
1
vote
1answer
31 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 g_{\...
0
votes
1answer
55 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 - \int_{...
1
vote
0answers
23 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
24 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
1answer
76 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 ...
2
votes
0answers
65 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 $$ S[\...
3
votes
1answer
65 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} j]l}=\frac{\kappa}{2}s_{ij}^{\...
0
votes
1answer
39 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 \bigg(\dfrac{dx^{\mu}}{d\...
3
votes
2answers
67 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
3answers
231 views

Problem in Euler-Lagrange imply Newton

I'm self-studying Mechanics and I have a little problem: We can see that in Landau's book or in Wikipedia that when we inject the lagrangian in Euler Lagrange equation the term $\frac{\partial v²}{\...
5
votes
1answer
95 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} \varphi(\...
1
vote
0answers
54 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
21 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
71 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' \mathcal{...
3
votes
1answer
82 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
36 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 $\dfrac{\...
3
votes
1answer
112 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
57 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
47 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 $$L={1\over2}(\epsilon_0E^2-{1\over\mu}B^2)-\rho\phi+\...
1
vote
1answer
60 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 ...
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.