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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 ...
29
votes
2answers
2k views

Rigorous underpinnings of infinitesimals in physics

Just as background, I should say I am a mathematics grad student who is trying to learn some physics. I've been reading "The Theoretical Minimum" by Susskind and Hrabovsky and on page 134, they ...
16
votes
2answers
572 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 ...
12
votes
4answers
976 views

Where can I find the full derivation of Helfrich's shape equation for closed membranes?

I have approximately 10 papers that claim that, from the equation for shape energy: $$ F = \frac{1}{2}k_c \int (c_1+c_2-c_0)^2 dA + \Delta p \int dV + \lambda \int dA$$ one can use "methods of ...
12
votes
4answers
1k views

What is the relation between (physicists) functional derivatives and Fréchet derivatives

I´m wondering how can one get to the definition of Functional Derivative found on most Quantum Field Theory books: $$\frac{\delta F[f(x)]}{\delta f(y) } = \lim_{\epsilon \rightarrow 0} \frac{F[f(x)+\...
12
votes
2answers
271 views

Is it possible to prove that planets should be approximately spherical using the calculus of variations?

Is it possible to use the Lagrangian formalism involving physical terms to answer the question of why all planets are approximately spherical? Lets assume that a planet is 'born' when lots of ...
9
votes
4answers
603 views

Is the Lagrangian of a quantum field really a 'functional'?

Weinberg says, page 299, The quantum theory of fields, Vol 1, that The Lagrangian is, in general, a functional $L[\Psi(t),\dot{\Psi}(t)$], of a set of generic fields $\Psi[x,t]$ and their time ...
9
votes
2answers
1k views

Introductory texts for functionals and calculus of variation

I am going to learn some math about functionALs (like functional derivative, functional integration, functional Fourier transform) and calculus of variation. Just looking forward to any good ...
8
votes
1answer
141 views

How to use Euler-Lagrange when Lagrangian is $L=\sqrt{t}\sqrt{1+(dy/dt)^2}$

In this Lagrangian problem, action is $$S = \int_{t_1}^{t_2} \sqrt{t}\sqrt{1+\dot{y}^2} \,\,dt$$ where $\dot{y} = dy/dt$ and $t_1$ and $t_2$ are some fixed points. I tried to solve this problem using ...
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. ...
6
votes
1answer
108 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{\...
6
votes
1answer
596 views

Lagrangian density for a Piano String

So I'm trying to do this problem where I'm given the Lagrangian density for a piano string which can vibrate both transversely and longitudinally. $\eta(x,t)$ is the transverse displacement and $\xi(x,...
5
votes
4answers
588 views

Is there a closed form solution to the Esdale river problem?

This is probably not well known problem but it looks like open problem. What kind of methods there are to find a closed form solution to the physical situation? Can you solve this problem? You're ...
5
votes
3answers
1k views

Is it safe to ignore derivatives of velocity w.r.t. position and vice versa?

In a certain textbook a function is given as: $$f=f(x(t))$$ And then this is differentiated w.r.t. $t$ to get: $$f_t=\dot{x}f_x$$ (Where the notation $f_u=df/du$, $f_{uu}=d^2f/du^2$, etc.) This ...
5
votes
1answer
126 views

Variational derivatives of strongly connected diagrams functional in gauge theory

Background In Jorge C. Romao's "Advanced Quantum Field Theory", at the end of page 218, Eq (6.266) reads: $$\tag{1} \left.\frac{\delta^{2}}{\delta \omega^{b}(y)\delta A_{\mu}^{c}(z)}\left[ \frac{\...
5
votes
1answer
854 views

Finding interplanetary flight trajectory using calculus of variations?

Consider two orbits $x(t),\space y(t)$ representing the origin and destination for some spaceflight of interest. These could be, for example, cycloids describing LEO and another orbit circling, say, ...
5
votes
0answers
140 views

Is this constraint non-holonomic?

I am working on a variational problem involving elastic stability of a beam. The deformation of the beam is given by six functions of the material coordinate along the beams longitudinal axis. The ...
5
votes
0answers
85 views

Optimal tunnel shape for travelling inside the earth [duplicate]

Say you were to travel from Paris to Tokyo by digging a tunnel between both cities. If the tunnel is straight, one can easily compute that the time for travelling from one city to the other (...
4
votes
2answers
2k views

Why are generalized positions and generalized velocities considered as independent of each other?

I'm confused how $$\dot{\mathbf{r}}_{j}=\sum_{k}\frac{\partial\mathbf{r}_{j}}{\partial q_{k}}\dot{q}_k+\frac{\partial\mathbf{r}_{j}}{\partial t}$$ leads to the relation, $$\frac{\partial\dot{\...
4
votes
1answer
93 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(\...
4
votes
1answer
431 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}_{\,\,...
4
votes
1answer
62 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(...
4
votes
0answers
237 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 ...
4
votes
0answers
247 views

How to calculate desired light path in continuous medium with gradient refraction index

See the Figure below. $O:(0,0)$ is the disk center of light source $\odot{O}$ with radius $3$. Then the profile light rays of disk $O$ from the view point $B:(-14,0)$ is defined by segments $DB$ ...
3
votes
3answers
590 views

Does the variation of the Lagrangian satisfy the product rule and chain rule of the derivative?

I have seen wikipedia use the product rule and maybe the chain rule for the variation of the Langragin as follows: \begin{align} \dfrac{\delta [f(g(x,\dot{x}))h(x,\dot{x})] } {\delta x} = \left( \...
3
votes
3answers
969 views

Lagrangian mechanics and time derivative on general coordinates

I am reading a book on analytical mechanics on Lagrangian. I get a bit idea on the method: we can use any coordinates and write down the kinetic energy $T$ and potential $V$ in terms of the general ...
3
votes
2answers
83 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 ...
3
votes
1answer
103 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
1answer
106 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 ...
3
votes
2answers
777 views

Path to obtain the shortest traveling time

Asume we have a particle sitting at the point A(0,0) in a gravitational field. (g=9.81) It is going to move along some path to the point B(a,b) Where a>0 and b<0. What is the curve the particle ...
3
votes
3answers
94 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_\...
3
votes
1answer
63 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}^{\...
3
votes
3answers
207 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 ...
3
votes
1answer
85 views

Free energy variations

In a paper, I found this: $\mathbf{h}=\mathbf{h}(\mathbf{r})$ is called molecular field and is defined as the variation field of the Frank free energy functional $F_{d}$ with respect to the ...
3
votes
2answers
45 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)...
3
votes
1answer
81 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 ...
3
votes
2answers
132 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} ...
3
votes
2answers
65 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, ...
3
votes
1answer
102 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 + \...
3
votes
1answer
62 views

What is a sufficient condition to determine if the action integral reaches an extremal through the trajectory described by $q(t)$?

As I understand it, the Euler-Lagrange equation is a necessary but not a sufficient condition to determine if the action integral reaches an extremal through the trajectory described by $q(t)$. If ...
3
votes
1answer
255 views

prove that flat shape minimizes a functional

The following functional arises in an information theoretic problem that I work on currently. $I(G(\omega)) = \int\limits_{-\kappa\pi}^{\kappa\pi}d\omega \frac{A}{G(\omega)+A}-\frac{| \int\limits_{-\...
3
votes
1answer
51 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 ...
3
votes
0answers
297 views

Question about an integration by parts in Feynman's Quantum Mechanics [closed]

I have begun reading Feynman & Hibbs Quantum Mechanics and Path Integrals. Knowing little about variational calculus or Lagrangians I found the following integration by parts opaque. I think if I ...
3
votes
0answers
960 views

Shape of a string/chain/cable/rope/wire?

The height of a string in a gravitational field in 2-dimensions is bounded by $h(x_0)=h(x_l)=0$ (nails in the wall) and also $\int_0^l ds= l$. ($h(0)=h(l)=0$, if you take $h$ as a function of arc ...
2
votes
2answers
820 views

Coulomb potential energy functional derivative

I'm having problem understanding how to compute a functional derivative when it's involved more than one integral, such as the coulomb potential energy functional: $$ J[\rho] = \frac 12\int \frac{\...
2
votes
2answers
70 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 $\frac{\delta}{\delta \phi}$ (...
2
votes
3answers
110 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 ...
2
votes
3answers
1k views

Does a four-divergence extra term in a Lagrangian density matter to the field equations?

Greiner in his book "Field Quantization" page 173, eq.(7.11) did this calculation: ${\mathcal L}^\prime=-\frac{1}{2}\partial_\mu A_\nu\partial^\mu A^\nu+\frac{1}{2}\partial_\mu A_\nu\partial^\nu A^...
2
votes
1answer
428 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 ...
2
votes
1answer
278 views

Finding the Shape of a Hanging Massive Wire

This came from a physics wave book. I have a static, massive wire on the x-axis, with a y-displacement due to a force per unit length $F_y$. I start with the equation $F_y = T_0\frac{\partial^2(\eta)}{...