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6
votes
1answer
101 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[ ...
9
votes
4answers
827 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} ...
1
vote
1answer
113 views

Subtlety in derivation of Noether's theorem by Di Francesco

In the book 'Conformal Field Theory' by Di Francesco et al, a derivation of Noether's theorem is demonstrated by imposing that, what I believe is said to be a more elegant approach, the parameter ...
2
votes
0answers
144 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
452 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
1answer
75 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 = ...
2
votes
0answers
54 views

Optimal Airplane trajectory

The last time I took a plane the following problem crossed my mind. Setting: take the Earth and neglect its rotation around the Sun. It then only rotates on itself with angular velocity $\Omega$. ...
1
vote
0answers
70 views

Noether's Theorem For Functionals of Several Variables

My question is on using a form of the single variable Noether's theorem to remember the multiple variable version. Does Noether's theorem, for functionals of a single independent variable, just say ...
2
votes
3answers
294 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 ...
4
votes
0answers
80 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 ...
9
votes
1answer
124 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 ...
0
votes
0answers
23 views

Optimization of a functional defined implicitly

I would like to minimize a functional of the type: $$L[\gamma]=\int_a^b F(T(\gamma(t))dt$$ on the space of paths $\gamma$, where $T=T(\gamma,t)$. Now, usually I would simply apply Euler-Lagrange's ...
2
votes
1answer
119 views

How to relate second variation to quadratic terms?

I am working on a physics problem, but my issue is math-related. My professor skips some steps based on 'intuition' that I lack: In a conservative system, to find out the nature of equilibrium ...
1
vote
2answers
88 views

With respect to what quantities do I vary Lagrangians in field theory?

I have recently been wondering, with respect to which quantities (covariant or contravariant) one should vary QFT Lagrangians and whether there is some rule regarding this. Let me give an example ...
2
votes
1answer
64 views

Question about “different” equations of motion in dependence of indices

Let's have the action $$ S = \int (\partial_{\mu}h^{\mu \sigma}\partial^{\nu}h_{\nu \sigma} - \Lambda h^{\mu \nu}T_{\mu \nu}) d^{4}x. $$ For definiteness, $$ h_{\mu \nu} = h_{\nu \mu} , \quad T_{\mu ...
20
votes
2answers
981 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 ...
1
vote
1answer
37 views

Double variation of Schwinger action principle

The Schwinger action principle is given by $$\delta_{1}\big\langle b\big|a\big\rangle= i\int_{t_{a}}^{t_{b}}\text{d}t\,\sum_{c,d}\big\langle b\big|c\big\rangle\big\langle ...
3
votes
1answer
69 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 ...
9
votes
4answers
404 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 ...
5
votes
4answers
484 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 ...
1
vote
0answers
50 views

path planning with invasive measurements

My background is not science, and I hope the question I am about to ask doesn't look like a homework kind of a thing! Anyway, I will try my best to make the point clear. imagine there is a lake, and ...
11
votes
4answers
582 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 ...
1
vote
1answer
177 views

Center of mass of a body on an incline

I am trying to reproduce a calculation by Carre et al. (1995) in which they calculate the shape of a droplet on an incline. My issue is in the derivation of the potential energy (essentially the ...
1
vote
1answer
83 views

Finding Hamilton's equations given a Hamiltonian

I am trying to find Hamilton's equations for a general Hamiltonian given by $$H[u]=\int_\mathbf{R} \phi(u,u_x)dx$$ Suppose $$\frac{\delta f[u]}{\delta u(x)}\equiv \frac{\partial f}{\partial ...
5
votes
0answers
81 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 ...
3
votes
1answer
247 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{| ...
5
votes
1answer
440 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 ...
0
votes
1answer
269 views

proper variation of action term

I have a term I want to vary by a field, $\phi$. $$ `S' = \frac{-1}{2}\,\sqrt{-g}\,g^{\mu\,\nu}\,\delta\left[h(\phi)\,\partial_{\mu}\phi\,\partial_{\nu}\phi \right]. $$ Is it correct to get this? ...
2
votes
3answers
417 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 ...
2
votes
1answer
138 views

Varying an action (cosmological perturbation theory)

I am stuck varying an action, trying to get an equation of motion. (Going from eq. 91 to eq. 92 in the image.) This is the action $$S~=~\int d^{4}x \frac{a^{2}(t)}{2}(\dot{h}^{2}-(\nabla h)^2).$$ ...
8
votes
2answers
676 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 ...
0
votes
0answers
38 views

Suggestions for a physics oriented book on Variational Calculus [duplicate]

I would like to buy a good book on Variational Calculus. Most of the books that I find seem to be rather formal in a mathematical sense, which is not necessarily bad, but makes the studying a bit ...
5
votes
1answer
369 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, ...
3
votes
2answers
349 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 ...
0
votes
1answer
822 views

Why is a cycloid path the fastest way to roll a ball downward? [duplicate]

Possible Duplicate: Path to obtain the shortest traveling time I've been told that if one would want to make a ramp to get a ball from point A to a lower point B (at a certain horizontal ...
0
votes
0answers
59 views

Help identifying an expression for the action

I found the following expression for the action of a (free, I think) relativistic particle in my notes but I can't remember from what it came from: $$ S = \int_{0}^{N} \left [ ...
1
vote
2answers
295 views

Can cos(x) or sin(x) be the function of stationary action?

Is there a way to express $\cos(x(t))$ (or $\sin(x(t))$) as the solution to the Euler-Lagrange equation, in other words is there a sense in which this function is the path of stationary action?
5
votes
3answers
648 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 ...
1
vote
5answers
546 views

Why do we automatically assume that the velocity vector $\vec{v}$ and location vector $\vec{r}$ are independent?

I'm not sure if it's relevant, but I'm talking about a situation where a particle is moving in an electro-magnetic field. As I understand, if we see the term $\nabla \cdot \vec{v}$ or $\nabla \times ...
3
votes
2answers
813 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, ...
0
votes
0answers
42 views

Variational Calculus or Tensor Calculus? [duplicate]

Possible Duplicate: Learning physics online? I'm a high school student, and I got fives in AP Calculus, Mechanics and Electricity and Magnetism exams, and I've taken Linear Algebra and ...
2
votes
2answers
557 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 ...