The variational-calculus tag has no wiki summary.
11
votes
4answers
286 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 ...
9
votes
4answers
266 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 ...
8
votes
2answers
498 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 ...
6
votes
0answers
81 views
When does a PDE solve a variational problem? [migrated]
I understand that for a functional $J[f]$ on the space of differentiable functions $f$ on some domain, setting $\delta J[f]|_{f=f_0} = 0$ yields a (possibly nonlinear) partial differential equation in ...
5
votes
3answers
437 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:
$$\frac{\delta F[f(x)]}{\delta f(y) } = \lim_{\epsilon \rightarrow 0} ...
5
votes
1answer
154 views
Finding interplanetary flight trajectory using calculus of variations?
Consider two orbits $x(t),\space y(t)$ representing the origin and destination for some spaceflight of interest. These could be, for example, cycloids describing LEO and another orbit circling, say, ...
5
votes
1answer
277 views
Lagrangian density for a Piano String
So I'm trying to do this problem where I'm given the Lagrangian density for a piano string which can vibrate both transversely and longitudinally. $\eta(x,t)$ is the transverse displacement and ...
5
votes
0answers
76 views
Optimal tunnel shape for travelling inside the earth [duplicate]
Say you were to travel from Paris to Tokyo by digging a tunnel between both cities. If the tunnel is straight, one can easily compute that the time for travelling from one city to the other ...
4
votes
3answers
271 views
Is it safe to ignore derivatives of velocity w.r.t. position and vice versa?
In a certain textbook a function is given as:
$$f=f(x(t))$$
And then this is differentiated w.r.t. $t$ to get:
$$f_t=\dot{x}f_x$$
(Where the notation $f_u=df/du$, $f_{uu}=d^2f/du^2$, etc.)
This ...
3
votes
2answers
581 views
Why are generalized positions and generalized velocities considered as independent of each other?
I'm confused how
$$\dot{\mathbf{r}}_{j}=\sum_{k}\frac{\partial\mathbf{r}_{j}}{\partial q_{k}}\dot{q}_k+\frac{\partial\mathbf{r}_{j}}{\partial t}$$
leads to the relation,
...
3
votes
1answer
241 views
prove that flat shape minimizes a functional
The following functional arises in an information theoretic problem that I work on currently.
$I(G(\omega)) = \int\limits_{-\kappa\pi}^{\kappa\pi}d\omega \frac{A}{G(\omega)+A}-\frac{| ...
3
votes
0answers
211 views
Shape of a string/chain/cable/rope?
The height of a string in a gravitational field in 2-dimensions is bounded by $h(x_0)=h(x_l)=0$ (nails in the wall) and also $\int_0^l ds= l$. ($h(0)=h(l)=0$, if you take $h$ as a function of arc ...
2
votes
2answers
386 views
Coulomb potential energy functional derivative
I'm having problem understanding how to compute a functional derivative when it's involved more than one integral, such as the coulomb potential energy functional:
$$ J[\rho] = \frac 12\int ...
2
votes
2answers
210 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 ...
2
votes
3answers
152 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
82 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
$$S~=~\int d^{4}x \frac{a^{2}(t)}{2}(\dot{h}^{2}-(\nabla h)^2).$$
...
1
vote
5answers
464 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 ...
1
vote
2answers
239 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?
1
vote
1answer
44 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
43 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 $$\frac{\delta f[u]}{\delta u(x)}\equiv \frac{\partial f}{\partial ...
0
votes
1answer
219 views
Why is a cycloid path the fastest way to roll a ball downward? [duplicate]
Possible Duplicate:
Path to obtain the shortest traveling time
I've been told that if one would want to make a ramp to get a ball from point A to a lower point B (at a certain horizontal ...
0
votes
1answer
97 views
proper variation of action term
I have a term I want to vary by a field, $\phi$.
$$
`S' = \frac{-1}{2}\,\sqrt{-g}\,g^{\mu\,\nu}\,\delta\left[h(\phi)\,\partial_{\mu}\phi\,\partial_{\nu}\phi \right].
$$
Is it correct to get this?
...
0
votes
0answers
34 views
Suggestions for a physics oriented book on Variational Calculus [duplicate]
I would like to buy a good book on Variational Calculus. Most of the books that I find seem to be rather formal in a mathematical sense, which is not necessarily bad, but makes the studying a bit ...
0
votes
0answers
43 views
Help identifying an expression for the action
I found the following expression for the action of a (free, I think) relativistic particle in my notes but I can't remember from what it came from:
$$ S = \int_{0}^{N} \left [ ...
0
votes
0answers
69 views
What are the details of this variational calculus solution?
This answer includes the problem:
Suppose you have a 2-d bullet going very fast through a 2-d gas. The gas molecules reflects specularly off the bullet, making glancing collisions. What shape of ...
0
votes
0answers
34 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 ...
