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23
votes
2answers
1k 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 ...
12
votes
2answers
197 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 ...
11
votes
4answers
757 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
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} ...
9
votes
4answers
492 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
1answer
130 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
2answers
928 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 ...
5
votes
4answers
538 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
121 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[ ...
5
votes
1answer
610 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
533 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
107 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
82 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
1k 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, ...
4
votes
1answer
66 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
3answers
357 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
2answers
513 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
1answer
77 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
1answer
37 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
253 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
113 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
0answers
214 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
679 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
700 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
3answers
619 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
3answers
89 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 ...
2
votes
1answer
142 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
1answer
68 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 ...
2
votes
3answers
549 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 ...
2
votes
1answer
47 views

Full time derivative of the Frank-Oseen energy. Mathematical problem

I am studying liquid crystal theory with the book Kleman, Lavrentovich, Soft Matter Physics. In the Ericksen-Leslie theory, Frank-Oseen energy density is: $$ f=0.5*(K_1*div^2 (n)+K_2 ...
2
votes
1answer
150 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 ...
2
votes
1answer
174 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 $$S~=~\int d^{4}x \frac{a^{2}(t)}{2}(\dot{h}^{2}-(\nabla h)^2).$$ ...
2
votes
0answers
40 views

Interchanging of variation and integration operator for holonomic systems

Meirovitch says in his "Principles and Techniques of Vibrations" (1997) on p.85: In the case of holonomic systems, the variation and integration processes are interchangeable (...) which means ...
2
votes
0answers
70 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
5answers
602 views

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

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
105 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 ...
1
vote
1answer
143 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 ...
1
vote
2answers
319 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
136 views

Calculus of variations and string theory

In Polchinski's String theory book, Vol 1., in chapter 1, p. 18, he is deriving the Lagrangian in the light cone gauge (that's not necessary to know in order to answer this question), and he gets ...
1
vote
1answer
117 views

Finding the magnetic vector potential by calculus of variations

Given the functional $$F[A]=\int_{\mathbb{R}^3}\{\frac{1}{2\mu(x)}|\nabla\times\vec{A}|^2-\vec{J}\cdot\vec{A}\}d^3x$$ with $\vec{A}$ is a candidate vector potential for the field ...
1
vote
1answer
49 views

Variation over complex function in Ginzburg-Landau theory

When deriving the Ginzburg-Landau equations, we minimize the following free energy over the complex function $\psi$: $$F = \int dV \left \{\alpha |\psi|^2 + \frac{\beta}{2}|\psi|^4 + \frac{1}{2m^*} ...
1
vote
1answer
84 views

The Euler-Poincare equation

Can anyone tell me very basically how the Euler-Poincare equation generalises the Euler-Lagrange equation? Further does anyone know if there is an "easy" relationship between the two, i.e. can anyone ...
1
vote
1answer
54 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 ...
1
vote
1answer
250 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
22 views

Components of acceleration in spherical polar co-ordinate

I wanted to calculate two component of acceleration in polar co-ordinate. Starting from the lagrangian $$L= \frac{1}{2}m( \dot{r} ^{2}+ r^{2} \dot{ \theta } ^{2} ) -V(r, \theta )$$ I ...
1
vote
0answers
54 views

How is Infinitesimal coordinate transformation related to Lie derivatives?

I am reading the book "Gravitaion and Cosmology" by S. Weinberg. In section 10.9, while discussing Lie derivatives of tensors of different ranks, he makes a general comment: The effect of an ...
1
vote
0answers
39 views

Explaining why planets are round [duplicate]

is it possible to prove that planets (and/or stars) are always round (elliptical if you consider the spin)? Is there a set of equation that demonstrate that fluids (after all, molten rocks "floating" ...
1
vote
1answer
35 views

Variational calculus, bending a stick and stationary states

We have a horizontal stick, one of its ends is on the wall, and we can apply a force to the other end. We assume that anything that we can do will leave this in the same plane. Our question is to ...
1
vote
1answer
51 views

Variational Principle to find Energy Eigenfunctions

In Quantum Mechanics one can estimate an upper bound for the ground state energy with the following functional: $$\mathcal{F}[\psi(x)] \equiv \int_{-\infty}^\infty \psi^*(x)\hat{H}\psi(x) \,\, dx ...