# Tagged Questions

**3**

votes

**1**answer

319 views

### Helmholtz equation Poynting vector integral

The Maxwell's equation for harmonic time dependent field in vacuum is
\begin{align}
\nabla \times B + i\omega E &= 0\\
\nabla \times E - i\omega B &= 0 \\
\nabla \cdot B &= 0 \\
\nabla ...

**4**

votes

**1**answer

74 views

### Explicit probability conserving solvers for Pauli equation?

I know that there exist probability conserving explicit solvers for time-dependent Schrödinger's equation, for example, Visscher's one.
But when I tried to add spin into account in this scheme, it ...

**2**

votes

**0**answers

82 views

### Diffusion equation on mixing of diffusing particles

I am trying to study mixing of diffusing particles like it was done by E. Ben-Naim On the Mixing of Diﬀusing Particles.
The picture below shows the idea how permutations and inversion numbers reflect ...

**3**

votes

**1**answer

322 views

### Path integrals for stochastic equations

Does there exist a rigorous mathematical proof for path integral representations given in the physics literature? See for example
http://arxiv.org/abs/hep-ph/9912209v1
For imaginary time rigorous ...

**19**

votes

**0**answers

275 views

### Infinitely many planets on a line, with Newtonian gravity

(I previously asked essentially this on physics.stackexchange, but was actually
hoping for answers with something closer to a proof than what I got there.)
Suppose we have a unit mass planet at each ...

**3**

votes

**1**answer

346 views

### conservation law and generalized Symplectic Monge-Ampere equation arising from 3-variables

If we have a Jacobi PDE system with conservation law $\theta \in \Omega^1(M)$ such that $d \theta$ is non-degenerate 2-form , then we know this fact that it can be written as symplectic 2D ...

**2**

votes

**1**answer

240 views

### symmetry of generationg function of PDE

We know that for finding the solutions of PDE equations, one of methods is "reduction of PDE", . For nonlinear equation
$v_t=(v^{-4/3}v_x)_x+\lambda v$ how can we compute the generators of Lie ...

**1**

vote

**0**answers

280 views

### a question about contact manifolds

Let $\mathfrak{g}=< X_{f_1},X_{f_2},...,X_{f_n}> $ be a lie algebra of contact symmetries , $\mathfrak{g}\subset Sym(E_\omega )$ , such that the orbit spact $ B_\mathfrak{g}=E_\mathfrak{g}/ ...

**1**

vote

**0**answers

157 views

### multivalued solution of a equation

Definition: A scalar k-th order differential equation on a smooth manifold $M$ ,
is $F(x,v,\frac{\partial {^\left | \sigma \right |}v}{\partial x^\sigma })=0 $
for $\left | \sigma \right |\leqslant ...

**1**

vote

**1**answer

150 views

### Bessel and Neumann functions:ordering the zeroes

I realize the abundance of the literature on this theme which is a disservice in this case, because there are too many formulas in all the books I've found and no explicit answers. Maybe someone could ...

**2**

votes

**0**answers

708 views

### Bessel functions in wave propagation and scattering

Is there a way to scale Bessel J(n,.) (Bessel of first kind) and Bessel H(n,.) (Bessel of third kind or Hankel)? I am having computer problems with higher orders (higher vlaues of n) and small ...

**3**

votes

**1**answer

448 views

### A name for PDE systems which are neither under- nor overdetermined?

The concepts of overdetermined and underdetermined PDE systems are well known. However, all sources I have so far looked into appear to avoid giving any name to PDE systems which are neither ...

**2**

votes

**2**answers

381 views

### Herpolhode equation

Poinsot’s construction describes the motion of a freely rotating rigid body in terms of an ellipsoid rolling on a plane. (http://www.phys.ttu.edu/~huang24/Teaching/Phys5306/CH5C.pdf), and the path of ...

**10**

votes

**3**answers

1k views

### Is there a Poincare-Hopf Index theorem for non compact manifolds?

Does Poincare-Hopf index theorem generalizes in any way to non compact manifolds ? In particular, I am interested in the case of a smooth vector field on a cylinder $\mathbb{T}_1\times\mathbb{R}$? If ...

**4**

votes

**1**answer

280 views

### Schrodinger's equation over a randomized grid

I am interested in solutions to
$$
\frac{d}{dt} \Psi = -iH \Psi
$$
for $H$ hermitian and time independent. This boils down to evaluating
$$
\Psi(t) = e^{-iHt}\Psi_0
$$
at points of interest $t_n$. I ...

**12**

votes

**9**answers

2k views

### Newton equations, second order equation and (im)possible motions

I am am currently studying Newtonian mechanics from a conceptional and axiomatic point of view. Now, if I am not mistaken, one (but surely not all) statement of Newtons second law about nature is, ...

**22**

votes

**11**answers

2k views

### What kind of Lagrangians can we have?

In any physics book I've read the Lagrangian is introuced as as a functional whose critical points govern the dynamics of the system. It is then usually shown that a finite collection of ...

**12**

votes

**5**answers

1k views

### 2- and 3-body problems when gravity is not inverse-square

Suppose that gravity did not follow an inverse-square law, but was instead a central force diminishing
as $1/d^p$ for distance separation $d$ and some power $p$.
Two questions:
Presumably the 2-body ...

**5**

votes

**3**answers

1k views

### Characterizing the harmonic oscillator creation and annihilation operators in a rotationally invariant way

I am interested in a characterization of the creation and annihilation operators that is in some sense invariant under $O(n)$ rotations of $\mathbb{R}^n$:
Background
The Harmonic Oscillator on ...

**2**

votes

**3**answers

1k views

### Justification for the matching condition for the wave function at potential jumps. Why is it both restrictive enough and sufficiently general?

Consider Schrödinger's time-independent equation
$$
-\frac{\hbar^2}{2m}\nabla^2\psi+V\psi=E\psi.
$$
In typical examples, the potential $V(x)$ has discontinuities, called potential jumps.
Outside ...

**7**

votes

**4**answers

2k views

### PDEs, boundary conditions, and unique solvability

I'm interested in a criterion that determines whether a linear scalar PDE (arbitrary order) has a unique solution given vanishing boundary conditions at spatial infinity. I'll try to formulate the ...

**2**

votes

**1**answer

390 views

### What kind of uniqueness can I conclude for solutions to a simple functional equation?

I'm going to ask a very vague question, and then give specifics for the version I particularly care about. I'm interested in answers at all levels of vagueness.
At the most vague version, I am in ...

**68**

votes

**0**answers

5k views

### Dropping three bodies

Consider the usual three-body problem with Newtonian
$1/r^2$ force between masses. Let the three masses start off at rest,
and not collinear. Then they will become collinear a finite time ...

**5**

votes

**3**answers

354 views

### Do there exist small neighborhoods in a classical mechanical system without pairs of focal points?

The question I will ask makes sense in much more generality, but I will leave the translation to the experts, since I'm only looking for a special case (and it would not surprise me if the answer does ...

**24**

votes

**7**answers

2k views

### Does every ODE comes from something in physics?

Not sure if this is appropriate to Math Overflow, but I think there's some way to make this precise, even if I'm not sure how to do it myself.
Say I have a nasty ODE, nonlinear, maybe extremely ...

**6**

votes

**1**answer

281 views

### Is there a theory of differential equations for smooth correspondences?

This question is very closely related to another one I just asked. The general question is to what extent there is a theory of differential equations for smooth correspondences (between a smooth ...

**2**

votes

**2**answers

451 views

### Is the Hessian of Hamilton's function positive-definite?

Background
Consider an electron with mass $1$ moving in $\mathbb R^n$ in under the influence of a static electromagnetic field. Up to identifying vector fields with differential forms, Maxwell's ...

**63**

votes

**11**answers

12k views

### What is an integrable system

What is an integrable system, and what is the significance of such systems? (Maybe it is easier to explain what a non-integrable system is.) In particular, is there a dichotomy between "integrable" ...

**10**

votes

**3**answers

464 views

### What happens to Newtonian systems as the mass vanishes?

This question is closely related to another one I asked recently, and may be thought of as a warm-up to that one.
Consider $\mathbb R^n$ with its usual metric, and pick a one-form $b$ and a function ...

**5**

votes

**5**answers

383 views

### What happens to the solutions of a fourth-order boundary-value problem as you turn off the fourth-order coefficient?

Background
Lagrangian mechanics on $\mathbb R^n$ is usually defined by picking a Lagrangian function $L: {\rm T}\mathbb R^n \to \mathbb R$, where ${\rm T}\mathbb R^n = \mathbb R^{2n}$ is the tangent ...

**7**

votes

**1**answer

339 views

### Is the space of nondegenerate classical paths connected?

I have a fairly specific question. My intuition says the answer is "yes", but there is a natural generalizations in which I take out all the "physics", and then I think the answer is "no".
Edit ...

**37**

votes

**9**answers

7k views

### Motivating the Laplace transform definition

In undergraduate differential equations it's usual to deal with the Laplace transform to reduce the differential equation problem to an algebraic problem.
The Laplace transform of a function $f(t)$, ...