5
votes
Accepted
Proving commutator of velocity differentiation and total time differentiation is position differentiation
Suppose $L$ is a function of $t$, $q$, and $\dot q$. (In Qmechanic's answer, it can depend on every derivative $q^{(n)}$, but the proof is the same.)
We have
$$
\frac d{dt}L(t,q,\dot q)=\frac \partial{...
- 17.2k
3
votes
Accepted
Topology for the resolution of identity equality
No, (2) can never hold in the norm topology. In more detail, suppose $V$ is an inner product space and that $\{x_i\}_{i\in I}$ is an infinite orthonormal set. Then, for any finite set $F\subset I$, ...
- 60.4k
2
votes
Name for a nonlinear first order Partial Differential Equation
To my knowledge, there is no universally accepted name for this equation.
Let $g\geq 0$. The steady state analogue of your equation (i.e. assuming the solution does not depend on $t$) is $$ \vert \...
- 7,604
1
vote
$L^p$ spaces and $L^2$
We say that $f\in L^p(\Omega)$ if $\int_{\Omega}|f|^pdx<\infty$, in your case you have to study the convergence of
$$\int_{\Bbb R} \left(\frac{e^{-x^2}}{\sqrt{|x|}}\right)^p dx $$
Note that as $|x|\...
- 6,855
1
vote
Accepted
Hamiltonian of the Ablowitz-Ladik NLS
1° The Hamiltonian might be interpreted as the energy of the system in many cases, but it isn't required ultimately; it shall be seen as a "state function", which describes a system by ...
- 11.7k
1
vote
Any electric field is necessarily potential?
I would say that in a potentially holey domain, you must have
$$\oint \vec{E}(\mathbf{r}) \cdot d \mathbf{r} = 0.$$
The potential difference is then
$$V(b)-V(a) = - \int_a^b \vec{E}(\mathbf{r}) \cdot ...
- 1,639
1
vote
Accepted
Derivative operator on $L_2([0, \, 1], \, \mathrm dx)$
Hints: $-i\int \psi'\phi =\int \chi \psi$ for all $\psi \in D(A)$. Let $h(x)=\int_0^{x}\chi (t)dt$, so that $-i\int \psi'\phi =\int h' \psi$. Integrating by parts we get $-i\int \psi'\phi =\int h \...
- 40.2k
1
vote
Accepted
Math_Physics - Regarding Conservation Laws
$\text{ }\text{ }\text{ }$ Divergence measures the tendency of a flow field to leave a point. So a field with positive divergence about a single point within a volume should be more aligned with the ...
- 1,066
1
vote
Math_Physics - Regarding Conservation Laws
'Time direction' is a critical point in first order evolution equations. The two signs make a difference between time forward and backward equations.
Generally a vector field $X$ is the non-exact ...
- 3,711
1
vote
Introduction to the replica trick
Two more recommendations.
Information, Physics, and Computation by Mezard and Montanari.
Self-contained book which introduces statistical mechanics & combintaorial optimization from scratch. ...
- 19
Only top scored, non community-wiki answers of a minimum length are eligible