All Questions
Tagged with ap.analysis-of-pdes hamiltonian-mechanics
9 questions
6
votes
0
answers
290
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A question in elementary differential geometry
Let $M$ be a finite dimensional manifold of constant curvature $\kappa$. Consider a solution of the Hamilton--Jacobi equation
$$ \partial_t u + |\nabla u|^2 = 0. $$
Can we give a precise estimate of a ...
- 39
4
votes
0
answers
141
views
Existence results for Lagrangian solutions to the Incompressible Euler Equation?
It is known that if a function (which we shall call the lagrangian flow, or lagrangian trajectory) $$X:(\mathbb{R}/\mathbb{Z})^3 \times [0,T] \to \mathbb{R}^3$$ with $X \in H^1_t$ (i.e. has weak time ...
- 989
3
votes
1
answer
429
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Integrability of Schroedinger's equation
Consider the periodic nonlinear Schrödinger equation
$$-i \partial_t u + \Delta u = f(|u|)u, \qquad u=u(t,x) \in \mathbb{C}, \; t\in \mathbb{R}, \; x\in \mathbb{T}^n,$$
where $\mathbb{T}:= \mathbb{R}/\...
- 191
3
votes
2
answers
369
views
Hamiltonian, energy, and conservation laws of nonlinear PDEs
In many PDEs, I see the papers mention the energy of the PDE. And some papers and books mention Hamiltonians. I know that integrable systems have infinitely many conservation laws and these laws are ...
- 159
3
votes
0
answers
352
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Proving that system is Hamiltonian
This question is moved from math stackexchange, seems like it is a more advanced question. Here the link from the original question: https://math.stackexchange.com/questions/2666194/proving-that-...
- 31
2
votes
1
answer
308
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Gradient descent relaxation dynamics of a Euler-Lagrange equation
I want to minimize the functional
$$
F=\int{L(u)}dx,
$$
where $L= u_x^2-u^2$ is the Lagrangian function of the functional. Even if its Euler-Lagrange equation is easily found and solved, I want to try ...
- 159
1
vote
1
answer
90
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Why is this Hamiltonian flow of the Vlasov equation well defined?
Lions and Paul claim in their 1993 paper "Sur les mesures de Wigner" that the Hamiltonian flow
$$\dot{x} = \xi, \quad \dot{\xi} = - \nabla V(x) $$
of the Vlasov equation
$$\partial_t f + \xi ...
- 469
1
vote
1
answer
239
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Infinitesimal generators and conserved quantities (Schrodinger type evolution)
First, I'm no expert in symmetry analysis of evolution equations and so I apologize if this post is a bit of a cobble. The question I have is about the evolution of $\psi: \mathbb{R}^{1+1}\to \mathbb{...
- 13
1
vote
1
answer
119
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Invertibility of the characteristic flow in Hamilton-Jacobi equations
We are in the context of Hamilton Jacobi equations, in particular I was reading the characteristic method. We want to solve the problem of the special form (Hamiltonian only depending on the "$p$" ...
- 111