Questions tagged [hamilton-equations]

Use this tag for questions related to Hamilton's equations.

101 questions
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Hamiltonian Flows and Heisenberg picture of quantum mechanics

I am a math bachelor student studying Quantum Mechanics and I was very briefly introduced to the Heisenberg picture. (Hence many of the following may be trivial) In particular what I know is that: ...
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What is proof of direction of Change in position vector $d \vec r$

How can I prove that the direction of an infinitesimal change in position vector $\mathrm d\vec r$ is the same as that of the instantaneous velocity $\vec v=\mathrm{d}\vec r/\mathrm{d}t$? What I ...
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Dynamical System is fixed point at origin hyperbolic or asymptotically stable and is the system Hamiltonian

I am given a system $\dot x = f(x,y) = (x^2 + y^2)(x^3 + y^2x -2y - x) \\ \dot y = g(x,y) = (x^2 + y^2)(y^3 + x^2y +2x - y)$ and I am asked if the fixed point at $(0,0)$ is hyperbolic or ...
I need to prove that the Hamiltonian system of the rigid body motion $$\begin{cases} \dot{R}_t=P_tJ^{-1},\\ \dot{P}_t=2R_t\Lambda,\quad\text{\Lambda is the Lagrange multiplier}\\ R_t^T R_t-I=0, \... 0answers 199 views When is a Divergence-Free Vector Field on the Tangent Bundle of a Riemannian Manifold Hamiltonian? Starting with a closed, connected Riemannian manifold (M^n, g), using the "product Euclidean metric" to form an associated Riemannian metric \tilde{g}(p,v)((v_1,w_1),(v_2,w_2))) = g(p)(v_1,v_2) + ... 2answers 109 views Find the Hamilton's equations Consider the functional given by$$ \ \large \ J(y)=\int_{a}^{b} \sqrt{(t^2+y^2) (1+\dot y^2) } \ dt .$$Find the Hamilton's equations . Answer: I am unable to find the Hamilton from the ... 1answer 106 views Gauge invariance of the Hamiltonian Consider a Lagrangian L(x,\dot x,t) and a corresponding Hamiltonian H=\dot xp-L where p=\partial L/\partial \dot x which satisfies Hamilton's equations$$\frac{\partial H}{\partial x}=-\dot p$$... 0answers 106 views How to find a Lax pair I've been getting into Hamiltonian PDE's lately and they give the KdV equation a lax pair to proof that it is integrable in some sense. My question is the following how does on find a lax-pair if we ... 2answers 91 views Hamiltonian from Lagrangian L= \frac{m}{2}(\dot{r}^2+r^2\dot{\theta^2})+ \frac{k\cos(\theta)}{r^2} I'm doing the first exercises with the Lagrangians and Hamiltonians. Let:$$L= \frac{m}{2}(\dot{r}^2+r^2\dot{\theta^2})+ \frac{k\cos(\theta)}{r^2}p_1=m\dot{r}p_2=mr^2\dot{\theta}H=\...
Suppose I have the following Hamiltonian $$H(q, p, x,y) = \frac{p^{2}}{2} - \frac{q^{2}}{2} \left(\left(\frac{y^{2}}{2} + \omega^{2}x^{2}\right)^{2} - \frac{q^{2}}{2}\right)$$ Where \$(q, p, x,y) \in \...