Hamilton Definition and 75 Threads

Hamilton's function in this case is the sum of potential and kinetic energy? But then I don’t remember or don’t understand what to do with e. I need to find Г, but I don't understand what to do with the field.

In Hamilton's "on a general method in dynamics", he starts with varying the function ##U## and writes the equation: $$\delta U=\sum m(\ddot x\delta x+\ddot y\delta y+\ddot z\delta z)$$ Then he defines ##T## to be: $$T=\frac{1}{2}\sum m (\dot x^2+\dot y^2+\dot z^2)$$ Then by ##dT=dU##, he...

Margaret Hamilton the Pioneering Software Engineer Who Saved the Moon Landing https://interestingengineering.com/margaret-hamilton-software-engineer-who-saved-the-moon-landing Margaret Hamilton began working with Edward Lorenz, the father of Chaos Theory, in MIT's meteorology department. As...

I really like the book for how much it covers. There's not a single topic that's missed that is relevant to nuclear reactor design/analysis. Often other books can miss a topic or two. It's just that the style is not to the point and often time is wasted talking about things that are irrelevant...

Here is an image of the problem: The problem consist in finding the moviment equation for the pendulum using Lagrangian and Hamiltonian equations. I managed to get the equations , which are shown insed the blue box: Using the hamilton equations, i finally got that the equilibrium angle...

I know that by extremizing lagrangian we get equations of motions. But what if we extremize the energy? I am just little bit of confused, any help is appreciated.

Studying Einstein's original Die Grundlage der allgemeinen Relativitätstheorie, published in 1916's Annalen Der Physik, I came across some equations which I couldn't verify after doing the computations hinted at. The first are equations 47b) regarding the gravity contribution to the...

Good Morning When we derive the Euler Lagrange equations using Hamilton's Principle, we make a point of varying the velocity and the position at the same time, (despite the fact that, normally, they are related through a derivative). I do understand that this is allowed: we are trying to find...

Classical mechanics is based on conservation laws which represent the symmetries of spacetime. The lagrangian function L is a function of position and velocity while the hamiltonian is a function of position and momentum. The velocity and momentum descriptions are related by a legendre...

Here is a graph. I wonder if it has hamilton path or circuit. In hamilton path we have to cross once and only once at an edge, and the start and the finish must be different locations. In hamilton circuit we have to start and finish the same edge. So the circle which B is rounded, which kind of...

Homework Statement Suppose the potential in a problem of one degree of freedom is linearly dependent upon time such that $$H = \frac{p^2}{2m} - mAtx $$ where A is a constant. Solve the dynamical problem by means of Hamilton's principal function under the initial conditions t = 0, x = 0, ##p =...

Homework Statement For a wavefunction ##\psi##, the variance of the Hamiltonian operator ##\hat{H}## is defined as: $$\sigma^2 = \big \langle \psi \mid (\hat{H} - \langle\hat{H}\rangle)^2 \psi \big\rangle$$ I want to prove that if ##\sigma^2 = 0##, then ##\psi## is a solution to the...

Let me show you part of a book "Mechanics From Newton’s Laws to Deterministic Chaos" by Florian Scheck. I do not understand why these integrands can differ by more than time derivative of some function M. Why doesn't it change the value of integrals? It seems this point is crucial for me to...

Homework Statement Hamiltonian of charged particle in magnetic field in 2D is ##H(x,y,p_x,p_y)=\frac{(p_x-ky)^2+(p_y+kx)^2}{2m}## where ##k## and ##m## are constant parameters. For separation of this system use ##S=U(x)+W(y)+kxy+S_t(t)##. Solve Hamilton - Jacobi equation to get ##x(t), y(t)## ...

Hi there guys, Currently writing and comparing two separate Mathematica scripts which can be found here and also here. The first one I've slightly modified to suit my needs and the second one is meant to reproduce the same results. Both scripts are attempting to simulate the trajectory of a...

Homework Statement I am asked to find the Hamilton equations for a block on an inclined plane (no friction) Homework EquationsThe Attempt at a Solution Please ignore the fact that my steps are written in French (sorry!) I am no longer sure of what I'm doing when it comes to finding the...

In case of Ising model we are working with effective Hamiltonian. So let's look to spins which interact. In a case of feromagnet energy function is defined by ## H=-JS_1S_2 ## We have two possibilities. ##S_1## and ##S_2## has different values. And ##S_1## and ##S_2## has the same value. In...

Hello, When doing a little internet search today on generalized coordinates I stumbled on this document: http://people.duke.edu/~hpgavin/cee541/LagrangesEqns.pdf If you are willing, would you be so kind as to open it up and look at the top of (numbered) page 6? OK, so the very existence of...

Yes, that is a serious title for the thread. Could someone please define GENERALIZED COORDINATES? In other words (and with a thread title like that, I damn well better be sure there are other words ) I understand variational methods, Lagrange, Hamilton, (and all that). I understand the...

Homework Statement This was supposed to be an easy question. I have a question here that wants you to describe a yoyo's acceleration (in one dimension) using Lagrangian mechanics. I did and got the right answer. Now I want to use Hamilton's equations of motion but I get a wrong number. Here is...

I'm a little confused about the hamiltonian. Once you have the hamiltonian how can you find conserved quantities. I understand that if it has no explicit dependence on time then the hamiltonian itself is conserved, but how would you get specific conservation laws from this? Many thanks

Hello everyone, Lately, I have been reading and studying the Maxwell's https://es.wikipedia.org/w/index.php?title=A_Treatise_on_Electricity_and_Magnetism&action=edit&redlink=1 https://es.wikipedia.org/w/index.php?title=A_Treatise_on_Electricity_and_Magnetism&action=edit&redlink=1 Thanks for...

Alright I was browsing through feynman's lectures VOL.1 When I came across Hamiltonian Optics I am not quite able to understand it in its entirety Could someone explain the basics of hamiltonian optics to me Help would be much appreciated !

Hey JO. The Hamiltonian is: H= \frac{p_{x}^{2}+p_{y}^{2}}{2m} In quantum Mechanics: \hat{H}=-\frac{\hbar^{2}}{2m}(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial x^{2}}) In polar coordinates: \hat{H}=-\frac{\hbar^{2}}{2m}( \frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}...

I have a problem (this is not homework) Based on covariant Lagrangian ## \mathcal {L} = \frac {m}{2} \frac{dx^{\mu}}{ds} \frac {dx _ {\mu}}{ds} ## record the equations of motion in Hamiltonian form for a particle in the Schwarzschild metric (SM). Based on Legandre transformations...

Homework Statement The motion of a free particle on a plane has hamiltonian $$H =E = \text{const} = \frac{1}{2m} (p_r^2 + \frac{p_{\theta}^2}{r^2})$$ Set up and find a complete integral for ##W##, the time independent generating function to canonical coordinates such that new coordinates are...

I want to obtain equation using Hamilton principle but I just couldn't figure it out; i have The kinetic energy : \begin{equation} E_{k}=\dfrac{1}{2}m_{z} \displaystyle\int\limits_{0}^{L}\ \left[ \left( \dfrac{\partial w(x,t)}{\partial t}\right)^{2}+\left( \dfrac{\partial v(x,t)}{\partial...

Show by direct calculation that Eqs. (4-134) and (4-137) in the textbook by Duderstadt and Hamilton hold, i.e.:(a) ∫ dΩΩiΩj= 4π/3 δij; i,j = x,y,z; 4π(b) ∫ dΩΩxΩyΩz = 0, if l, m, or n is odd. 4π The integrals are over 4π. This is part of the derivation of the diffusion equation...

Casual talk. Constrained Hamilton systems. Dirac-Poisson brackets. Casual talk. Constrained Hamilton systems. Dirac-Poisson brackets. Hi guys, I think I have finally succeeded in understanding the ideas which Dirac explained in the two first chapters of his book "Lectures on Quantum...

Definition/Summary Hamilton's equations of motion is a very general equation of a system evolving deterministically in phase space. Equations \left( {\begin{array}{*{20}{c}} {\dot q}\\ {\dot p} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 0&1\\ { - 1}&0 \end{array}}...

Hi, I was wondering about the interpretation of the Hamilton Jaccobi equation. Naively we have H + \partialS/\partialt = 0 where H is the Hamiltonian and S is the action. But the action is the time integral of the Lagrangian so you would expect \partialS/\partialt = L Thus H + L = KE +...

Given a Matrix A = [a,b;c,d] and it's characteristic polynomial, why does the characteristic polynomial enables us to determine the result of the Matrix A raised to the nth power?

I was given the (general) following form for the Hamilton and Heisenberg Equations of motion \dot{A} = (A, H)_{}, which can represent the Poisson bracket (classical version) or \dot{A} = -i/h[A,H] (Quantum Mechanical commutator). I was given the general solutin for A(t) = e^{tL}A...

Homework Statement Given the following matrix A = [3 -1; -1 3] Find C = (0.5*A - I)100 Homework Equations Using the knowledge that the Cayley - Hamilton Theorem must satisfy its own characteristic polynomial. The Attempt at a Solution Here the characteristic polynomial is λ2 -...

Author: James Douglas Hamilton Title: Time Series Analysis Amazon Link: https://www.amazon.com/dp/0691042896/?tag=pfamazon01-20

if lagrangian mechanics is available the why we needed hamiltonian mechanics. where they differ from each other?

Here is the graph in question: The edges in red are the paths that must be included in the cycle, since vertices a, k, e, and o have only degree 2. I listed all possible routes starting from vertex b, and showed that they all routes closes a cycle while leaving some vertex disconnected. Is...

I know there are no algorithms for finding one, but what are some guidelines? One tip I came up with is that if you have a vertex with degree 2, there is only one way to go through that vertex. Are there any others?

For a Lagrangian L(x^k,\dot{x}^k) which is homogeneous in the \dot{x}^k in the first degree, the usual Hamiltonian vanishes identically. Instead an alternative conjugate momenta is defined as y_j=L\frac{\partial L}{\partial \dot{x}^j} which can then be inverted to give the velocities as a...

I have a system described by the Langevin equation da/dt = - dF/da* + r where a are complex amplitudes of electromagnetic modes (and r is the white noise). How if F was real, it would be the Hamiltonian of the system, but in my case (and in general), F is complex (because the a are...

In my linear algebra course, we just finished proving the cayley hamilton theorem (if p(x) = det (A - xI), then p(A) = 0). The theorem seems obvious: if you plug in A into p, you get det (A-AI) = det (0) = 0. But, of course, you can't do that (this is especially clear when you consider what...

(First of all apologies for the long wall of text) I am to study BRST transformations, for which I'm currently trying to understand constrained Hamiltonian dynamics to treat systems with singular Lagrangians. The crude recipe followed is Lagrangian -> Hamiltonian -> Dirac brackets and their...

Hello! General Question about the H-J equation. What are the steps to be followed if we are in a conservative system? And while answering my question, please in the step after we find S, and when you derive S wrt alpha and place it equals to β. When is alpha Energy? When it is not? i.e is it...

Can anyone help me with the following exercise from Dummit and Foote? ============================================================ Describe the centre of the real Hamilton Quaternions H. Prove that {a + bi | a,b R} is a subring of H which is a field but is not contained in the centre...

Homework Statement The Hamilton-operator is given as \hat{H} and describes the movement of a free rigid object that has the moments of inertia I_{i} Under what circumstances is <\Psi|\hat{L_{1}}|\Psi> time-independent? Homework Equations...

Homework Statement The problem is taken out of Landau's book on classical mechanics. I must find the Hamilton function and the corresponding Hamilton equations for a free particle in Cartesian, cylindrical and spherical coordinates.Homework Equations Hamilton function: H(p,q,t)= \sum p_i \dot...

the title is pretty clear, so we have this F=q( E+V/c X B) the force for a point charge q in an electromagnetic field. and we have the formula to the hamiltonian of electromag force: H= (P+ q/c.A)/2m _ e\phi. the question is how can I get the F from the hamilton principle, using the...

Are there any nice applications of the Cayley Hamilton Theorem. I am looking for a real life application which would motivate students.

Could anyone give me a simple explanation as to why the Fermat/Hamilton principle would be called more general than the Jacobi least time principle? I am trying to understand what differences would result from using the one principle vs. the other; eg: where/in what way would the Jacobi least...

Homework Statement I'm working with a complex scalar field with the lagrange density L= \partial_{\mu} \phi^{\ast} \partial^{\mu} \phi - m^2 \phi^{\ast} \phi And I've shown that's its hamilton density H= \int d^3 x ( \pi^{\ast} \pi + \nabla \phi^{\ast} \cdot \nabla \phi + m^2 \phi^{\ast}...