# Tagged Questions

The Hamiltonian formalism is a formalism in Classical Mechanics. Besides Lagrangian Mechanics, it is an effective way of reformulating classical mechanics in a simple way. Very useful in Quantum Mechanics, specifically the Heisenberg and Schrodinger formulations. Unlike Lagrangian ...

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### Hamilton's equations of motion

One of hamilton's equations is $(\frac{\partial H}{\partial q} )_t = -(\frac{\partial p}{\partial t}) _q$. But isn't it $\frac{\partial L}{\partial q} = \frac{dp}{dt}$? If H = L(i.e. V = 0), what ...
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### How does this quantisation relation come about?

I'm currently doing a course in string theory and in the lecture notes it is stated: $$[x^-, p^+]~=~-i \tag{1}$$ I am fine with this. However, after trying (and failing) a question, I looked at ...
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### Time derivative of a function in Phase Space

Consider a function $\mathcal{H}(q_i,p_i;t)$ such that it obeys the equation: $$\frac{d\mathcal{H}}{dt}=\frac{\partial\mathcal{H}}{\partial t}$$ What does this equation imply (read: mean), physically?...
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### Hamilton's Equations

The last step of this derivation of Hamilton's Equations is what's making me doubt it. It is as follows: Assuming the existence of a smooth function $\mathcal{H}(q_i,p_i)$ in $(q_i(t), \,p_i(t))$ ...
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### A physical system is described by the following Lagrangian: $L = \frac{m}{2} (\dot{\rho}² + \rho ² \dot{\phi} ² + \dot{z} ²) + a \rho² \dot{\phi}$ [closed]

Where $a$ is a constant and $(\rho,\phi,z)$ are cylindrical coordinates. I found the following Hamiltonian $H =\frac{m}{2}(\dot{\rho}² + \dot{z}² + \rho²\dot{\phi}²)$. The problem asked me to find ...
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### Energy conservation Hamiltonian dependency

Suppose the a system has a Hamiltonian $H = H(q,p)$, and suppose $H$ does not depend explicitly on time. If $H\neq E$ the total energy of the system, does this necessarily say that $E$ is not ...
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### Spacetime in Loop quantum gravity

In LQG, does spacetime consist of interconnected loops? Are those loops real? If spacetime does not consist of the aforementioned loops, what it consists of? The definition of real would be: they ...
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### Derive statistical entropy from information entropy

I'm trying to compute the entropy of an Hamiltonian system with a fixed energy from the information entropy. Consider a random variable $X$ (state) to be uniformly distributed in a domain $\Omega$ (...
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### Birkhoff Method for Harmonic Oscillator Perturbation

Problem: Given Hamiltonian $$H = \frac12 (p^{2}+q^{2})+q^{3}-3qp^{2}$$ make a perturbative canonical transformation $(q,p) \rightarrow (Q,P)$ such that the new Hamiltonian, apart from terms of degree ...
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### Constructing a Hamiltonian from a mass matrix?

I was solving some questions regarding the Hamiltonian, which required a lot of algebra, but as I finished and looked professor answer I saw that he constructed a matrix from the kinetic energy and ...
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### Is time-1 map of a Hamiltonian vector field on a cylinder always twist?

I have a one degree of freedom analytic Hamiltonian $H(q,p)$ defined on a semi-infinite cylinder, i.e. $(q,p) \in \mathbb{T} \times \mathbb{R}^{+}$, such that all level sets $H(q,p)=c$ are closed ...
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### Canonical transformation question

Let $(\vec{r},\vec{p})$ denote set of canonical variables. Assume a system is described by the following Hamiltonian $$H(r,p) = \frac{1}{2m}(p_1^2 + (p_2 - \beta*x_1)^2 + p_3^2),$$ where $\beta$ is ...
I am reading goldstein there is some comment I don't understand. Consider the following hamiltonian $$H = \frac{p^2}{2m} + \frac{kq^2}{2}$$, which can be rewritten as follows H = \frac{1}{2m}(p^2 +...