# Questions tagged [hamiltonian-formalism]

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 Mechanics, this formalism relies on a "Hamiltonian" instead of a Lagrangian, which differs from the Lagrangian through a Legendre transformation.

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### Meaning of a canonical transformation “preserving” a differential form?

In Chapter 9 of Arnold's Mathematical Methods of Classical Mechanics, we find the following definition: Let $g$ be a differentiable mapping of the phase space $\mathbb R^{2n}$ to $\mathbb R^{2n}$. ...
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### Noise spectrum of two systems and interacting Hamiltonian

I've been discovering recently the concept of noise spectrum, defined as: $$S_{xx}[\omega] = \int dt \langle x(t)x(0)\rangle \text{e}^{-i\omega t}$$ Roughly the Fourrier transform of the two-point ...
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### Operator Ordering Ambiguities

I have been told that $$[\hat x^2,\hat p^2]=2i\hbar (\hat x\hat p+\hat p\hat x)$$ illustrates operator ordering ambiguity. What does that mean? I tried googling but to no avail.
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### Potential Energy tends to infinity on the N-Body Problem

I need help to solve this problem related with the N-Body problem, i dont understand quite well what I need to define or to express in order to solve it. We assume a particular solution to the N-Body ...
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### Why don't we use the concept of force in quantum mechanics?

I'm a quarter of the way towards finishing a basic quantum mechanics course, and I see no mention of force, after having done the 1-D Schrodinger equation for a free particle, particle in an ...
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### Why the Hamiltonian and the Lagrangian are used interchangeably in QFT perturbation calculations

Whenever one needs to calculate correlation functions in QFT using perturbations one encounters the following expression: $\langle 0| some\ operators \times \exp(iS_{(t)}) |0\rangle$ where, ...
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### What makes an equation an 'equation of motion'?

Every now and then, I find myself reading papers/text talking about how this equation is a constraint but that equation is an equation of motion which satisfies this constraint. For example, in the ...
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### Relativistic Hamiltonian Formulations [duplicate]

Possible Duplicate: Hamiltonian mechanics and special relativity? The Hamiltonian formulation is beautifully symmetric. It's a shame that the explicit time derivatives in Hamilton's equations ...
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### Poisson brackets: prove that they are canonical invariants

I need a clarification about Poisson brackets. I'm studying on Goldstein's Classical Mechanics (1 ed.). Goldstein proves that Poisson brackets are canonical invariants for any functions F and G. ...
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### What exactly are Hamiltonian Mechanics (and Lagrangian mechanics)

What exactly are Hamiltonian Mechanics (and Lagrangian mechanics)? I want to self-study QM, and I've heard from most people that Hamiltonian mechanics is a prereq. So I wikipedia'd it and the entry ...
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### Integrable vs. Non-Integrable systems

Integrable systems are systems which have $2n-1$ time-independent, functionally independent conserved quantities ($n$ being the number of degrees of freedom), or $n$ whose Poisson brackets with each ...
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### Dirac equation as Hamiltonian system

Let us consider Dirac equation $$(i\gamma^\mu\partial_\mu -m)\psi ~=~0$$ as a classical field equation. Is it possible to introduce Poisson bracket on the space of spinors $\psi$ in such a way that ...
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### Is the geometric formulation of Hamiltonian mechanics really necessary? [duplicate]

Possible Duplicate: Why is the symplectic manifold version of Hamiltonian mechanics used in Newtonian mechanics? I was sitting around with some friends the other day trying to come up with an ...
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### Dirac equation as canonical quantization?

First of all, I'm not a physicist, I'm mathematics phd student, but I have one elementary physical question and was not able to find answer in standard textbooks. Motivation is quite simple: let me ...