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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Energy operator

Does the Hamiltonian always translate to the energy of a system? What about in QM? So by the Schrodinger equation, is it true then that $i\hbar{\partial\over\partial t}|\psi\rangle=H|\psi\rangle$ ...
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Physical interpretation of Poisson bracket properties

In classical Hamiltonian mechanics evolution of any observable (scalar function on a manifold in hand) is given as $$\frac{dA}{dt} = \{A,H\}+\frac{\partial A}{\partial t}$$ So Poisson bracket is a ...
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Other application of Liouville's theorem besides thermodynamics

Are there any other important practical and theoretical consequences of Liouville's theorem on the conservation of phase space volume besides the calculation of the microcanonical potential in ...
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formal framework for talking about 'minimal couplings'

usually on physical theories one would have Lagrangians or Hamiltonians with multiple fields; say, a vector $A_{\mu}$ and a scalar $\phi$ and one would postulate ad hoc a coupling between the fields ...
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Yang Mills Hamiltonian: why do we use the Weyl's temporal gauge?

Do you know why in the quantization of SU(2) Yang Mills Gauge Theory, it is always chosen the Weyl (temporal) gauge to derive the Hamiltonian? Is it possible to fix another gauge?
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Why are we living in the $q$ part of the phase space?

In Hamilton mechanics and quantum mechanics, $p$ and $q$ are almost symmetric. But in the real world, the $p$ space isn't as intuitive as the $q$ space. For example, We can uniquely identify a person ...
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What is Maupertuis' principle good for?

The strength of Hamilton's principle is obvious to me and I see the advantage. Now, for conservative systems we also have Maupertuis' principle that says: $$ \delta \int p dq =0$$ and I am not sure ...
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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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What would happen if energy was conserved but phase space volume wasn't? (and vice-versa)

I'm trying to understand the relationship between the two conservation laws. As I understand, Liouville's result is a weaker condition: it relies merely on the particular form assumed by Hamilton's ...
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Why don't we use Hamilton-Jacobi method in QM?

In classical mechanics, we usually try to find a set of coordinates by Hamilton-Jacobi method to transform the Hamiltonian to zero such that the coordinates are conservations. However, we never try ...
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Is Feynman talking about the Zeroth Law of Thermodynamics?

In Volume 1 Chapter 39 of the Feynman Lectures on Physics, Feynman derives the ideal gas law from Newton's laws of motion. But then on page 41-1, he puts a caveat to the derivation he has just ...
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The consistency conditions of constrained Hamiltonian systems

I am studying the Hamiltonian description of a constrained system. There are some questions puzzled me for days, which I have been stuck on it. From the lagrangian, we can obtain the primary ...
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Writing $\dot{q}$ in terms of $p$ in the Hamiltonian formulation

In the Hamiltonian formulation, we make a Legendre transformation of the Lagrangian and it should be written in terms of the coordinates $q$ and momentum $p$. Can we always write $dq/dt$ in terms of ...
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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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The number of independent variables in the Lagrangian and Hamiltonian methods in Classical Mechanics

It's told in Landau - Classical Mechanics, that in the Hamiltonian method, generalized coordinates $q_j$ and generalized momenta $p_j$ are independent variables of a mechanical system. Anyway, in the ...
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Why is the symplectic manifold version of Hamiltonian mechanics used in Newtonian mechanics?

Books such as Mathematical methods of classical mechanics describe an approach to classical (Newtonian/Galilean) mechanics where Hamiltonian mechanics turn into a theory of symplectic forms on ...
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The string Poisson bracket

Where does the factor $\frac{1}{T}$ ($T$ is the string tension) in this Poisson bracket come from? $$ \{X^{\mu}(\tau,\sigma),\dot{X}^{\nu}(\tau,\sigma')\} ~=~ ...
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Hamilton formalism for Dirac spinors

Let's have the Dirac free lagrangian: $$ L = \bar {\Psi} (i\gamma^{\mu}\partial_{\mu} - m) \Psi . $$ I can rewrite it as $$ L = i\Psi^{\dagger}\partial_{0}\Psi - H_{d}, \quad H_{d} = ...
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Why does a cyclic coordinate reduce the dimension of the cotangent manifold by 2?

Our professor's notes say that "In general, in Hamiltonian dynamics a constant of motion will reduce the dimension of the phase space by two dimensions, not just one as it does in Lagrangian ...
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An example of non-Hamiltonian systems

I am preparing for the exam. And I need to know the answer to one question which I can't understand. "Give an example of non-Hamiltonian systems: in case of infinite number of particles; for a finite ...
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What are some mechanics examples with a globally non-generic symplecic structure?

In the framework of statistical mechanics, in books and lectures when the fundamentals are stated, i.e. phase space, Hamiltons equation, the density etc., phase space seems usually be assumed to be ...
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Which transformations are canonical?

Which transformations are canonical? Why do canonical transformations preserve the measure of integration in phase space?
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Constraints of massive relativistic point particle in hamiltonian mechanics

I try to understand constructing of Hamiltonian mechanics with constraints. I decided to start with the simple case: free relativistic particle. I've constructed hamiltonian with constraint: ...
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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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Hamiltonian mechanics and special relativity?

Is there a relativistic version of Hamiltonian mechanics? If so, how is it formulated (what are the main equations and the form of Hamiltonian)? Is it a common framework, if not then why? It would be ...
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Generalizing Heisenberg Uncertainty Priniciple

Writing the relationship between canonical momenta $\pi _i$ and canonical coordinates $x_i$ $$\pi _i =\text{ }\frac{\partial \mathcal{L}}{\partial \left(\frac{\partial x_i}{\partial t}\right)}$$ ...
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To what extent is the “minimal substitution” or “minimal coupling” for the EM vector potential valid?

In all text books (and papers for that matter) about QFT and the classical limit of relativistic equations, one comes across the "minimal substitution" to introduce the magnetic potential into the ...
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Mathematica to help for an Hamiltonian problem

I have an Hamiltonian problem whose 2D phase space exhibit islands of stability (elliptic fixed points). I can calculate the area of these islands in some cases, but for other cases I would like to ...
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Lagrangian mechanics vs Hamiltonian mechanics

First of all, what are the differences between these two: Lagrangian mechanics and Hamiltonian mechanics? And secondly, do I need to learn both in order to study quantum mechanics and quantum field ...
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Why is the Legendre transformation an application of the duality relationship between points and lines?

When I read the Wiki about Legendre transformation, there is a statement The Legendre transformation is an application of the duality relationship between points and lines. What's the meaning of ...
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Do we have a fundamental Hamiltonian for the system of H$_2$O molecules?

From the quantum mechanics(QM) viewpoint, does there exist a Hamiltonian $H$ for the system of H$_2$O molecules? Assume that the number of H$_2$O molecules is fixed. Imagine that by calculating the ...
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Peculiar Hamiltonian Phase space

I was solving an exercise of classical mechanics : Consider the following hamiltonian $H(p,q,t) = \frac{p^2}{2m} + \lambda pq + \frac{1}{2}m\lambda^2\frac{q^6}{q^4+\alpha^4}$ Where ...
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How to get Hamiltonian of QED from lagrangian?

I have the QED lagrangian: $$ L = \bar {\Psi}(i \gamma^{\mu }\partial_{\mu} + q\gamma^{\mu}A_{\mu} - m)\Psi + \frac{1}{16 \pi}F_{\alpha \beta}F^{\alpha \beta} . $$ I tried to get hamiltonian by ...
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Does the Renormalization of QFT Contradict Canonical Quantization?

Does the renormalization of QFT contradict canonical quantization? In canonical quantization, you take the classical fields and canonical momenta and turn them into operators, and you require that ...
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Connections of iterative solvers for large systems of equation in Physics?

I am trying to find the domains in physics where solving large systems of equations is computationally expensive. The sparse systems are of my particular interest, where the input matrix A is in GBs ...
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Some questions about spacetime topology, causality structures and other GR businesses

1) What are the exact conditions required for the canonical transformation? Most papers just assume away with global hyperbolicity, but is there a more general condition for it? "Quantum gravity in ...
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Hamiltonian function for classical hard-sphere elastic collision

I'm trying to find the Hamiltonian function for a system consisting of a single particle in one dimension colliding elastically with a wall at x = 0. Everything I've read on the topic (e.g. this ...
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Is a particle subject to dissipation proportional to its velocity a Hamiltonian system?

Why or why not? I'm pretty sure that this isn't a Hamiltonian system because it involves a dissipation term, but using the Hamiltonian flow it gives me that the system is Hamiltonian.
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Find the Hamiltonian given $\dot p$ and $\dot q$

I have these equations: $$\dot p=ap+bq,$$ $$\dot q=cp+dq,$$ and I have to find the conditions such as the equations are canonical. Then, I have to find the Hamiltonian $H$. To answer to the first ...
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First class and second class constraints

Hello I am working on a project that involves the constraints. I checkout the paper of Dirac about the constraints as well as some other resources. But still confuse about the first class and second ...
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Two components of angular momentum conserved $\Rightarrow $ All three components are conserved?

I was wondering whether it is correct to say that if two components of the angular momentum are conserved, then all three Cartesian coordinates of the angular momentum are conserved? I would regard ...
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Phase space in quantum mechanics and Heisenberg uncertainty principle

In my book about quantum mechanics they give a derivation that for one particle an area of $h$ in $2D$ phase space contains exactly one quantum mechanical state. In my book about statistical physics ...
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Gauge Invariance of the Hamiltonian of the electromagnetic field

The Hamiltonian for an electron of mass $m$ and charge $e$ in an exterior electromagnetic field is $$H=\frac{1}{2m}(p-(e/c)A)^2+e\varphi.$$ The corresponding (via canonical quantization) quantum ...
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Why does Quantum Field Theory use Lagrangians rather than Hamiltonains? [duplicate]

Why does Quantum Field Theory use usually Lagrangians rather than Hamiltonains? I heard many reasons, but I'm not sure which is true. Some say it's just a matter of beauty, so Lagrangians are more ...
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Relating generalized momentum, generalized velocity, and kinetic energy: $2T~=~\sum_i p_{i}\dot{q}^{i}$

According to equation (6) on the first page of some lecture notes online, the above equation is used to prove the virial theorem. For rectangular coordinates, the relation $$ 2T~=~\sum_i ...
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What is the difference between manifest Lorentz invariance and canonical Lorentz invariance?

I often read that the Lorentz symmetry is manifest in the path integral formulation but is not in the canonical quantization - what does this really mean?
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Is there a Hamiltonian for the (classical) electromagnetic field? If so, how can it be derived from the Lagrangian?

The classical Lagrangian for the electromagnetic field is $$\mathcal{L} = -\frac{1}{4\mu_0} F^{\mu \nu} F_{\mu \nu} + J^\mu A_\mu.$$ Is there also a Hamiltonian? If so, how to derive it? I know how ...
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primary constraints for constrained Hamiltonian systems

I would be most thankful if you could help me clarify the setting of primary constraints for constrained Hamiltonian systems. I am reading "Classical and quantum dynamics of constrained Hamiltonian ...
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Good book for Analytical Mechanics

What is a good book for Analytical Mechanics? To be more specific, I would prefer a book that: Is written "for mathematicians", i.e. with high mathematics precision (for example, with less emphasis ...
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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 ...