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 ...

learn more… | top users | synonyms (1)

5
votes
0answers
336 views

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 ...
5
votes
0answers
73 views

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 ...
4
votes
3answers
362 views

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 ...
4
votes
1answer
888 views

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 ...
4
votes
2answers
189 views

Why is the phase space a symplectic manifold rather than a manifold with a metric?

Why does phase space require a symplectic geometry rather than a metric? Is there some scenario where a metric is unable to describe the notion of length in phase space, specifically in relation to ...
4
votes
1answer
241 views

Strange definition of microcanonical partition function

I always thought that the microcanonical partition function would measure the number of states that correspond to some fixed energy. Despite, I found in this paper (equation 3.4) that we integrate ...
4
votes
1answer
639 views

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 ...
4
votes
2answers
311 views

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')\} ~=~ ...
4
votes
2answers
452 views

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 ...
4
votes
2answers
261 views

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} = ...
4
votes
2answers
128 views

Why does a cyclic coordinate reduce the dimension of the cotangent manifold by 2?

Our professor's notes read, "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 dynamics." ...
4
votes
2answers
2k views

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 ...
4
votes
1answer
396 views

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: ...
4
votes
3answers
1k views

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 ...
4
votes
2answers
725 views

Counting degrees of freedom in presence of constraints

In a $N$ dimensional phase space if I have $M$ 1st class and $S$ 2nd class constraints, then I have $N-2M-S$ degrees of freedom in phase space. How can I calculate the degrees of freedom in ...
4
votes
1answer
126 views

Legendre transform for non-local Lagrangians, or Hamiltonian of non-local Lagrangian and their properties

This is sort of a multi-part question, mostly dealing with how to treat non-local Hamiltonians and how the corresponding properties of Hamiltonians work in a non-local framework. I proposed an example ...
4
votes
1answer
383 views

Non-integrability of the 2D double pendulum

Context: For a system with $n$ degrees of freedom (DOF), one has to deal with $2n$ independent coordinates ($2n$ dimensional phase space), of position $q$ and $\dot{q}$ in Lagrangian formulation, ...
4
votes
1answer
209 views

Why is the Hamiltonian the Legendre transform of the Lagrangian?

So, as the title says, why is the Hamiltonian the Legendre transform of the Lagrangian? I know that from quantum mechanics, one can start with the Hamiltonian defined as the generator of time ...
4
votes
1answer
169 views

Which transformations are canonical?

Which transformations are canonical? Why do canonical transformations preserve the measure of integration in phase space?
4
votes
3answers
482 views

Generalizing Heisenberg Uncertainty Priniciple

Writing the relationship between canonical momenta $\pi _i$ and canonical coordinates $x_i$ $$\pi _i =\text{ }\frac{\partial \mathcal{L}}{\partial \left(\frac{\partial x_i}{\partial t}\right)}$$ ...
4
votes
1answer
563 views

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 ...
4
votes
2answers
2k views

Lagrangian mechanics vs Hamiltonian mechanics [duplicate]

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 ...
4
votes
1answer
109 views

Some question about symplectic transformation

I read Arnold's book Mathematical Methods of Classical Mechanics and come across with three problems in page 229. 1.Let $\lambda$ and $\bar{\lambda}$ be simple (multiplicity 1) eigenvalues of a ...
4
votes
1answer
168 views

A question about canonical transformation

I have posted this question in math.stackexchange before with no answer till now. It may be more suitable to post here. There is a problem in Arnold's Mathematical Methods of Classical Mechanics ...
4
votes
1answer
72 views

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 ...
4
votes
1answer
197 views

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 ...
4
votes
3answers
166 views

About constraints of the first class and electrodynamics

Consider a theory in the Hamiltonian formalism and assume that it has constraints between canonical variables $Q, \pi$. By the Dirac terminology, the set of constraints $F_{a}(Q, \pi) \approx 0$ of ...
4
votes
1answer
371 views

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 ...
4
votes
1answer
211 views

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 ...
4
votes
1answer
39 views

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 ...
4
votes
1answer
115 views

Dimension agreement in canonical transformation

In this Physics.SE post, there is a transformation: $$Q = q,$$ $$P = \sqrt{p} - \sqrt{q}.$$ for Hamiltonian $H = \frac{p^2}{2}$. The post discusses the validity of this transformation as a canonical ...
4
votes
0answers
57 views

CM: Need to recover the Hamiltonian, knowing conserved quantities and information about the EOM, possibly via action-angle coordinates

Statement of the problem: I have a system with 2 degrees of freedom and I have found two independent conserved quantities, without knowledge of the Hamiltonian. I'm looking for a method to recover a ...
4
votes
0answers
750 views

Show that the Laplace-Runge-Lenz vector is conserved using poisson brackets

(I realise similar Phys.SE questions already exist but there is no answer with the Poisson bracket notation, I'll take this down if someone lets me know I should have commented in the existing ...
4
votes
0answers
107 views

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 ...
4
votes
1answer
137 views

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.
3
votes
3answers
683 views

Liouville's theorem and conservation of phase space volume

It can be proved that the size of an initial volume element in phase space remain constant in time even for time-dependent Hamiltonians. So I was wondering whether it is still true even when the ...
3
votes
4answers
932 views

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 ...
3
votes
1answer
231 views

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 ...
3
votes
3answers
600 views

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 ...
3
votes
3answers
689 views

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?
3
votes
2answers
130 views

Symplectic geometry in thermodynamics

There seems to be analogues between Hamiltonian dynamics and thermodynamics given the Legendre transforms between Lagrangian and Hamiltonian functions and all of Maxwell's relations. Poincarè tried to ...
3
votes
1answer
174 views

Lagrangian and Hamiltonian EOM with dissipative force

I am trying to write the Lagrangian and Hamiltonian for the forced Harmonic oscillator before quantizing it to get to the quantum picture. For EOM $$m\ddot{q}+\beta\dot{q}+kq=f(t),$$ I write the ...
3
votes
1answer
225 views

Missing terms in Hamiltonian after Legendre transformation of Lagrangian

Short question Given any Lagrangian density of fields one could possibly conceive, is it the case that after one has performed a Legendre transformation, if the Hamiltonian is then expressed in terms ...
3
votes
2answers
392 views

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 ...
3
votes
5answers
1k views

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 ...
3
votes
1answer
400 views

How to explain the different forms of the Hamilton-Jacobi equation?

In Arnold's Mathematical Methods of Classical Mechanics, he derives the Hamilton-Jacobi equation (HJE) using a generating function $S_1(Q, q)$ to get $$ H\left(\frac{\partial S_1(Q, q)}{\partial q}, ...
3
votes
2answers
134 views

Non-relativistic QFT Lagrangian for fermions

Take the ordinary Hamiltonian from non-relativistic quantum mechanics expressed in terms of the fermi fields $\psi(\mathbf{x})$ and $\psi^\dagger(\mathbf{x})$ (as derived, for example, by A. L. Fetter ...
3
votes
1answer
190 views

Geometric mechanics - Symplecticity

I am just trying to wade through literature on classical mechanics and I really don't know where to start, everything is Fibre bundle this or manifold that, and doesn't really ease you in to the ...
3
votes
2answers
237 views

$p\ dq$ is the “tautological” one-form?

The one-form $$\theta=\sum_i p_i\, \text dq^i$$ is a central object in hamiltonian mechanics. It has a bunch of applications: $\omega=\text d\theta$ is the symplectic structure on phase space, ...
3
votes
1answer
507 views

How do you derive Lagrange's equation of motion from a Routhian?

Given a Routhian $R(r,\dot{r},\phi,p_{\phi})$, how do you derive Lagrange's equation for $r$? Do you just solve the following for $r$? $$\frac{d}{dt}\frac{∂R}{∂\dot{\phi}}-\frac{∂R}{∂\phi}=0$$ And ...