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
2answers
316 views

Momentum is a cotangent vector?

Imagine we have a particle described by $x \in M$, where $M$ is some manifold, then it is very intuitive I think that a velocity is an element of the tangent space at $x$, so $x' \in T_{x}M.$ Thus, by ...
10
votes
1answer
448 views

What is the physical interpretation of the Poisson bracket [duplicate]

Apologies if this is a really basic question, but what is the physical interpretation of the Poisson bracket in classical mechanics? In particular, how should one interpret the relation between the ...
3
votes
2answers
285 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 ...
2
votes
2answers
126 views

Can I take the partial derivative of the Lagrangian with respect to a constant?

I've got a system where I know that the derivative of one of the generalized coordinates is constant. So to find the Hamiltonian of the system I need to take the partial derivative with respect to ...
2
votes
0answers
110 views

Hamiltonian linearly proportional to momentum

In this question, it is discussed why, in Lagrangians we usually stick to first derivatives and quadratic terms we never see higher derivatives. The selected answer shows that, if a Lagrangian $L(q, \...
1
vote
2answers
210 views

Phase space Lagrangian?

Reading out of this lecture series we define a phase space Lagrangian $\mathcal L$ to be a function of $4n+1$ variables namely $q,\dot q,p,\dot p,t$. My question is, what space is this function ...
2
votes
1answer
134 views

Hamiltonian field equations constraints

Let's consider the Lagrangian $$\mathcal{L}~=~-\frac{1}{2}(\partial_\mu\phi^\nu)^2+\frac{1}{2}(\partial_\mu\phi^\mu)^2+\frac{1}{2}m^2\phi_\mu \phi^\mu,$$ with Minkowski metric $\eta_{\mu\nu}={\rm ...
1
vote
1answer
173 views

Path integral in quantum mechanics

I am confused by the derivation in Srednicki QFT's chapter 6 from (6.8) to (6.9). In (6.8), we have $$<q'',t''|q',t'>~=~\int DqDp \exp[i\int_{t'}^{t''}dt(p\dot{q}-H(p,q))],\tag{6.8}$$ and (6....
3
votes
1answer
113 views

When can an autonomous system be written using a Hamiltonian?

If I have an autonomous series of differential equations $$\tag{1} \frac{dx_i}{dt} ~=~ A_i(x_1,...,x_n)$$ with the condition that $$\tag{2} \sum_{i=1}^n\frac{\partial A_i}{\partial x_i}~=~0$$ in all ...
1
vote
0answers
156 views

When to use Hamiltonian vs Lagrangian?

I currently studying the Lagrangian and Hamiltonian formalisms in classical mechanics, but something I'm not seeing is how do I know which one to use in a given problem? After I find the Lagrangian, ...
1
vote
1answer
163 views

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 ...
0
votes
1answer
205 views

Is it possible to formulate a Hamiltonian for a damped system? [duplicate]

I recently found out that it is possible to formulate a Hamiltonian for a system with time-dependent coordinates such that the Hamiltonian is not the same as the energy When is the Hamiltonian of a ...
1
vote
2answers
145 views

Metric and the Lagrangian [duplicate]

Does the Lagrangian formalism require a metric on the configuration manifold $Q$ in order to define a Lagrangian $L$ on the tangent bundle $TQ$? Further, if we specify a metric on the tangent bundle ...
1
vote
1answer
68 views

Formulating a symplectic integrator for a non-local Hamiltonian

I recently asked two questions, Q. [1] and Q. [2], regarding reformulating non-local Lagrangians as Hamiltonians. In these questions, the Hamiltonian is formulated as an integral because of it's non-...
1
vote
2answers
154 views

An inconsistency in Hamiltonian formulation for non-local Lagrangian: what am I doing wrong?

This question is based on a previous question I asked, Q. [1] In this question, I proposed an example of a non-local Lagrangian (functional), I'm revisiting it here: $$\mathbb{L}=\frac{1}{2}\int^t_0 ...
4
votes
1answer
179 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 ...
1
vote
1answer
100 views

Geometry of Hamilton-Jacobi Equation

I'm trying to understand the geometry of the Hamilton-Jacobi equation (working from Gelfand + Fomin), but I'm stuck. I know that: If we define the function $S(t,y;t_0, y_0)$ as: $$S(t,y;t_0,y_0) = \...
1
vote
1answer
87 views

How to calculate the Hamiltonian from the Lagrangian for a non-relativistic charged point particle in an EM field?

I was given the equation of the Lagrangian: \begin{equation} L~=~\frac{1}{2}m \dot{x}^2+\frac{e}{c}\vec{\dot{x}}\cdot \vec{A}(\vec{x},t)-e\phi (\vec{x},t). \end{equation} I proceeded to use the ...
3
votes
1answer
304 views

Lapse and shift in ADM decomposition

Poisson in Relativist's Toolkit and also other authors in various papers state explicitly that after one does the 3+1 decomposition, the lapse and shift $N$ and $N^a$ are non-dynamical variables, and ...
3
votes
2answers
375 views

Hamiltonian from a Lagrangian with constraints?

Let's say I have the Lagrangian: $$L=T-V.$$ Along with the constraint that $$f\equiv f(\vec q,t)=0.$$ We can then write: $$L'=T-V+\lambda f. $$ What is my Hamiltonian now? Is it $$H'=\dot q_i p_i -L'~?...
1
vote
0answers
79 views

Does the additivity property of Integrals of motion and Lagrangians valid in all situations?

I would like to know if the additivity property of an integral (constant) of motion valid in all situations ? It works for energy but does it work for all other integrals of motion in all kinds of ...
0
votes
2answers
135 views

Interaction Hamiltonian in the interaction picutre

The Schrodinger and Heisenberg pictures make sense to me. But the interaction picture which is a hybrid of the two does not. Author of this text first splits the Hamiltonian up as $$H=H_0+H_{int}$$...
1
vote
1answer
172 views

Hamilton-Jacobi theory and initial value problem?

Having read through some recent posts regarding the Lagrangian formulation being interpreted into an initial value problem rather than the familiar boundary condition problem we are familiar with, I ...
6
votes
2answers
399 views

Does the $\frac12mv^2$ law apply to quantum mechanics?

Consider the classical Hamiltonian for a spring: \begin{equation} H = \frac{1}{2}\frac{p^2}{m} + \frac{1}{2}kx^2 \end{equation} This is one of those simple cases where when you work out the math we ...
0
votes
3answers
206 views

Energy of system in eigenstate of Hamiltonian

I know how to find the spectrum of the Hamiltonian to get the allowed energies for a system. If the spectrum is quantized, I can get definite values for each energy level. But when the system is in ...
2
votes
2answers
270 views

Hamiltonian mechanics really useful for numerical integration? Lagrangian can become 1st-order

(I'm talking about the classical mechanics.) Many texts say that Euler-Lagrange equations are difficult to treat numerically because they are second-order ODEs, ${f_i(\boldsymbol{q, \dot{q}, \ddot{q}}...
4
votes
1answer
147 views

Quantization of a free field: Klein-Gordon case

I am a beginner and reading this course text on QFT. The author first introduces the KG equation: $$\partial_\mu\partial^{\mu}\phi+m^2\phi=0$$ [with Minkowski signature $(+,-,-,-)$]. Then the ...
1
vote
4answers
380 views

Help understanding what the Hamiltonian signifies for the action compared with the Euler-Lagrange equations for the Lagrangian?

Consider the Lagrangian for a simple harmonic oscillator \begin{equation} L (x,\dot{x}) = \frac{1}{2}m\dot{x}^2 - \frac{1}{2}kx^2 \end{equation} Obviously we have \begin{align} \frac{\partial L}{\...
0
votes
0answers
51 views

What justification is necessary for convolutional variational principles to be considered legitimate?

I recently asked a related question and was interested in why/or why we cannot use convolutional variational principles in practice or in theory. Summarizing the points I made in the earlier post: ...
0
votes
2answers
134 views

According to Liouville's theorem, why is the measure on an energy-surface different from the measure on the phase space in general

I recently read Khinchin's derivation of Liouville's theorem. I was able to follow the math for the most part, however I was hoping for an intuitive understanding about why the form of the measure on ...
1
vote
2answers
172 views

Vanishing of conjugate momentum $\Pi^0$ and non-existence of propagator

We know that if we try to quantize the free electromagnetic field without a gauge fixing term added to the Lagrangian, then one of the conjugate momentum density $\Pi^0$ vanishes. We also find that ...
0
votes
0answers
52 views

Why are functional representations of systems important in physics or computational physics?

This was an addendum to a previous question I asked, but I figured I should make it it's own discussion. Assuming I am able derive a functional representation for any dynamical system (dissipative, ...
4
votes
1answer
545 views

How to formulate variational principles (Lagrangian/Hamiltonian) for nonlinear, dissipative or initial value problems?

Although this questions is very much math related, I posted it in Physics since it is related to variational (Lagrangian/Hamiltonian) principles for dynamical systems. If I should migrate this ...
0
votes
1answer
172 views

Simple explanation of first and second class constraints with an example

Can someone give a simple physical example of first class and second class constraints? I mean, if you were giving a classical mechanics lecture for undergraduates, how would you explain this concept ...
2
votes
2answers
118 views

Symplectic structure and isomorphisms

In his book Mathematical Methods of Classical Mechanics, V.I. Arnold writes To each vector $\xi$, tangent to a symplectic manifold $(M^{2n},\omega^2)$ at the point $\mathbf{x}$, we associate a 1-...
6
votes
2answers
249 views

Can the momentum eigenstates be non-orthogonal?

Consider the Hilbert space of a particle, whose position domain is confined to $q\in[0,1]$ (e.g. a particle in a box with unit width). Using $$ 1=\int_0 ^1 dq |q\rangle\langle q| $$ and the position ...
0
votes
1answer
152 views

Infinitesimal transformations and Poisson brackets [duplicate]

I want to understand how bracket operations in general are related to symmetry and infinitesimal transformations (in hindsight of quantumfieldtheory), so I calculated an example with a particle that ...
5
votes
0answers
70 views

Hamiltonian System Outside Physics [closed]

What are good examples of Hamiltonian systems outside physics? I heard there are financial systems that can be described by a Lagrangian, and was interested to see some examples
2
votes
1answer
192 views

Given a QFT Hamiltonian, is there a unique Lagrangian?

Consider a QFT in one spatial dimension specified by the following Hamiltonian density: $\mathcal{H} = -i \phi^\dagger \frac{\partial}{\partial x} \phi + V(\phi^\dagger,\phi)$ where $\phi$ is a ...
1
vote
1answer
374 views

What is “momentum density” and why it important to QFT?

I am reading Quantum Field Theory for the Gifted Amateur. On page 98, they provide a summary of a basic canonical quantization procedure: Step I: Write down a classical Lagrangian density in ...
0
votes
1answer
50 views

Transformation of $q_k$ and $p_k$ from invariance of Hamiltonian

This is a step in Nakahara's Geometry, Topology and Physics, 2nd edition, 2003, on pages 7-8: Given that $q_k ' = q_k +\epsilon f_k(q)$, we have that $$\Lambda_{ij} = \frac{\partial q_i'}{\partial ...
2
votes
1answer
86 views

Separability of Hamilton Jacobi Equation

When we talk about integrability of classical systems in terms of Hamiltonian or Lagrangian mechanics, it's all to do with counting independent conserved quantities. Then when we move to the Hamilton-...
2
votes
1answer
81 views

Is Liouville's theorem valid for dimensionally restricted systems?

Liouville's theorem states that the phase space volume of a system is conserved over time. Intuitively, this seems to imply that if a system is at some time constrained to, say, a curve in phase space,...
3
votes
2answers
87 views

Why does choosing a time break covariance?

I'm reading that in EM theory, in hamiltonian formalism, we choose a specific reference frame with a specific time, and that this breaks covariance. Why? Surely it's simple because it's just stated ...
1
vote
0answers
176 views

Legendre Transformation for multiple variables

I need to show that for $F(x_1, .., x_n)$, the Legendre transformation is, $$G(s_1, ..., s_n) = \sum_{i}^{N} x_i s_i - F$$ where $$s_i = \frac{\partial F}{\partial x_i}$$ and has the property that $$...
1
vote
1answer
310 views

Lapse function definition

Let $t$ be a time function and $t^a$ the time flow vector such that $t^a\nabla_a t=0$. Let $\Sigma_t$ be a hypersurface of constant $t$ with unit normal $n^a$, $n^a n_a=-1$. Wald (1984), p. 255 ...
3
votes
2answers
272 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 ...
1
vote
1answer
109 views

Conservation of probability in phase space flow

In J.Binney's notes on classical mechanics, under the section 'Liouville's theorem', he states that (paraphrasing): the conservation of probability requires that $\frac{df}{dt} = 0.$ where $f$ ...
2
votes
1answer
76 views

conservation of volume in phase space

I was reading through a proof of Liouville's theorem on conservation of volume in phase space from David Tong's lecture notes (Chapter 4: "Hamiltonian formalism") and on page 89 it says that $\det(J)=...
2
votes
1answer
196 views

Bopp operators and Wigner-Weyl representation

I am learning about the Wigner-Weyl transformations to move a $c$-number Lindblad operator $A(x,p)$ back into operator form. As far as I know, to move back and forth normally requires a four variable ...