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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Doubt on the geometry of "quantum phase space"

In Jose & Saletan's "Classical Dynamics", they show the global structure of Hamiltonian mechanics: you then have a $Q$ manifold (configuration space), and the phase space structure is ...
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On-shell Poisson brackets and time derivative

In classical statistical mechanics, the information about a given system is given by a distribution of probability over phase space $\rho(p,q,t)$. Let $H(p,q, t)$ be the hamiltonian of the system and $...
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Why is Hamilton's equations sometimes written with a gradient? [closed]

I am used to seeing Hamilton's written as: $$\frac{dq_j}{dt} = \frac{\partial H}{\partial p_j}\\ \frac{dp_j}{dt} = - \frac{\partial H}{\partial q_j}.$$ However I have also seen it written as $$\frac{...
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The different generators of canonical transformations

Consider the phase space of a one degree of freedom mechanical system. We can pass from one phase space coordinates to another phase space coordinates via a canonical transformation. I want to focus ...
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Integrability of one-dimensional system of motion?

How can I prove that every one-dimensional system is integrable (meaning that there is a constant of motion)? It is clear that if $H$ does not depend explicitly on time then $H$ is indeed a constant ...
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Integration by Parts in Liouville's Theorem

I am looking at a proof of Liouville's Theorem, which states that for $F, G \in C_0^\infty$ and a Hamiltonian $H$, the operator $$D_H = \sum_{i=1}^n\Big(\frac{\partial H}{\partial p_i} \frac{\partial}{...
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Hamiltonian density of Abelian-Higgs Lagrangian

Given the Lagrangian density $$\mathcal{L}=-(\nabla^{\mu}\phi)^\dagger(\nabla_\mu \phi)-\frac{\lambda}{4}\left(\phi^{\dagger}\phi-v^2\right)^2-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\,,\quad \nabla_\mu\phi=\...
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In Hamilton-Jacobi theory, how is the new coordinate $Q$ time-independent when Hamilton's principal function separates?

Following the notation in Goldstein, the solution to the Hamilton-Jacobi equation is the generating function $S$ for a canonical transformation from old variables $(q,p)$ to new variables $(Q,P)$ ...
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How to define and find separatrix for a Hamiltonian system with more than one degree of freedom?

For a Hamiltonian system with $n>1$ pairs of positions and momenta, how do we define a separatrix? Once defined, is there a theorem (involving action-angle variables (AAVs)) that we can use to find ...
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How can I show that applying Hamiltonian dynamics recovers the original wave equation?

Problem Consider the wave equation: $$ \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}\tag{1}$$ with $ u = u(t, x)$ over domain $x \in [0, l] = \Omega$. This can be ...
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About characteristics of smearing function

I met the word smearing function (or test function) when I was learning ADM formalism in GR books. What makes me scratch my head is the reason of introducing such a smearing function when we calculate ...
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Angular momentum and energy conservation for a given Hamiltonian

Assume that the evolution of a system is defined by a Hamiltonian $H$ given by $$ H= a \,\mathbf{p} \cdot \mathbf{p} + b \,\mathbf{p} \cdot \mathbf{q} + c \,\mathbf{q} \cdot \mathbf{q}. $$ Here $a,...
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Finding a new hamiltonian from a given canonical transformation

Let us suppose we have a given Hamiltonian $$H = \frac{P_1^2}{4}+\frac{P_2^2}{2}+V(Q_2)$$ and $q_1 = Q_1 + Q_2/2$ with $q_2 = Q_1 - Q_2/2$. I need to find the new hamiltonian by using these canonical ...
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Liouville's theorem on the tangent bundle [duplicate]

One interpretation of Liouville's theorem is the determinism and reversibility of classical mechanics, i.e. the mechanical states can't converge or diverge. The theorem is often formulated on the ...
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Hamiltonian and Lagrangian in Condensed Matter

Can metal can be described as sum of many particle Lagrangian rather than Hamiltonian?
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Symmetry of the second partial derivatives of the Hamiltonian

A dynamical system with generalised particle position $q$ and generalised momentum $p$, described by: $$\dot{q}=F_1(q,p)\quad\text{and}\quad\dot{p}=F_2(q,p)\tag{1}$$ is a Hamiltonian system if: $$\...
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Derivation of Hamiltonian $H=T+V$ from Lagrangian $L=T-V$

I understand that the Hamiltonian is the Legendre transform of the Lagrangian: $$ \begin{split}H(q,p,t) &= \frac{\partial L}{\partial \dot{q}}\dot{q} - L(q,\dot{q},t) \\ \implies H&=p\dot{q} -...
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Would it be more insightful to teach students Lagrangian/Hamiltonian mechanics before Newtonian mechanics? [closed]

What benefits would it bring to teach analytical mechanics before Newtonian (vector) mechanics? I began thinking in this after I saw a 2019 article in the magazine Physics Today that advocates ...
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Confusion about the action variable definition

Suppose we have an integrable system consisting of a $2n$-dimensional phase space $M$ together with $n$ independent functions $f_{1\leq j \leq n }$ in involution. Suppose the level set $$M_f = \{ (p,q)...
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Why is the anticommutation relation for the Dirac field between fields? [duplicate]

The commutation relation for neutral Klein Gordan field is $$[\phi(x,t),\pi(x',t)]=i\delta^3(x-x')$$ with all other commutators zero; The commutation relation for charged Klein Gordan field is $$[\phi(...
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What is the correct general form of Hamilton's equation?

Usually, Hamilton's equations of motion are given by: $$ (1)\;\; \frac{dp}{dt} = -\frac{\partial H}{\partial q} \;\;\; \text{ and } \;\;\;(2)\;\; \frac{dq}{dt} =\frac{\partial H}{\partial p}.$$ ...
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Time-dependent canonical transform step in Hamiltonian perturbation theory (Percival problem 8.20)

In Percival and Richards's great book, "Introduction to Dynamics", problem 8.20 asks the following question. Any insight on how to solve this would be appreciated: Consider a system with ...
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Is this hamiltonian of the form of some well-known physical system?

I'm doing a homework exercise and I'm asked whether some hamiltonian (that is the result of a canonical transformation of some other hamiltonian) is reminiscent of the hamiltonian of some well-known ...
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Can a primary constraint contain spatial derivative of the field?

I am currently studying the Hamiltonian formulation of GR and I have problems understanding this definition of primary constraint. In the textbooks, primary constraint occurs when a momentum conjugate ...
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Problem with derivation of the Dirac Hamiltonian

I'm having son trouble when obtaining the Dirac equation. I am working in (1+1)-dimensional Minkowski spacetime with signature $(-, +)$ in coordinates $(t, x)\equiv(1, 2)$. I can think of two ways to ...
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In canonical transformation, is there any rules or methods for finding the transformation $(q,p)\to(Q,P)$?

If we get two different Hamiltonian by using two methods of canonical formulation of theory and these two Hamiltonian are equivalent. How can I find the canonical transformation from which we can ...
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Interpretations of Lagrangian vs. Hamiltonian mechanics

This might seem like a duplicate question; however, rest assured, it is not. My question is pointed and particular: Some background: Given a system we describe its Lagrangian $L$ as $T-V$, where each $...
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How to find canonical transformation to achieve desired Hamiltonian?

I am trying to find a way to transformation that will turn a Hamiltonian from one form into another form: $$(1)\;\;\;H=p^2+e^x\rightarrow\bar{H}=p'^2.$$ I don't know of any systematic ways to do this. ...
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How to calculate commutator between fermionic terms and bosonic conjugate momenta?

Consider a system which has a three bosonic scalar fields and non-relativistic fermionic part. Let the field operator for the bosonic part be $\phi_k(x)$, $k=1,2,3$, with conjugate momentum $\pi_k(x)$....
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Can conservation of phase space volume be viewed as a consequence of some symmetry via Noether's theorem? [duplicate]

Liouville's theorem says that for the Hamiltonian evolution of a system, the flow of points on the phase space with time is like that of an incompressible fluid i.e. the phase space density is ...
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Is the Hamiltonian of interacting systems integrable if the interaction is linear?

Suppose we allow two integrable systems with Hamiltonians $H_1$ and $H_2$ to interact. Then their combined dynamics can be described by a joint Hamiltonian, $$H = H_1(\mathbf{q}_1,\mathbf{p}_1) + H_2(\...
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Why Hamiltonian for a Solid is periodic in Bloch Theorem?

It is as simple as the title and more general to Bloch theorem treatment. In any periodic infinite solid lattice, we say that the potential will be periodic. This makes sense, but how do we know that ...
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Existence of a unitary transform $(q,p) \rightarrow (-q, p)$

If $q$ and $p$ are the canonical position and momentum operators of a quantum harmonic oscillator, is there a unitary that transforms $(q,p)$ into $(-q, p)$? For instance, denoting the annihilation ...
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Functional derivative acts on covariant derivative

I'm confusing about how functional derivatives act on a covariant derivative. I'm doing such a calculation: In ADM formalism, let $h_{ij}(x)$ be the spatial metric while $\pi^{ij}(x)$ is its momentum ...
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Wave equation for lightcone coordinate $X^-$

A quick question from Polchinski volume.1 : He claims in p.20 that the worldsheet lightcone coordinates $X^\pm$ also (i.e. in addition to the transverse coordinates $X^i$) satisfy the wave-equation. ...
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Why is this the requirement for invertibility within the context of canonical transformations in mechanics?

I'm reading "Analytical Mechanics", by Hand and Finch. In page 210, there's the following statement, regarding the generating function $F$ for some lagrangian such that $L'=\lambda L-\frac{\...
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Canonical transformation of 2-body hamiltonian into center of mass and translation components. How to gain an expressions for the conjugate momenta?

I am trying to transfom a quantum hamiltonian as detailed in this website. It starts of by using the classical hamiltonian and conjugate momenta: $$ H\equiv \frac{p_1^2}{2m_1}+\frac{p_2^2}{2m_2}+V\...
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Two Liouville's theorem

Within the context of Hamiltonian mechanics and phase spaces I have learnt that the phase space distribution function is constant, in other words, that the "volume" of any region is constant,...
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General Proof that Noether Charge Generates a Symmetry [duplicate]

I am trying to understand how one proves that a charge derived from the Noether procedure generates the corresponding symmetry. That is, I would like to prove that $$[Q, \phi(y)] = i\delta\phi(y)$$ I ...
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2 answers
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Any reason why the Lagrangian just happens to show up in the path integral?

I saw that derivation in which they start with the Schrodinger equation propagator and introduce a resolution of identity between each term. And boom ! The lagrangian showed up in the phase. But.......
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When is the motion of the perturbative term resonant with the Hamiltonian?

I am given the following Hamiltonian, $H$, which is a perturbed version of $H_0$, $$ H(\theta,I) = H_0(I) -\epsilon \cos(\theta-\Omega t)$$ where $H_0 = \frac{I^2}{2}$, $\epsilon << 1$ and $(I,\...
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Conjugate momentum vs translation generator with non-standard kinetic term

I am reading this paper and for equation (2.5) (associated with the Lagrangian in eq 2.1) there is the claim that for a Lagrangian $L(\varphi,A,\dot{\varphi},\dot{A})$ containing an extra non-standard ...
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Ability to represent canonical transformation generating functions in different forms?

In Goldstein's Classical Mechanics (3rd edition) section 9.1, we introduce the generating function method of describing a canonical transformation. We then introduce four types of generating functions:...
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Does the Legendre transformation describe two views on the same physical system or different physical systems?

In mechanics we perform the Legendre transform to go from the Lagrangian $L(q, \dot{q})$ to the Hamiltonian $H(q, p)$. This seems to be describing the same physical system. $L$ and $H$ both describe ...
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How are conjugate variables in mechanics and stat mech related to duality in convex optimization?

I recently studied duality in optimization where a primal optimization problem can be casted as a dual problem which provides meaningful lower bounds on the primal. There is also a notion of conjugate ...
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About the curvature of solutions of Hamilton's equations

I am a math major and have recently stumbled on the Hamilton's system of equations in the context of Hamiltonian Monte Carlo Markov chains on a continuous state space, say $\mathbb{R}^d$. I am trying ...
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Hamiltonian from Lagrangian defined as an integral

I need to derive the Hamiltonian from the Lagrangian defined in the following way: $$L[x, \dot{x}] = \int_{t_0}^{t_1} f(x(t), t) \sqrt{1 + \dot{x}^2} \mathrm{d}t.\tag{1}$$ The usual method is to ...
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Why inverse flow of separable Hamiltonian with even kinetic energy can be written like this?

Why is it true that the inverse of the flow of a separable Hamiltonian with even kinetic energy can be written as $\phi_N \circ \varphi_t \circ \phi_N$ where $\varphi_t$ is the flow of the Hamiltonian ...
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Does compactification of a Nambu-Goto string in one direction break Diff invariance?

Assume we have a Nambu-Goto action, in phase space, for a closed string. If I compactify one coordinate of the target space, do I reduce the diff invariance of the system. We have $$S=\int d^2\sigma \...
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Definition of time-reversibility of flow of Hamilton's equations

I cannot find a good, simple definition of time-reversibility of the flow $\phi_t$ of Hamilton's equations $$ \dot{z} = J^{-1}\nabla_z H(z) \quad \text{where} \quad z = (q, p)^\top \quad \text{and} \...
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