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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Reference request: Scattering from action

Consider a separable solution to a Hamilton-Jacobi equation of an $n$-dimensional autonomous system of the form $$W(\alpha_1,...,\alpha_n,x^1,...,x^n) = \sum_j \int_0^{x^j} w^j(\alpha_1,...,\alpha_n,x'...
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Is there a way to construct a Hamiltonian from a set of DE?

Let's say I have a set of first-order differential equations for set of position $x_i$ and conjugate momenta $p_i$, which might be complicated and time-dependent $$ \dot{x}_i = f_i(x_j,p_j,t)$$ $$ \...
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Path integrals and classical paths obtained via saddle point integration

EDIT: Focussed my question more based on @octonion's comments Say one is interested in the following action, $$S[\{\phi_i(x,t), \tilde{\phi_i(x,t)}\}] = \int \mathrm{d}t\mathrm{d}^d x\left(\tilde{\...
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How does one “invert” such infinite-dimensional sympletic form?

In the book "Lectures on the IR structure of gravity and gauge theories" by Strominger the author considers the sympletic form for free electrodynamics: $$\Omega_\Sigma[A;\delta_1 A,\delta_2A]=-\frac{...
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The power of exterior differential calculus in CM

Classical Mechanics has a beautiful interpretation in terms of symplectic geometry and exterior differential calculus. For example, the Hamilton equations of motion can be written in a coordinate free ...
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Energy of a Hamiltonian that is the sum of commuting Hamiltonians

Suppose that $$H=\sum _i H_i\quad\textrm{with}\quad [H_i,H_j]=0 \quad\textrm{for} \quad j\neq i$$ Let $\psi_i$ be a solution to $H_i$ so that $$H_i\psi_i=E_i\psi_i$$Does this always imply that the ...
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Elastic Pendulum on accelerated base (Hamiltonian Approach)

Lets consider an elastic pendulum with $k$ spring constant and $\ell$ its natural length. The base of the pendulum is moving with constant acceleration $a$. a) Get the hamiltonian of the system and ...
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On the derivative of the Hamiltonian of a particle in quantum mechanics

The Hamiltonian function for a particle of mass $m$ moving along the $x$-axis, subject to a potential energy $V(x)$, the Hamiltonian function is: $$H = \frac{p_x ^2}{2m} + V(x).$$ My book states ...
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On- & off-shell conserved charges/constants of motion

I am trying to understand how conserved charges generate symmetry transformations via the Poisson bracket, but I am missing something in one part of the derivation. The part I am struggling with is ...
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Principled approach to arrive at geodesic Hamiltonian $H = g^{\mu \nu} p_\mu p_\nu$?

The background: If we have a spacetime path $x^\mu(t)$ parameterized by arbitrary parameter $t$, the proper time along the path between $t_1$ and $t_2$ is $$ \int_{t_1}^{t_2} (g_{\mu \nu} \dot x^\mu \...
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Drawing the orbits in the phase space of a particle with the following properties

Considering a 1D space, I'm given the graph where $U(\pmb{q})$ is the potential energy as a function of position $\pmb{q}$, and each $E_0, E_1,E_2$ is a different amount of total energy in the system....
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Transformation between a dynamical system to a Hamiltonian system [duplicate]

Consider a dynamical system characterized by these equations $$\dot{x}=x-xy \\ \dot{y}=-y+xy$$ If we transform $\ln(y)=q$ and $\ln(x)=p$, the system can be changed into a Hamiltonian system with $q$ ...
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When can you simplify partition function and Hamiltonian for a classical fluid?

I am new to statistical mechanics and classical mechanics. A set of $N$ classical particles has a Hamiltonian \begin{equation} \mathcal{H} = K_N(q^{3N},p^{3N}) + U_N(q^{3N},p^{3N}) \end{equation} ...
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Classical Hamiltonian of a free falling particle

I am stuck with the basic question on the classical Hamiltonian for free fall particle (let's say from the infinity). The Hamiltonian can be represented as the total energy of the system, which is the ...
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Anticommutation relations for fermionic fields imply that Hamiltonian / Lagrangian can at most be linear?

Fermionic field operators do obey anticommutation relations, so for a chosen Field operator (and the field momentum), we have: $$ \{\Psi_a, \Psi_b\} = \{\pi_a, \pi_b\}= 0 $$ with the $\Psi_a$ being ...
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Hamilton-Jacobi Equation: Why does any $F(q,Q,0)=f(q,Q)$ lead to a solution?

I) Given the Hamilton-Jacobi equation,$$\frac{\partial F(q,Q,t)}{\partial t}+H\left(q,\frac{\partial F(q,Q,t)}{\partial q},t\right)=0 $$ it is stated that any function $$F(q,Q,t=0)=f(q,Q)\tag{22}$$ ...
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Hamiltonian mechanics on irregular graphs - application to the physics of intelligence?

Are there physical models that describe mechanical/Hamiltonian dynamics of some particle/subject on irregular graphs? Apparently, the dynamics on the more or less regular lattices can be described ...
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Classical action is zero in Klein-Gordon theory for a particle wavepacket

I'm interested in rewriting actions in the form $$ S = -\int H dt + \int p_i dx^i, $$ (where $H$ is the Hamiltonian and the $p_i$ are conjugate momenta) and then evaluating them along a classical ...
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Why we are not using Lagrangian instead of Hamiltonian in non-relativistic quantum mechanics? [duplicate]

Why we are not using Lagrangian instead of Hamiltonian in non-relativistic quantum mechanics?
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Can the ADM mass be defined locally inside a closed surface?

To calculate the ADM mass, one has to specify a closed surface inside which all the mass is located. Are there any ways to calculate the mass inside a closed surface outside of which other mass still ...
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Uniqueness of Lagrangian and Hamiltonian [duplicate]

Is a Lagrangian unique in the same field? Is Hamiltonian unique? If it is unique then please explain why is it so and if it is not then please explain why is it not so.
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Choice of conjugate momentum for Ostrogradsky instability

I was reading this post and I don't understand why chosing: $Q_1=q\ $ and $\ Q_2=\dot{q}$ implies that $$P_1=\dfrac{\partial L}{\partial \dot{q}}-\dfrac{\mathrm{d}}{\mathrm{d}t}\dfrac{\partial L}{\...
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Differentiate the Lagrangian wrt. momentum?

Given $$ L=L(t, x_i, \dot x_i) $$ as a function of generalized coordinates/velocity, and $$ p_i:=\frac{\partial L}{\partial \dot x_i}, $$ how can we calculate $$\frac{\partial L}{\partial p_i}?$$
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Verifying completeness of constants of motion

I can find constants of motion by looking at the null space of the Poisson Bracket operator $ \{H, \cdot\} $ over a polynomial space by brute force with symbolic algebra (code). This scales terribly ...
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Reparametrization invariance in FRW

It is usually pointed out that FRW metric is invariant under time reparametrization. Consider the flat case for simplicity $$ds^2=N(t)^2dt^2-a(t)^2dr^2-a(t)^2r^2d\Omega^2$$ The choice of function $N(...
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Geodesics and first fundamental form

If we consider Hamiltonian $H(p,q)=(1/2)p^TAp$. where $p=(p_1,p_2)$ and $A$ is a symmetric matrix $A_{ii}=e^{q_i^2}+1$ and $A_{12}=A_{21}=1$. Then how I proceed to find the first Fundamental form of a ...
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How did Hamilton come up with his idea of summing up the kinetic energy and potential energy?

When we browse through any classical physics textbook, the Lagrangian comes first where the potential energy is subtracted from Kinetic energy. Was Hamilton playing around with energy functions and he ...
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Time Dependence on Landau & Lifshitz's Proof of Poisson's Bracket Canonical Invariance

I'm reading Landau & Lifshitz's Mechanics and, at a certain point when discussing canonical transformations, they prove that Poisson brackets are canonical invariants. The proof starts with ...
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von Neumann on the Hamiltonian

I'm reading von Neumann's book on QM and I'm slightly confused by a simple point. He writes, "The energy is a given function of the coordinates and their time derivatives: $E = L(q_1,..,q_k;\dot{q}...
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Some clarifications about ADM Hamiltonian constraints

I have some trouble with refreshing ADM split and Hamilton formalism of GR in context of introducing Wheeler-de-Witt equation. Having Lagrangian in form: $$\mathcal{L}_{ADM}=\sqrt{h}N(G^{abcd}K_{ab}...
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Undefined Hamiltonian for this particular Lagrangian [duplicate]

So, this a question from a Phd qualifying examination. Given the following Lagrangan $$L=\frac{1}{2}\dot{q}\text{sin}^2q,$$ what is the Hamiltonian for this system? So, finding the canonical momentum $...
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Adiabatic Invariant of the simple pendulum

I don't understand something about the adiabatic invariant in Tomonaga book that I discuss below. (Page 22&23)in calculating the increase of energy(5.9) we only consider the oscillation energy ...
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How to check if a generating function produces an identity transformation without substituting the CT equations in the Hamiltonian?

In chapter 9, Goldstein ($3^{rd}$ ed.) includes a discussion and a few "trivial special cases" of Canonical Transformation which keeps the form of the Hamiltonian unchanged and named it Identity ...
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Using direct conditions to check if a gauge transformation is canonical

Gauge transformations are known to be canonical transformations, see here and here. Using direct conditions to check if they are, however, imposes a constraint on the gauge parameter $\Lambda(t,\...
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Why are so many (all?) Key concepts in thermodynamics derived from a legendre transformation? [duplicate]

Given a function $f(x,y)$, a legendre transform w.r.t. $x$ is $f^*(p,y)=p x - f(x,y) |_{p=\frac {\partial f(x,y)}{\partial x}}$. E.g. , the various free energies, enthalpy, etc are all legendre ...
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Are first Integrals the same in Lagrangian and Hamiltonian formalism?

A first integral in Lagrangian formalism is defined as a function which is constant along the solutions $(q,\dot{q})$ where $q$ are the generalized coordinates; while a first integral in Hamiltonian ...
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What exactly is the Legendre transformation? [duplicate]

Goldstein et al's Classical Mechanics states that: The Hamiltonian $H(q,p,t)$ is generated by the Legendre transformation $$ H(q,p,t) = \dot{q}_i p_i - L(q,\dot{q},t). \tag{8.15} $$ But I don't ...
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Question about using Liouville's theorem to calculate time evolution of ensemble average

With the Liouville's theorem $$\frac{{d\rho }}{{dt}} = \frac{{\partial \rho }}{{\partial t}} + \sum\limits_{a = 1}^{3N} {(\frac{{\partial \rho }}{{\partial {p_a}}}\frac{{d{p_a}}}{{dt}} + \frac{{\...
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Canonical Transformations that are Complex

I'm self studying through a book that has the following question. The book gives the answer, but I'm trying to understand why: Under what condition is the following transformation NOT canonical? $$Q =...
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A proof of Liouville’s theorem

I have found a proof of Liouville's theorem on the internet, which fits me very well except one step I don't understand, the derivation is as follows: In the derivative, it must have used the ...
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The complex form of Hamilton canonical equations

I found an excerpt on page 171 of "The variational principles of mechanics" written by Cornelius Lanczos stated that If, however, the conjugate variables $q_k$, $p_k$ are replaced by the complex ...
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Coupled oscillators in Hamiltonian formalism - problem with diagonalization

I have a problem with simple coupled oscillator system. I tried to solve single oscillator with Hamiltonian, and then coupled system of two, but when I try to put coupling constant $k^\prime=0$ in my ...
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Hamilton's equation of motion with other momentum

I wrote here a problem couple days ago. I figured out what was the problem there, but now it made another problem. Sorry for similiar question. I'm trying to draw phase portrait for my ODE and for my ...
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Time-reversibility symmetry in classical mechanics

Newton's laws are invariant under time reversal transformation $$ t \longrightarrow -t $$ for time-independent potentials. But Hamilton-Jacobi equation is too an equivalent description of classical ...
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How does a Lagrangian with delta potential transform to a Hamiltonian?

Suppose the Lagrangian was given as: $$L = \frac{1}{2}\int_{-\infty}^{\infty} \underbrace{\left(\dot{A(}z)^2-A(z)^2\right)+\left(\dot{Q(}z)^2\delta(z)-Q(z)^2\delta(z)\right)+2\dot{Q(}z)\cdot A(z) \...
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What is the Lagrangian for FRW universe?

A straightforward calculation to the Lagrangian for FRW as $\mathscr{L}=\sqrt{-g}R$ gets me the following result: $$\mathscr{L}\sim a^3\big(\frac{\ddot{a}}{a}+(\frac{\dot{a}}{a})^2+\frac{k}{a^2} \big)...
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Legendre transform and coordinate system independence

I'm self-learning analytical mechanics. Consider a classical mechanical system. Even if it's clear to me that via (the usual) Legendre transform we can get a unique Hamiltonian function from a ...
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How is it possible to have four types of generating functions?

Since the Hamilton's equations of motion remain unchanged in form under a canonical transformation $(q,p)\to (Q,P)$, the Lagrangians must differ by a total time derivative of a function of $q,t$. In ...
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Why does the Hamiltonian of a Lagrangian that consists of only a coupled term become zero?

Let's say our Lagrangian looks something like this: $$L = \int dz\, Q\cdot \dot{A},\tag{1}$$ where $Q$ and $A$ are two generalized coordinates and $\dot{Q}$ and $\dot{A}$ would be the respective ...
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Justification of dropping term in Hamiltonian and expectation Values

While reading Sakurai's Modern QM, I was stuck at the point where he explains the absorption and emission of light quanta in atoms. He proceeds with Hamiltonian: $$H= p^2/2m + e\phi(x) -e/mc A\cdot p$...