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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Deriving Hamilton's equations from KdV Hamiltonian

Let $f=f(q,p)$, $g=g(q,p)$ and Possion bracket $$\{f,g\}=\frac{\partial f}{\partial q}\frac{\partial g}{\partial p}-\frac{\partial f}{\partial p}\frac{\partial g}{\partial q}. \tag{1}$$ Then Hamilton'...
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Meaning of centrifugal term in the mechanical energy of a orbiting planet [duplicate]

For a planet under the effect of gravitational force the mechanical energy can be written as $$E=\frac{1}{2}\mu {\dot{r}}^2+\frac{L^2}{2\mu r^2}-\gamma \frac{m M}{r^2} \tag{1}$$ Where $\mu$ is the ...
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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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Non-null hessian condition for regular dynamical systems

I'm "researching" on unquantised Yang-Mills theory. For that I'm studying the Dirac's method for singular constrained systems and having problems to follow the first considerations on that matter. I ...
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52 views

Possible duality between Harmonic oscillator and free particle?

There is some connection between classical non-interacting harmonic oscillator (OH) and the free particle in higher dimensions? I was studying statistical mechanics and I came across the idea that ...
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40 views

Show that Newtonian orbits are closed and periodic

I want to prove to show that the change of the rotation angle of a body in a two-body-problem is exactly $\Delta \phi = 2\pi$. I know that the whole energy of the system is given by $$ E = \left(\...
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Deriving kinetic energy in cylindrical coordinate constraints

Consider a mass $m$ which is constrained to move on the frictionless surface of a vertical cone $\rho = cz$ (in cyclindrical polar coordinates $\rho, \theta, z$ with $z>0$) in a uniform ...
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Who developed the phase space path integral?

The original path integral introducted by Feynman is $$ \lim_{N\to +\infty} \int \left\{\prod_{n=1}^{N-1} \frac{\mathrm{d}q_n}{\sqrt{2 \pi i \hslash \varepsilon}} \right\} \exp\left[{\frac{i}{\hslash}...
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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: $$S=-m\...
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Hamiltonian and Energy of a charged particle in an Electromagnetic field

The Lagrangian of a charged particle of charge $e$ moving in an electromagnetic field is given by $$L=\frac{1}{2}m\dot{\textbf{r}}^2-e\phi-e\textbf{A}\cdot \textbf{v}$$ where $\phi(\textbf{r},t)$ is ...
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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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Ostrogradski’s theorem's proof

I am looking for a proof of the next theorem: "If the higher order time derivative Lagrangian is non-degenerate, there is at least one linear instability in the Hamiltonian of this system." Where ...
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Hamilton's equations from the action with boundary conditions involving position and momentum

Generally, when you are given the action $$ S=\int_{t_1}^{t_2}\mathrm dt (p\dot q - \mathcal H )$$ the boundary conditions are $q(t_1)=q_1$ and $q(t_2)=q_2$. This is useful because to calculate $\...
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In what cases do we use the Routh Function? [duplicate]

As many of you, I studied Lagrangian Mechanics and Hamiltonian Mechanics, with the so famous functions called Lagrangian $\mathcal{L}$ and Hamiltonian $\mathcal{H}$ related by: $$\mathcal{H}(q_i, p_i,...
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Non-holonomic constraints in Dirac-Bergmann theory

The Dirac-Bergmann algorithm effectively isolates the physical degrees of freedom of a system, by changing from Poisson brackets $\{\cdot,\cdot\}_\mathrm{PB}$ to Dirac brackets $\{\cdot,\cdot\}_\...
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90 views

What does Liouville's Theorem actually mean?

Basically, the mathematical statement of Liouville's theorem is: $$\frac{\partial \rho }{\partial t}= -\sum_{i}\left(\frac{\partial \rho}{\partial q_i}\,\dot{q_i}+\frac{\partial\rho}{\partial p_i}\,\...
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Usage of total time derivative within first Euler-Lagrange equation

As one key argument in Introductoryt QM Class, we've been taught to use a Lagrangian and Hamiltonian generalized description of a dynamic systems, which follows the Euler-Lagrange or Hamilton equation ...
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Does the conservation of the Wronskian follow from Noether's principle?

Noether's principle is the paradigm that symmetries of Hamiltonian and Lagrangian systems correspond to conservation laws of various kinds. Consider a one-dimensional harmonic oscillator $$\tag{*} \...
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Density of states of classical harmonic oscillator in phase space

Since all classical harmonic oscillators are ellipses in phase (position-momentum) space, and since the entire phase trajectory of a given system (with a fixed rigidity and mass factor) can be ...
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39 views

Classical Field Theory: Physical meaning of various terms in total Hamiltonian

In a classical field theory problem Lagrangian density is given as $${\cal L}=\frac{1}{2}\dot{\phi }^{2}-\frac{1}{2}\left ( \bigtriangledown \phi \right )^{2}-\frac{1}{2}m^{2}\phi ^{2}\tag{2.6}$$ ...
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Coordinates from action-angle variables

I'm interested into getting the original coordinates, $q(t)$ and $p(t)$, from the action, $J=\oint p dq$, and angle, $w(t)=\frac{dH}{dJ}t+\beta$, variables for a 1-D, one particle system. I know that ...
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How do I obtain the SUSY Transformations from Poisson Brackets?

In Friedman's and Van Proyen's Supergravity textbook it is explained how one can get the supersymmetry transformations using the conserved currents. Specifically this is in section 6 where we are ...
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Finding geodesics: Lagrangian vs Hamiltonian

I have a question referring to how to compute geodesics of a given spacetime (say, Kerr). I know that the direct way is via the geodesic equation $$\frac{d^{2}x^{\mu}}{d\lambda^{2}}+\Gamma^{\mu}_{\...
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kinetic energy in analytical mechanics

I have a question about the derivation of kinetic energy in analytical mechanics. What is the physical diffrence bettwen the 3 components (T0 T1 and T2), and when should we use those components ...
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Why Liouville's theorem is obvious?

In Florian Scheck's Mechanics, he stated the local form of Liouville's theorem as follows: Let $\Phi_{t,s}(x)$ be the flow of the differential equation $$-J\frac{d}{dt}x=H_{x}.$$ Then for all $x,t,s$ ...
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Motivation for covariant phase space

The covariant phase space idea, in one sentence, is that there is a natural symplectic structure on the space of the classical trajectories of a system and that the usual $(q,p)$ coordinates just ...
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Least Action Principle (Classical and Quantum Theory)

I) My first question would be "why should classical systems obey the principle of least action ?" When we find out the propagator in quantum physics, we find the amplitude to be equal to the sum over ...
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Classical particle in a box [closed]

I'm trying to work out some of the details for this system. A particle with mass $\mu$, initial velocity $v_0$ at $x_0$ and moving freely between two walls located at $\pm L/2$, with which it bounces ...
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Statistical Physics of a System with Friction inside a Hot Bath

If you have a classical system (i.e obeying Newton's equations of motion) with Hamiltonian $H(x,p) = \frac{p^2}{2m} + U(x)$ then the statistical behaviour of this system is described by the ...
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Why is the Lagrangian approach preferred over the Hamiltonian approach in QFT? [duplicate]

Going from non-relativistic quantum mechanics(QM) to QFT there is a marked change in the approach used. QM almost exclusively uses Hamiltonains. Lagrangian based methods like the path-integrals are ...
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Why are the integrability conditions necessary and sufficient for the existence of a canonical transformation's generating function?

Consider a canonical transformation $(p,q) \rightarrow (P,Q)$ under a generating function $F$. The condition for form invariance of Hamiltonian equations of motion looks like : $$\sum_{s}P_s\dot{Q_s} ...
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Hamilton's equations of motion

One of hamilton's equations is $(\frac{\partial H}{\partial q} )_t = -(\frac{\partial p}{\partial t}) _q$. But isn't it $\frac{\partial L}{\partial q} = \frac{dp}{dt}$? If H = L(i.e. V = 0), what ...
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How does this quantisation relation come about?

I'm currently doing a course in string theory and in the lecture notes it is stated: $$ [x^-, p^+]~=~-i \tag{1}$$ I am fine with this. However, after trying (and failing) a question, I looked at ...
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How to scale variables in a classical Hamiltonian?

So I looked at some research articles where one has a classical Hamiltonian $H(p,q,t) = p^{2}/2 + V(q,t)$. If one introduces the scaling transformation $$t \mapsto t/\sqrt{s}, \quad H \mapsto Hs, \...
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Phase Portraits Given Hamiltonian

Given a Hamiltonian say $$ H = 5p^2 $$ What is the correct procedure for producing a phase portrait. My initial thoughts were to solve the system of equations $\frac{dq}{dt} = 0$ and $\frac{dp}{dt} ...
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Counting number of degrees of freedom in constrained system

Following Counting degrees of freedom in presence of constraints, we know that there would be N-2M-S dofs if we have M 1st-class constraints and S 2nd-class constraints in N-dim phase space. I don't ...
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Weyl exponential form of the Canonical Commutation Relations

What is the physical meaning of the $c$-numbers $Q, P\in \mathbb{R}$ in the exponent of the Weyl system $\exp\left[\frac{i}{\hbar} Q \hat{p}\right]$ and $\exp\left[\frac{i}{\hbar}P\hat{q}\right]$? ...
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Proof of Liouville's theorem: Relation between phase space volume and probability distribution function

I understand the proof of Liouville's theorem to the point where we conclude that Hamiltonian flow in phase-space is volume preserving as we flow in the phase space. Meaning the total derivative of ...
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Canonical Transformation [duplicate]

How can I prove that the following transformation is canonical: $\begin{cases}\overline{q}_i=\dfrac{q_i}{Q} \\ \overline{p}_i=Qp_i-2Pq_i \end{cases},\ i\in\overline{1,n}$ where $Q=\sum_{i=1}^n q^2_i$...
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Recovering a symmetry transformation from a conserved charge

I'm going through some notes on how to apply the Hamiltonian formalism to systems with gauge invariance and I found a derivation of Noether's theorem I had never seen before. The idea is roughly that ...
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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 ...
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Can one write down a Hamiltonian in the absence of a Lagrangian?

How can I define the Hamiltonian independent of the Lagrangian? For instance, let's assume that i have a set of field equations that cannot be integrated to an action. Is there any prescription to ...
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Find the error: If $L_x$ and $L_y$ are zero, then $L_z$ is conserved

From Goldstein's Classical Mechanics (2nd ed.), problem 38 of chapter 9 basically says the following: It's been shown that the Poisson bracket of two constants of the motion is also a constant of ...
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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'~?...
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Time derivative of a function in Phase Space

Consider a function $\mathcal{H}(q_i,p_i;t)$ such that it obeys the equation: $$ \frac{d\mathcal{H}}{dt}=\frac{\partial\mathcal{H}}{\partial t}$$ What does this equation imply (read: mean), physically?...
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Hamilton's Equations

The last step of this derivation of Hamilton's Equations is what's making me doubt it. It is as follows: Assuming the existence of a smooth function $\mathcal{H}(q_i,p_i)$ in $(q_i(t), \,p_i(t))$ ...
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A physical system is described by the following Lagrangian: $ L = \frac{m}{2} (\dot{\rho}² + \rho ² \dot{\phi} ² + \dot{z} ²) + a \rho² \dot{\phi}$ [closed]

Where $a$ is a constant and $(\rho,\phi,z)$ are cylindrical coordinates. I found the following Hamiltonian $ H =\frac{m}{2}(\dot{\rho}² + \dot{z}² + \rho²\dot{\phi}²)$. The problem asked me to find ...
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Energy conservation Hamiltonian dependency

Suppose the a system has a Hamiltonian $H = H(q,p)$, and suppose $H$ does not depend explicitly on time. If $H\neq E$ the total energy of the system, does this necessarily say that $E$ is not ...
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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 ...
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Hamiltonian form: inverting the general momenta

I am studying analytical mechanics and am confused with the part about Hamiltonian form. My textbook develops the whole theory of Legendre transforms, in order to invert the formula $$\mathbf{p}=\...