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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Confusion regarding properties of Poisson Brackets

I have just started learning about Poisson Brackets, and came across the following property $$\{q_i,q_j\}=0$$ And $$\{p_i,p_j\}=0.$$ Where $p$ and $q$ are respectively the momentum and position ...
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Why is the change in Hamiltonian for an active infinitesimal canonical transformation defined the way it is?

I'm trying to understand infinitesimal canonical transformations and conservation theorems (section 9.6 Goldstein ed3). My specific problem is with understanding eq 9.104, $\partial H = H(B) - ...
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Hamiltonian of non-regular Lagrangian is well-defined on phase space

In section 1.1.3 of Quantization of Gauge Systems by Henneaux and Teitelboim, it is stated that the Hamiltonian $$H=\dot{q}^np_n-L,\tag{1.8}$$ although trivially a function of $q$ and $\dot{q}$, can ...
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Transform from Cartesian coordinates to Delaunay elements

Place a central mass $M = 1/G$ at the origin $(0,0,0)$. We know that for a test particle moving in the $xy$-plane, we can transform $(x,y,p_x,p_y)$ to the Delaunay elements $(\theta_\lambda, \theta_\...
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Hamilton Constraint of the WdW equation

Can someone explain specifically what the surface term of the hamilton constraint in quantum cosmology actually describes and how it creates time even though we start with a timeless universe? And why ...
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Extension of classical Liouville operator

Let us consider a classical Hamiltonian system described by the Hamiltonian \begin{equation} H(q,p) =\frac{p^2}{2m}+V(q) \end{equation} where we stick to the case of single particle for simplicity. I ...
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ADM decomposition of the general scalar tensor theory Lagrangian

I have question about ADM decomposition of some general scalar-tensor theory of gravity. Starting with ADM form of the metric: $ds^2=-N^2dt^2+h_{ij}(dx^i+N^idt)(dx^j+N^jdt)$ provided with extrinsic ...
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Online courses on Analytical Mechanics [duplicate]

I'm having trouble understanding some concepts in Analytical Mechanics (particularly, the usage of lagrangean multipliers to get to the equations of motions and Noether's theorem) and I'm wondering if ...
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Poisson bracket of momentum constraints in general relativity

I wish to compute the Poisson bracket of the momentum constraints in general relativity. Unfortunately, I am not able to do it correctly and the answer I am getting is not a linear combination of the ...
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Why are positions and momenta independent variables in the Hamiltonian formulation?

Lifshitz and Landau's Vol. $1$ explicitly states that $$ \cfrac{\partial{q_k}}{\partial{p_i}} = 0$$ And seems to imply also that $$ \cfrac{\partial{p_k}}{\partial{q_i}} = 0.$$ I guess that whenever ...
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The Hamiltonian and differentials

From Lifshitz and Landau Vol.$1$: From the equation in differentials $$ \mathrm{d} H=-\sum \dot{p}_{i} \mathrm{d} q_{i}+\sum \dot{q}_{i} \mathrm{d} p_{i} $$ in which the independent variables ...
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Generator of translations in classical mechanics

The generator of spatial translations is momentum. In quantum mechanics this makes a lot of sense to me and so we can write the translation operator like this: $$\hat{T}(\Delta \textbf{r}) = e^{\frac{...
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Poisson brackets of three dimensional angular momentum and its Lie lagebra

I've recently noticed that the Poisson brackets of the three dimensional angular momentum $$\{L_i,L_j\}$$ in classical mechanics follow the same commutator relations as the standard basis of the Lie ...
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Quantising action to find evolution of wavefunction

I am given the following action functional $$S=\int dt \left[p_1\dot q_1+p_2\dot q_2-\frac{p_1p_2}{m}-\sum_{n=0}^\infty\frac{\partial^nV(q_1)}{\partial q_1^n}q_2^n\right]$$ where $p_i$ is the ...
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A question on Liouville's theorem and time-dependence of the Hamiltonian

The condition of equilibrium in statistical mechanics is $\frac{\partial \rho}{\partial t}=0$ where $\rho$ is the phase space density. By virtue of Liouville's theorem, this is equivalent to the ...
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Assignment of energy functions to flows is “equivariant”?

I am trying to understand the 2012 blog post What is a symplectic manifold, really? It says (with correction of a typo in the second point): If $f: M \to \mathbb{R}$ is a smooth compactly ...
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Geometric vs Physical meaning of conjugate momentum [closed]

I'm studying mechanics and wondering about what are the physical and geometric meanings/interpretations of the conjugate momentum. My study started with the Newtonian mechanics, and from what I ...
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What happens to the configuration manifold when one quantizes the Hamiltonian?

A system in classical mechanics can be described by a configuration manifold $Q$ and a Lagrangian \begin{equation} L:TQ\rightarrow \mathbb{R} \end{equation} where $TQ$ is the tangent bundle or a ...
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Which mechanics is best? [duplicate]

Is there any specific reason for using Newton mechanics instead of Hamiltonian or any other theory of mechanics. I am asking just out of curiosity (I know nothing about Hamiltonian mechanics)
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Hamiltonian dependence of variables

How can one say that in Hamilton mechanics the $q$'s are independent of the $p$'s while if I have the Lagrangian $L = \frac{1}{2}\dot{x}^2 + \frac{1}{2}x^2\dot{y}^2$ then $p_y = \frac{\partial{L}}{\...
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Is it useful to define canonical pressure?

Pressure is defined as: $$ P = \frac{\partial U}{\partial V}$$ where $V$ is the volume and $U$ is the internal energy. Does the quantity $P'$: $$ P' = \frac{\partial U}{\partial V'}$$ where $V'$ ...
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Can any sum of infinitesimal canonical transforms on phase space be obtained from evolution under a static Hamiltonian?

Suppose I have a canonical transformation on phase space, which is obtained by evolving a classical Hamiltonian system from time $t=0$ to $t=T$, with some arbitrary time-dependent Hamiltonian $H(t)$. ...
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General question about quantization procedure

As far as I understood, when we want to quantize a system, the procedure can be the following (but probably not the most general one): We start by writing down the Lagrangian of the system. We ...
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Generating function for canonical transformation

Short version: I've been reading through some notes on integrable systems/Hamiltonian dynamics, and am stuck on a problem: Show that the coordinate transformation derived via the generating function ...
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1answer
106 views

Action-angle variables for anharmonic oscillator

I have an equation of potential given: $$U = U_0\tan^2( \alpha(t)q)$$ I need to find a motion rules for that potential in terms of action-angle variables. Using the fact that Hamiltonian is equal to ...
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Hamiltonian energy density, energy flux, and Poisson brackets

I came across an old paper (Hardy, Energy Flux Operator for a Lattice. Physical Review. Volume 132 Number 1, 1963) that states a relation between the time rate of energy density, the "energy density ...
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Constrained Hamiltonian dynamics [closed]

Suppose that I have a state $q$ that is constrained to satisfy $\mu(q)=0$. Assuming that $\nabla_q \mu(q)$ is a matrix of full-rank, I can use the Jacobian of the constraint function to define tangent ...
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Reversibility of Hamiltonian dynamics

I'm trying to understand a very basic property of Hamiltonian dynamics. I don't have a physics background but I do know some mathematics. I want to understand why negating the momentum is equivalent ...
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37 views

Equations of motion for a certain constrained system

As an exercise in Lagrangian and Hamiltonian mechanics, I am looking at a system with the following Lagrangian: $$L=\dot R \cdot\dot R-\theta\dot R\cdot (SR)+\lambda(R\cdot R-1) $$ $R$ is a vector ...
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Leapfrog integrator, kill off lower order terms [closed]

Let $H=A+B$ be the Hamiltonian, let $L_A(\cdot):=[\cdot,A], L_B(\cdot):=[\cdot,B]$ denote the Lie derivatives of $A, B$ respectively. We can show that the leapfrog integrator $$ \exp(\frac{\tau}{2}...
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Rigorous canonical coordinate definition

I would like to know what is the definition of a canonical coordinate. Let's assume we have a Lagrangian $\mathcal{L}(q,\dot{q})$. Are the canonical coordinate simply the set $(q,p)$ where $p=\frac{\...
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Changing sign of Lagrangian & Hamiltonian: how to interpret energies then?

In Lagrangian mechanics, it is possible to multiply the Lagrangian by a constant $a$. Let's assume I take $a=-1$. Then, the Hamiltonian will have its sign changed as well. And it will represent the ...
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Performing Legendre transformation when conjugate momentum is independent of time derivative of generalised co-ordinates [duplicate]

Suppose there is a Lagrangean $L = \frac{1}{2} m c \dot x - \frac{1}{2}kx^2$ where $c$ is a constant to keep the dimensions right. The conjugate momentum is then $ p_x = \frac{\partial L} {\...
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Delaunay variables

I've read a little bit about Delaunay variables, but I can't understand what they are good for. Do they make calculations easier? What is the advantage of using them? Where can I read a bit more about ...
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Why is it justified to discard off-shell momenta contributions in the exponent of the expression for a path integral amplitude?

Let us consider a free field theory with one field $\phi$. The Lagrangian density is $L(\phi, \partial_{\mu} \phi)$ and the corresponding Hamiltonian density is $H(\phi,\pi,\partial_{\mu \neq 0}\phi)$....
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Conceptual difficulty with the Legendre transformation

I feel like I have a fairly muddled understanding of function transformations, and so I'm hoping someone can clarify things for me a bit. I think this might be in part to do with confusion surrounding ...
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Resource recommendation for studying Elastostatics and Elastodynamics using Lagrangian and Hamiltonian formulation

I've been self studying Elastostatics and Elastodynamics from Kip S. Thorne's Modern Classical Physics (Chapter 11 & 12). The whole topic is discussed based on Newtonian mechanics, vector & ...
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Lagrangian formulation of classical spin chains

Is there a way to construct a Lagrangian formulation of the classical dynamics of a spin chain - say a Heisenberg or XY chain? The Hamiltonians here are obvious.
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135 views

Quantum mechanics of projectile motion [closed]

In classical mechanics, the Lagrangian of a particle undergoing projectile motion without drag in a constant gravitational field is $$L=\frac{1}{2}m\left(\dot{x}^2+\dot{y}^2\right)-mgy.$$ Performing a ...
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Degenerate perturbation theory in classical mechanics

I would like to know if there is a way to properly do time-independent degenerate perturbation theory in classical mechanics. Any answer or pointer to a good source would be appreciated. The issue of ...
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Why I find two possible $S$ of a free particle by solving the Hamilton-Jacobi Equation?

For a free particle, the Hamiltonian is $$H(p)=\frac{p^2}{2m}.$$ The corresponding H-J equation thus can be written as $$\frac{1}{2m} \left(\frac{\partial S(q,t)}{\partial q}\right)^2=- \frac{\...
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How to find the canonical transformation if given the new Hamiltonian we want?

If we know the new Hamiltonian $H^{\prime}$ we want, and $H^{\prime} \neq 0$, how can we find the canonical transformations? For example, we want to transform the $$H(p,q)=\frac{p^2}{2m}+\frac{1}{2}...
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Hamilton equations and phase space

Let $(M,\omega)$ with $M$ a $2n$-dimensional manifold and $\omega$ a $2$-form on $M$, furthermore let $(M,\omega)$ be a symplectic manifold (smooth, differentiable, continuous, whatever is needed for ...
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Physical Systems whose Phase Space is not a Cotangent Bundle

I'm trying to justify the full power of symplectic mechanics yet I keep finding examples of physical systems which are only trivial examples of symplectic mnaifolds, cotangent bundles. What physical ...
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Understanding the Staeckel conditions for separability of the Hamilton-Jacobi equation

I'm trying to understand the so-called Staeckel conditions, specifically the fifth one, for separability of the Hamilton-Jacobi equations, as described in Goldstein's Classical Mechanics (3:rd edition)...
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Poisson Bracket of a Quantity Involving a Differential

I am working through Warren Siegel's "Fields" and have come across the following exercise on p. 58 involving an action measure and a symmetry generator: Exercise IA4.1. For general ...
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Hamiltonian formulation in Ashtekar's Variable [duplicate]

Is there is a reference where I can learn how to do ADM or hamiltonian formulation in Ashtekar's variables?
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Solving a simple classical Hamiltonian problem

Suppose you've been given this Hamiltonian: $H = m\ddot{x}$ and are asked to find the equations of motion. (This is a simplified version of a question given here on page 3.) This isn't a homework ...
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How do we derive the Hamiltonian of the Wilson Loop Action?

I'm reading Fradkin and Susskind's 1978 paper "Order and disorder in gauge systems and magnets" to try and understand how they derive the Hamiltonian for the U(1) compact lattice gauge theory. The ...
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Applications and limitations of the Hamilton-Jacobi formalism

It was my understanding that the Hamiltonian formalism was inadequate to describe systems that are invariant under time reparametrization or that have gauge symmetries. However, I see in Classical ...

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