# 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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### 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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### 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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### 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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### 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 ...