Questions tagged [poisson-brackets]
In phase-space classical Hamiltonian mechanics, the Poisson bracket is an antisymmetric binary operation acting like a derivative.
339 questions
4 votes
2 answers
277 views
A (probably wrong) proof that the fundamental Poisson brackets are independent of the special choice of the canonical variables
In Wolfgang Nolting's book Theoretical Physics 2 - Analytical Mechanics, the following theorem is stated: While the formal definition of canonical trasformation is given only several pages later in ...
- 41
3 votes
1 answer
151 views
Infinitesimal canonical transformations and Lie algebras
I'm currently studying classical mechanics, partly from Goldstein's book. I'm reading the part about infinitesimal canonical transformations (ICT) in the Poisson bracket formulation (section 9.6). ...
- 735
2 votes
1 answer
148 views
Why does the boundary behavior of Lagrange multipliers matter for gauge vs global symmetries?
In Henneaux & Teitelboim (Quantization of Gauge Systems, p. 30), they discuss the variation of a dynamical variable $$ \delta F = \int d^nx\, u(x)\,\{F, C(x)\}_{PB},\tag{1.62} $$ where $C(x)$ is a ...
- 529
-3 votes
2 answers
270 views
Why is the Poisson bracket $\{q_i,p_j\}=\delta_{ij}$? [closed]
Why is $\{q_i,p_j\}=\delta_{ij}$? Here the Poisson bracket $$\{F,H\}=\sum_{i=1}^f\frac{\partial F}{\partial q_i}\frac{\partial H}{\partial p_i}-\frac{\partial F}{\partial p_i} \frac{\partial H}{\...
1 vote
0 answers
126 views
Finite Symmetry Transformations in Classical Mechanics
This is not a homework exercise. I graduated from univerisity more than 10 years ago. I ask questions from my self-study. There're two types of symmetry transformations in classical mechanics. One is ...
- 3,237
2 votes
1 answer
151 views
Difference between the Euler-Arnold equation and the Poisson bracket equation
From this blog the Euler–Arnold equation is defined as follows: $\textbf{Theorem 1 (Euler–Arnold equation)}$ Let $\gamma: \mathbb{R} \to M$ be a geodesic flow on $M$ using the right-invariant metric $...
- 1,586
3 votes
2 answers
262 views
From the Poisson bracket equation to the Euler equations in fluid dynamics
In the book Introduction to Mechanics and Symmetry (Marsden, 1998, pp. 20–21), it is stated that the Euler equations for an incompressible, ideal fluid: $$ \frac{\partial \mathbf{v}}{\partial t} + (\...
- 1,586
0 votes
1 answer
119 views
What are the conserved quantities for the Euler top? And how does the symplectic formalism differ from the Poisson?
I am currently trying to explain why the free rigid body with a fixed point is an integrable system. My questions are the following: A $2n$-dimensional system is called integrable if there are $n$ ...
- 3
5 votes
1 answer
200 views
Commutation Relations in BRST Quantization
In Chapter 15 of Weinberg's The Quantum Theory of Fields, Weinberg states that the commutation relations for the creation and annihilation operators associated with the $A^\mu$ gauge fields of pure ...
- 345
5 votes
2 answers
339 views
Sign issue in the Nambu-Goto constraint algebra
When trying to show that the momentum constraint in the Nambu-Goto string actually generates world-sheet spatial diffeomorphisms, I encountered the following sign issue which I was not able to resolve:...
- 116
1 vote
0 answers
96 views
Possible Typo in Poisson Brackets in Weinberg QFT Vol. 1, Section 8.3
In Chapter 8.3 of Weinberg's The Quantum Theory of Fields-Volume I, he gives the following Poisson bracket relations: $$\begin{aligned} \left[A^i(\mathbf{x}), \chi_{1\,\mathbf{y}}\right]_P &= -\...
- 345
4 votes
5 answers
286 views
Explicit Hamiltonian without legendre transform in symplectic geometry
Most textbooks derive the Hamiltonian through the Legendre transform, so for simple mechanical systems one lands at $H = T + V$. I am interested in whether the time-evolution generator $H$ can be ...
7 votes
1 answer
503 views
Equivalence of "different" versions of Noether's theorem
There is a (perhaps?) less-well-known version of Noether's theorem in John Lee's Introduction to Smooth Manifolds, i.e. Theorem 22.22, which roughly states that if a vector field $V$ such that $${\cal ...
- 71
2 votes
1 answer
175 views
Quantum Operators and Hamiltonian Mechanics
I've lately started to study the modern quantum theory and I've obviously run into the concepts of Operators. More specifically here I would like to discuss about all the type of evolution operators, ...
- 41
3 votes
1 answer
412 views
Questions about the definition of Poisson bracket
Let's start with some preliminaries. Let $(M, \omega)$ be a symplectic manifold where $M=T^\ast X$ is $2d$-dimensional phase space for $d$-dimensional configuration space $X$, and $\omega:TM\times TM\...
- 1,095