Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

Recent 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.

The Hamiltonian

The Hamiltonian can be interpreted as an “energy input”, as opposed to a Lagrangian, which is the "energy output". The Euclidean Hamiltonian, which is used in Classical Mechanics is given by:

$$H = \frac{{{p^2}}}{{2m}} + U$$

The Euclidean Lagrangian, on the other hand, has a minus instead of a plus. Notice that

$$L + H = p\frac{{{\text{d}}x}}{{{\text{d}}t}}$$

This shows that the two are related by a Legendre transformation.

The Poisson Bracket relationships and the Dynamic Hamiltonian Relationships

The Poisson Bracket relations are algebraic relationships between phase space variables, and without the presence of any dynamical Lagrangian or Hamiltonian. Thus, the Poisson Bracket relations would obviously (to someone with a basic knowledge of Lagrangian Mechanics) be :

$$ \begin{gathered} \{ {{p_i},{x_j}} \} = {\delta _{ij}} \\ \{ {{p_i},{p_j}} \} = 0 \\ \{ {{x_i},{x_j}} \} = 0 \\ \end{gathered} $$

The Dynamical Relationships, however, are obviously changed. It is clear that the new relationshipjs are that:

$$\begin{gathered} \frac{{\partial H}}{{\partial {\mathbf{x}}}} = - \frac{{{\text{d}}{\mathbf{p}}}}{{{\text{d}}t}} \\ \frac{{\partial H}}{{\partial {\mathbf{p}}}} = \frac{{{\text{d}}{\mathbf{x}}}}{{{\text{d}}t}} \\ \end{gathered}$$

Compare this to the dynamical Lagrangian Relations:

$$\begin{gathered} \frac{{\partial L}}{{\partial {\mathbf{x}}}} = \frac{{{\text{d}}{\mathbf{p}}}}{{{\text{d}}t}} \\ \frac{{\partial L}}{{\partial {\mathbf{p}}}} = \frac{{{\text{d}}{\mathbf{x}}}}{{{\text{d}}t}} \\ \end{gathered}$$

The central equation of Hamiltonian Mechanics is the Hamilton Equation:

$$\frac{{{\text{d}}A}}{{{\text{d}}t}} = \{A,H \}$$
+ 1 like - 0 dislike
0 answers 186 views
+ 0 like - 0 dislike
0 answers 279 views
+ 1 like - 0 dislike
2 answers 740 views
+ 0 like - 0 dislike
0 answers 2468 views
+ 3 like - 0 dislike
0 answers 550 views
+ 1 like - 0 dislike
2 answers 2731 views




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...