Conserved charge generates symmetry transformation in Hamiltonian Mechanics

In summary, the conversation discusses the proof that q+\epsilon\delta q satisfies the equation of motion, with \delta q = \{q, Q\}. The proof involves manipulating the equations to show that \dot{q'} = \{q,H\}|_{q'}, where q' = q+\epsilon\delta q. The key step is to consider the change in momentum, which leads to \{\dot{q},Q\} = \partial_q\{q,H\}\{q,Q\}.
  • #1
kakarukeys
190
0
[tex]Q[/tex] is a conserved charge if [tex]\{Q, H\} = 0[/tex]
Show that [tex]q+\epsilon\delta q[/tex] satisfies the equation of motion.
[tex]\delta q = \{q, Q\}[/tex]

I couldn't find the proof. Anybody knows?
My workings:
[tex]\delta q = \{q, Q\}[/tex]
[tex]\delta\dot{q} = \{\{q,Q\},H\} = - \{\{Q,H\},q\} - \{\{H,q\},Q\}[/tex]
[tex]\delta\dot{q} = \{\{q,Q\},H\} = \{\{q,H\},Q\}[/tex]
[tex]\delta\dot{q} = \{\{q,Q\},H\} = \{\dot{q},Q\}[/tex]
Let [tex]q' = q+\epsilon\delta q[/tex]
Show that [tex]\dot{q'} = \{q,H\}|_{q'}[/tex]
L.H.S. is [tex]\dot{q}+\epsilon\delta\dot{q}[/tex]
R.H.S is [tex]\{q,H\}+\epsilon\partial_q\{q,H\}\delta q[/tex]
Therefore we have to prove [tex]\{\dot{q},Q\} = \partial_q\{q,H\}\{q,Q\}[/tex]
...
No clue how to continue
 
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  • #2
Solved! I neglected the change in p
 

1. What is a conserved charge in Hamiltonian mechanics?

A conserved charge is a physical quantity that remains constant over time in a system described by Hamiltonian mechanics. It is associated with a symmetry transformation, meaning that the system's equations of motion are invariant under that transformation.

2. How does a conserved charge generate a symmetry transformation?

A conserved charge generates a symmetry transformation by acting as the generator of that transformation. This means that the transformation can be expressed as an exponential function of the conserved charge, and the conserved charge itself can be derived from the transformation using Noether's theorem.

3. What is the relationship between conserved charges and symmetries in Hamiltonian mechanics?

In Hamiltonian mechanics, conserved charges are intimately connected to symmetries. Each conserved charge corresponds to a symmetry transformation, and vice versa. This relationship is described by Noether's theorem, which states that for every continuous symmetry of a physical system, there exists a conserved quantity.

4. Can a conserved charge generate more than one symmetry transformation?

Yes, a conserved charge can generate multiple symmetry transformations. This is possible because different symmetry transformations can correspond to the same conserved charge. For example, in rotational symmetry, there are infinitely many possible transformations that can be generated by a single conserved angular momentum.

5. How are conserved charges used in Hamiltonian mechanics?

Conserved charges are used in Hamiltonian mechanics to study the dynamics of physical systems. They provide a way to analyze symmetries and their corresponding transformations, which can reveal important information about the behavior of a system. Additionally, conserved charges are often used to simplify the equations of motion and make calculations more tractable.

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