What is a Hamiltonian vector field in General Relativity?

[unchained1978]

I'm researching General Relativity and have stumbled upon a bit of Hamiltonian mechanics. I roughly understand the idea behind the Hamiltonian of a system, but I'm utterly confused as to what the hell a Hamiltonian vector field is. I've taken ODE's, PDE's, Linear Algebra, and I'm just being introduced to Differential Geometry so I can handle the math, but every article I read on this subject is entirely too abstract or jargon filled for me to understand. Can someone please explain to me what this vector field is and what it represents?

 

[A. Neumaier]

I'm researching General Relativity and have stumbled upon a bit of Hamiltonian mechanics. I roughly understand the idea behind the Hamiltonian of a system, but I'm utterly confused as to what the hell a Hamiltonian vector field is. I've taken ODE's, PDE's, Linear Algebra, and I'm just being introduced to Differential Geometry so I can handle the math, but every article I read on this subject is entirely too abstract or jargon filled for me to understand. Can someone please explain to me what this vector field is and what it represents?

Please look at the example in http://en.wikipedia.org/wiki/Hamiltonian_vector_field and tell us what you don't understand.

 

[homology]

When we're doing Hamiltonian mechanics we're in phase space and so have local coordinates $$q^i, p_i$$. What are Hamilton's equations? They're a 'list' of derivatives, and you could write it as:
$$\begin{pmatrix}<br /> \dot{q}^1\\<br /> \dot{q}^2\\<br /> ...\\<br /> \dot{q}^n\\<br /> \dot{p}_1\\<br /> \dot{p}_2\\<br /> ...\\<br /> \dot{p}^n\\<br /> \end{pmatrix}=<br /> \begin{pmatrix}<br /> \partial H/\partial p_1\\<br /> \partial H/\partial p_2\\<br /> ...\\<br /> \partial H/\partial p_n\\<br /> -\partial H/\partial q^1\\<br /> -\partial H/\partial q^2\\<br /> ...\\<br /> -\partial H/\partial q^n\\<br /> \end{pmatrix}$$
So, just like any ordinary vector field you've seen before, if you're at point 'm' it tells you where to go next to follow along the vector field. Differential equations are vector fields.

What's your background in mechanics?

 

[unchained1978]

Haven't taken any mechanics courses. I mainly don't understand what a hamiltonian vector field physically represents. Would the components be the velocities and forces of a particle or what? If someone could just give me an example it would go a long way. Thanks

 

[A. Neumaier]

Haven't taken any mechanics courses.

Without a good understanding of mechanics it makes little sense to study general relativity. So you first need to improve your background.

I mainly don't understand what a hamiltonian vector field physically represents. Would the components be the velocities and forces of a particle or what? If someone could just give me an example it would go a long way. Thanks

It is the vector field F(z) in phase space (whose points are the pairs z=(q,p)) that tells in which direction a particle at point z will move: zdot = F(z). Post #3 tells you more explicitly how F looks like, given the Hamiltonian.