equations of motion

Hamiltonian Mechanics

Hamilton’s equations of motion

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For a Hamiltonian H, given by

where T and U are the total kinetic energy and total potential energy of the system, respectively; q is a generalised position (such as x, y, or r), and; p is a generalised momentum. Using this notation, Hamilton’s equations of motion are

Notice that q with a ‘dot’ represents a generalised velocity and p with a ‘dot’ represents a generalised force.

Equations of motion derivation

We know that the kinetic energy is classically expressed as

where

is the velocity. Recalling that p = mv, we find that

Let’s now consider the partial derivative of the Hamiltonian (in equation 1) with respect to this generalised momentum p:

Clearly, U doesn’t depend on p so does not factor into this equation. Now we may evaluate the partial derivative of the kinetic energy to find

recalling that

we have

Finally,

which we recognise as the Hamiltonian equation of motion for generalised velocity!

Now, we have a potential energy, given by

where W is the work done. Given that

i.e. Newton’s Second Law of Motion, we find that

Thus, our Hamiltonian is

which has an infinitesimal change

In the following steps we divide through by the dq element, keeping in mind that the two are linearly independent and therefore do not depend on one-another:

We recognise this as the Hamiltonian equation of motion for generalised force!

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For those of us who are Earthbound, it’s easy to think of liquids and gases as being the most common fluids. But plasma–the fourth state of matter–is a fluid as well. Plasmas are essentially ionized gases, which, thanks to their freely flowing electrons, are electrically conductive and sensitive to magnetic fields. Their motions are described by a combination of the Navier-Stokes equations–the usual equations of motion for a fluid–and Maxwell’s equations–the equations governing electricity and magnetism. Studies of plasma motion often fall under the subject of magnetohydrodynamics and can include topics like planetary auroras, sunspots, and solar flares. (Video credit: SciShow)

Standing in the shower, water running down my face, pretending to be in a music video n shit. Every movement you make has to be slow to equate to being in slow motion. Next level shit.