What does the Hamiltonian depend on in a classical system?

[mieral]

1. In Classical Hamiltonian, it's equal to the kinetic energy plus potential energy.. but I read it that for a free particle, it doesn't even depend on position.. i thought the potential energy depends on position. If it doesn't depend on position, what does it depend on?

2. Since the Hamiltonian is equal to the potential energy plus kinetic energy. The Hamiltonian can be measured as when you measured the electric field which is a potential energy. But I read the Hamiltonian can't be observed or measured.. how come?

This is an entirely classical question so please do NOT forward this to the QM forum. Thank you.

 

[Vanadium 50]

for a free particle...potential energy

If the particle is bound by a potential, is it free?

 

[mieral]

If the particle is bound by a potential, is it free?

ah.. so a free particle not bound by potential has no position but only momentum...

but if the particle has neither momentum nor position preferred.. what would happen to Hamiltonian = potential energy + kinetic energy?

Can anyone mention any thing that has potential energy and kinetic energy yet without any position or momentum?

 

[Vanadium 50]

so a free particle not bound by potential has no position

I never said that.

 

[mieral]

I never said that.

isn't it a free particle has plane waves and only momentum and no position?

 

[muscaria]

isn't it a free particle has plane waves and only momentum and no position?

yet in your OP, you say

This is an entirely classical question so please do NOT forward this to the QM forum.

Does a classical particle have wave solutions? Does it have definite position and momentum?

 

[mieral]

yet in your OP, you say

Does a classical particle have wave solutions? Does it have definite position and momentum?

Oh.. but please don't forward this thread to the quantum because all my messages there are locked immediately hehe.. they told me to focus on the classical dynamics first before going into quantum...

so let's discuss Classical Hamiltonian.. that is.. purely waves ingredients..

 

[muscaria]

Oh.. but please don't forward this thread to the quantum

1) That was not my intention and 2) I do not have such privileges anyway.

I would refer you back to the 2 questions of my previous post, but maybe in the opposite order.
1) Does a classical particle have a definite position, a definite momentum, neither, both?

Once you have the answer to this, the next question:
2) Does a classical particle have wave-like solutions?

should then become clear.

 

[mieral]

1) That was not my intention and 2) I do not have such privileges anyway.

I would refer you back to the 2 questions of my previous post, but maybe in the opposite order.
1) Does a classical particle have a definite position, a definite momentum, neither, both?

yes a classical particle has definite position and definite momentum like a baseball..

Once you have the answer to this, the next question:
2) Does a classical particle have wave-like solutions?

it doesn't have... so let's discuss waves in classical Hamiltonian.. does a wave in classical Hamiltonian have both position and momentum?

should then become clear.

 

[Nugatory]

isn't it a free particle has plane waves and only momentum and no position?

In quantum mechanics, yes. (More precisely, an infinite plane wave is a solution of the time-independent Schrödinger's equation when $V=0$).

But this is the classical physics subforum, and you've asked us to avoid quantum mechanics in this thread... So how about not mentioning waves any more?

 

[mieral]

In quantum mechanics, yes. (More precisely, an infinite plane wave is a solution of the time-independent Schrödinger's equation when $V=0$).

But this is the classical physics subforum, and you've asked us to avoid quantum mechanics in this thread... So about not mentioning waves any more?

Why.. are there no waves in classical Hamiltonian?

Anyway what is the best everyday example where Newtonian mechanics can't solve something that can easily be solved by the classical Hamiltonian.. and i'd like to understand if it is true that even classical Hamiltonian can't be measured or observed.. because according to Demystifier.. the Hamiltonian in quantum can't be measured or observed.. is this only a quantum thing and not true in classical Hamiltonian or the same? and Why?

 

[muscaria]

yes a classical particle has definite position and definite momentum like a baseball..

Indeed. Do you see now why the question

isn't it a free particle has plane waves and only momentum and no position?

does not make sense?

so let's discuss waves in classical Hamiltonian

Sure, we can discuss classical waves :).

does a wave in classical Hamiltonian have both position and momentum?

A wave propagates through a medium. In the case of a sound wave in air, the air is the medium. If you had a 1-d chain of masses connected by springs (mass-spring-mass-spring etc..), the springs are the medium and pulling one of the masses upwards with you fingers will stretch the springs around and when you let go a wave/ripple propagates through the medium as the masses move upwards and downwards pulling each other up and down in succession through the springs. What would be the "position" of the wave for this case in your opinion?

 

[Nugatory]

Why.. are there no waves in classical Hamiltonian?

You can use Hamiltonian methods to solve problems involving classical waves... but first you have to learn to use these methods, and that is way easier with problems involving particles.

Anyway what is the best everyday example where Newtonian mechanics can't solve something that can easily be solved by the classical Hamiltonian..

Google for "Hamiltonian mechanics examples" and you'll find many. The Hamiltonian solution to the Kepler problem (two massive bodies orbiting one another) might be one of the more interesting examples.

and i'd like to understand if it is true that even classical Hamiltonian can't be measured or observed.. because according to Demystifier.. the Hamiltonian in quantum can't be measured or observed.. is this only a quantum thing and not true in classical Hamiltonian or the same? and Why?

The Hamiltonian is a mathematical function, so of course it cannot be measured or observed, any more than you could measure/observe the quadratic formula. You can differentiate it, you can do algebra on it, you can evaluate it for particular values of its parameters, you can write it down and talk about it, you can do calculations on it that lead to relationships between its parameters.