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Published from the Principia Machina source

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	{{DISPLAYTITLE:The Hamiltonian}}		{{DISPLAYTITLE:The Hamiltonian}}
	{{Technical\|Hamilton's principle/Mathematics/The Hamiltonian and Hamilton's equations}}		{{Technical\|Hamilton's principle/Mathematics/The Hamiltonian and Hamilton's equations}}
			{{Hamilton's principle map}}

	This page explains a reformulation of mechanics in terms of position and		This page explains a reformulation of mechanics in terms of position and

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	Apply the swap from		Apply the swap from
	[[Hamilton's principle/Explainers/Reading a curve two ways]] and you get a		[[Hamilton's principle/Explainers/Reading a curve two ways\|Reading a curve two ways]] and you get a
	description tracking '''where it is''' and '''its momentum''' instead.		description tracking '''where it is''' and '''its momentum''' instead.

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	Pushed to its conclusion, that search produces a single equation replacing all the		Pushed to its conclusion, that search produces a single equation replacing all the
	others, and it turns out to describe waves. It is at		others, and it turns out to describe waves. It is at
	[[Hamilton's principle/Explainers/Waves and rays]].		[[Hamilton's principle/Explainers/Waves and rays\|Waves and rays]].

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	others, and it turns out to describe waves. It is at		others, and it turns out to describe waves. It is at
	[[Hamilton's principle/Explainers/Waves and rays]].		[[Hamilton's principle/Explainers/Waves and rays]].

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hamiltons-principle>HPVBuild: Initial import of B-12

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Initial import of B-12

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{{Technical|Hamilton's principle/Mathematics/The Hamiltonian and Hamilton's equations}}

This page explains a reformulation of mechanics in terms of position and
momentum. It uses no mathematics. The technical page it goes with is named above.

== A different pair of quantities ==

[[File:OPL-HP-F-11.png|thumb|right|400px|Phase portrait of a pendulum.]]

The usual description of a moving object tracks '''where it is''' and '''how fast
it is going'''.

Apply the swap from
[[Hamilton's principle/Explainers/Reading a curve two ways]] and you get a
description tracking '''where it is''' and '''its momentum''' instead.

For a simple object momentum is just mass times speed, so this looks like a
relabelling. It is not, and the reason is what happens to the equations.

== Fewer steps, more equations ==

[[File:OPL-HP-F-24.png|thumb|right|400px|Lagrangian and Hamiltonian descriptions compared.]]

Newton's laws, and everything in this volume so far, involve
'''accelerations''' — how the speed is changing. Acceleration is a rate of change
of a rate of change. Two steps removed from position.

In the new description, that never happens. You get twice as many equations, but
each involves only '''one''' rate of change.

The trade is real. Equations with two steps of change are harder to handle,
harder to solve numerically, and harder to reason about, than twice as many with
one step each.

The two forms have the same content. Neither knows anything the other does not.

== A space where a point is a whole state ==

Here is the change that actually matters.

In the usual description, knowing where an object is does not tell you what it
will do. A ball at a given place might be rising or falling. Position alone is not
the state.

In the new description, position and momentum '''together''' are the state, and
they are treated as coordinates of equal standing. Every possible state of the
system is a single point in a space with both.

That space is called '''phase space'''.

The consequence is the useful one: '''a single point in phase space determines the
entire future.''' There is no ambiguity left over, because the information that
was missing is now part of the location.

== Paths that never cross ==

From that follows a fact with real force.

Trajectories in phase space '''never cross'''.

If two trajectories met at a point, that point would be one state with two
futures. Impossible, because the state fixes the future completely.

So the motion of a system becomes a flow, like a fluid, with streamlines that
never meet. Every point moves, nothing collides, nothing merges.

This gives you a way to reason about a system without solving anything.
Trajectories are hemmed in by their neighbours; regions of phase space get carried
along and deformed but cannot be torn.

== Reading off the answer ==

For a system where energy is conserved, the payoff is immediate.

Energy has a fixed value along any trajectory. So each trajectory stays on a
single surface of constant energy.

Draw those surfaces and you have the shape of the motion, without solving
anything.

For a pendulum swinging gently, the surfaces are closed loops. A trajectory going
round a closed loop returns to where it started: the motion repeats. That is the
swing.

For a pendulum with enough energy to go over the top, the surfaces are not closed
loops but curves running off in one direction. The pendulum keeps rotating the same
way and never comes back.

Between the two lies a special curve separating the swinging region from the
spinning one. On it, the pendulum takes infinitely long to creep up to the vertical
and balance.

'''All of that came from looking at the shape of the energy surfaces.''' No
equation was solved. That is the characteristic move of this description, and it is
why phase space is the right picture for asking what a system does in general
rather than what it does in one particular case.

== A caution about energy ==

The quantity that plays the leading role here is usually the energy, and it is
worth knowing when it is not.

For ordinary systems — masses, springs, gravity, fixed constraints — it is the
familiar energy, kinetic plus potential.

For systems where the constraints themselves are moving, it can differ. Something
is still conserved, and it is still the quantity that runs the equations, but
calling it the energy will mislead you.

'''Conserved and equal to the energy are two separate claims.''' Most treatments
merge them because for the standard examples they coincide.

== Where this leads ==

Phase space is not just tidier. It is the setting for most of what came next.

The idea that a region of phase space keeps its volume as it flows became the
foundation of statistical mechanics — the physics of systems with too many parts to
track individually.

The pairing of position with momentum became, in quantum mechanics, the pairing
that cannot be measured simultaneously to arbitrary precision.

And the search for descriptions in which the flow is as simple as possible leads
directly to the next page.

== Where to go next ==

Pushed to its conclusion, that search produces a single equation replacing all the
others, and it turns out to describe waves. It is at
[[Hamilton's principle/Explainers/Waves and rays]].

Hamilton's principle/Explainers/The Hamiltonian - Revision history

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