Hamiltonian Mechanics: why paths in state space never cross each other

In summary, the author is discussing how Hamilton's equations of motion for a system define a vector field that gives a vector for every q,p in phase space. If there is a point on the curve where it intersected itself, then it would mean that the vector tangent to the curve at that point would have two directions at once, which is mathematically impossible.
  • #1
dRic2
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I'm reading a book about analytical mechanics and in particular, in a chapter on hamiltonian Mechanics it says:

"In the state space (...) the complete solutionbof the canonical equations is pictured as an infinite manifold of curves which fill (2n+1)-dimensional space. These curves never cross each other. Indeed, such crossing would mean that two tangents are possible at the same point of the state space , but that is excluded because of the canonical equations which give a unique tangent at any point of the space."

I'm not following very well the argument of the author. Can someone help me, please ?

Thanks
Ric
 
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  • #2
Suppose paths could cross, and further suppose I prepare the system in the state at the crossing point. How does the state evolve? Which path does it take?
 
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  • #3
The canonical equations basically say from any point in phase space which direction you go. Each point has only one direction. If there is a crossing then there would be two directions to go from the same point.
 
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Dale said:
If there is a crossing then there would be two directions to go from the same point.
Adding to that, there would also be two directions in past time to come from. Alternate histories like that are not allowed in physics.
 
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Should I open a mini can of worms here and through the question whether the universe is deterministic or non deterministic. If the universe is non deterministic then we can certainly have paths in the state space crossing each other.
BUT since we are in the classical physics subforum and the book the OP is referring to is for classical Hamiltonian mechanics, the universe is always deterministic for classical physics. Not sure if we can say the same for quantum physics though.
 
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So basically, if I set a point, giving all the coordinates, according to Hamilton's equation I can then predict the motion, but if two or more paths intersected in that point, geometrically I wouldn't be able to choose one path: I would need an other information. But that contradicts Hamilton's equations which assure me that the motion is uniquely defined if I gave all the coordinates of that point. Did I get it ?

If I can bother a little more, the line just below this one says:

"The geometrical and analytical picture we get here is in complete analogy with the motion of a fluid"

Can you provide some more hints please?

Thank you very much
Ric
 
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The kind of differential equations that can describe some real-world physics always have a unique solution. I'm not sure, though, whether it could be possible to construct a sequence of Hamiltonian systems with an increasing number of degrees of freedom, such that in the limit of infinite dimensions the equations of motion fail to have a unique solution.
 
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dRic2 said:
So basically, if I set a point, giving all the coordinates, according to Hamilton's equation I can then predict the motion, but if two or more paths intersected in that point, geometrically I wouldn't be able to choose one path: I would need an other information. But that contradicts Hamilton's equations which assure me that the motion is uniquely defined if I gave all the coordinates of that point. Did I get it ?

If I can bother a little more, the line just below this one says:

"The geometrical and analytical picture we get here is in complete analogy with the motion of a fluid"

Can you provide some more hints please?

Thank you very much
Ric

That's overcomplicating it just a little, although it is correct. Here's what I would say:

Hamilton's equations of motion for a system define a vector field that gives a vector for every value of (q,p) in phase space. When supplied with initial conditions, Hamilton's equations can be solved for a curve that represents the motion of the system in phase space, in the sense that we can say that the state of the system is represented as a particle that moves along this curve with its location on the curve parameterized by time.

The vectors defined by Hamilton's equations are tangent to the curve. If there was a point on the curve where it intersected itself, then it would mean that the vector tangent to the curve at that point would have two directions at once, which is mathematically impossible (how can a vector have two directions?).

This article might help to clarify:
https://en.wikipedia.org/wiki/Integral_curve
 
  • #10
Thanks for the reply, but I don't find it that complicated :biggrin:

BTW I solved this other issue some days ago and forgot to mention
dRic2 said:
"The geometrical and analytical picture we get here is in complete analogy with the motion of a fluid"

Can you provide some more hints please?

It think all my doubts regarding the question have been answered. Thank to you all :)
 

Related to Hamiltonian Mechanics: why paths in state space never cross each other

1. Why is it important to understand Hamiltonian Mechanics?

Hamiltonian Mechanics is a fundamental theory in classical mechanics that describes the motion of particles in a system. It is important because it allows us to predict the behavior of physical systems and has applications in fields such as physics, engineering, and astronomy.

2. What is the difference between Hamiltonian Mechanics and Lagrangian Mechanics?

Hamiltonian Mechanics and Lagrangian Mechanics are two different approaches to studying classical mechanics. While Lagrangian Mechanics uses generalized coordinates to describe the motion of particles, Hamiltonian Mechanics uses the concept of energy to describe the system. Hamiltonian Mechanics also takes into account the time evolution of a system, while Lagrangian Mechanics does not.

3. Why do paths in state space never cross each other in Hamiltonian Mechanics?

In Hamiltonian Mechanics, the state of a system is described by a set of coordinates and their corresponding momenta. These coordinates and momenta are related through Hamilton's equations, which dictate the evolution of the system over time. Since the equations are deterministic, the paths in state space can never cross because this would imply multiple possible outcomes for the same initial conditions, which violates the principles of classical mechanics.

4. Can Hamiltonian Mechanics be applied to systems with multiple particles?

Yes, Hamiltonian Mechanics can be applied to systems with multiple particles. In this case, the Hamiltonian function takes into account the coordinates and momenta of all the particles in the system, and Hamilton's equations describe the evolution of the entire system. This allows for a more comprehensive understanding of the behavior of complex systems.

5. What are some real-world applications of Hamiltonian Mechanics?

Hamiltonian Mechanics has numerous real-world applications, including predicting the motion of planets and satellites, analyzing the behavior of mechanical systems such as pendulums and springs, and understanding the dynamics of chemical reactions. It is also used in fields such as quantum mechanics, optics, and fluid mechanics.

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