- #1

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If that is so, then, for a classical system, is there only one path in the phase portrait between two states

*A*and

*B*over a fixed time interval from

*t*to

_{i}*t*?

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- Thread starter crastinus
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- #1

- 67

- 7

If that is so, then, for a classical system, is there only one path in the phase portrait between two states

- #2

wrobel

Science Advisor

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I believe that the best way to grasp such things is to use theorems.

Consider an initial value problem $$\dot x=v(x),\quad x(0)=\hat x,\quad x=(x^1,\ldots,x^m)\in\mathbb{R}^m.\qquad (*)$$ We also assume for simplicity that the vector field ##v## is smooth in ##\mathbb{R}^m##. This assumption is sufficient to guarantee existence and uniqueness of the solution by the Cauchy theorem. The phase space is ##\mathbb{R}^m##.

Let ##g^t(\hat x)## stand for solution to problem (*)

Theorem. Let ##K\subset \mathbb{R}^m## be any bounded open domain. Then there exists a positive constant ##T## such that the solution ##g^t(\hat x)## is defined for all ##t\in[-T,T]## and for all ##\hat x\in K##. Moreover (this is the answer to your question) for any ##t\in[-T,T]## the mapping ##\hat x\mapsto g^t(\hat x)## is a diffeomorphism between ##K## and ##g^t(K)##.

If ##t,s,t+s\in[-T,T]##then ##g^{t+s}(\hat x)=g^t(g^s(\hat x))##

Consider an initial value problem $$\dot x=v(x),\quad x(0)=\hat x,\quad x=(x^1,\ldots,x^m)\in\mathbb{R}^m.\qquad (*)$$ We also assume for simplicity that the vector field ##v## is smooth in ##\mathbb{R}^m##. This assumption is sufficient to guarantee existence and uniqueness of the solution by the Cauchy theorem. The phase space is ##\mathbb{R}^m##.

Let ##g^t(\hat x)## stand for solution to problem (*)

Theorem. Let ##K\subset \mathbb{R}^m## be any bounded open domain. Then there exists a positive constant ##T## such that the solution ##g^t(\hat x)## is defined for all ##t\in[-T,T]## and for all ##\hat x\in K##. Moreover (this is the answer to your question) for any ##t\in[-T,T]## the mapping ##\hat x\mapsto g^t(\hat x)## is a diffeomorphism between ##K## and ##g^t(K)##.

If ##t,s,t+s\in[-T,T]##then ##g^{t+s}(\hat x)=g^t(g^s(\hat x))##

Last edited:

- #3

- 67

- 7

Thanks again!

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