Mixing
Monoid in general dynamic system definition
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0
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I am a newbie in this field but what difference does taking monoid or group in the following definition of dynamic system make?
A tuple
beginequation
(T,M,phi)
endequation
is called dynamic system, where $T$ is additively written monoid (time), $M$ is a phase space and $phi$ is an evolution operator
beginequation
phi = Usubseteq Ttimes M rightarrow M
endequation
of the system.
I have found another stronger definiton in which $T$ is said to be additive group.
Does it matter? Is the addition necessarily commutative?
dynamical-systems monoid
asked Jul 22 at 7:44
Jan Filip
1033
add a comment |Â
up vote
0
down vote
favorite
I am a newbie in this field but what difference does taking monoid or group in the following definition of dynamic system make?
A tuple
beginequation
(T,M,phi)
endequation
is called dynamic system, where $T$ is additively written monoid (time), $M$ is a phase space and $phi$ is an evolution operator
beginequation
phi = Usubseteq Ttimes M rightarrow M
endequation
of the system.
I have found another stronger definiton in which $T$ is said to be additive group.
Does it matter? Is the addition necessarily commutative?
dynamical-systems monoid
asked Jul 22 at 7:44
Jan Filip
1033
1
I am at odds with $T$ being a group. How do you turn back the clock?
– scaaahu
Jul 22 at 7:54
I am confused with group structure too. It is the reason of the question actually. In two trustworthy reference books (in czech) written by experts, there is a groupt $T$ used.
– Jan Filip
Jul 22 at 8:15
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am a newbie in this field but what difference does taking monoid or group in the following definition of dynamic system make?
A tuple
beginequation
(T,M,phi)
endequation
is called dynamic system, where $T$ is additively written monoid (time), $M$ is a phase space and $phi$ is an evolution operator
beginequation
phi = Usubseteq Ttimes M rightarrow M
endequation
of the system.
I have found another stronger definiton in which $T$ is said to be additive group.
Does it matter? Is the addition necessarily commutative?
dynamical-systems monoid
asked Jul 22 at 7:44
Jan Filip
1033
I am a newbie in this field but what difference does taking monoid or group in the following definition of dynamic system make?
A tuple
beginequation
(T,M,phi)
endequation
is called dynamic system, where $T$ is additively written monoid (time), $M$ is a phase space and $phi$ is an evolution operator
beginequation
phi = Usubseteq Ttimes M rightarrow M
endequation
of the system.
I have found another stronger definiton in which $T$ is said to be additive group.
Does it matter? Is the addition necessarily commutative?
dynamical-systems monoid
asked Jul 22 at 7:44
Jan Filip
1033
edited Jul 22 at 7:49
asked Jul 22 at 7:44
Jan Filip
1033
asked Jul 22 at 7:44
Jan Filip
1033
asked Jul 22 at 7:44
Jan Filip
1033
1033
1
I am at odds with $T$ being a group. How do you turn back the clock?
– scaaahu
Jul 22 at 7:54
I am confused with group structure too. It is the reason of the question actually. In two trustworthy reference books (in czech) written by experts, there is a groupt $T$ used.
– Jan Filip
Jul 22 at 8:15
add a comment |Â
1
I am at odds with $T$ being a group. How do you turn back the clock?
– scaaahu
Jul 22 at 7:54
I am confused with group structure too. It is the reason of the question actually. In two trustworthy reference books (in czech) written by experts, there is a groupt $T$ used.
– Jan Filip
Jul 22 at 8:15
1
1
I am at odds with $T$ being a group. How do you turn back the clock?
– scaaahu
Jul 22 at 7:54
I am at odds with $T$ being a group. How do you turn back the clock?
– scaaahu
Jul 22 at 7:54
I am confused with group structure too. It is the reason of the question actually. In two trustworthy reference books (in czech) written by experts, there is a groupt $T$ used.
– Jan Filip
Jul 22 at 8:15
I am confused with group structure too. It is the reason of the question actually. In two trustworthy reference books (in czech) written by experts, there is a groupt $T$ used.
– Jan Filip
Jul 22 at 8:15
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
0
down vote
accepted
The difference is whether the time $t$ can only forward or backward as well. A group has inverses, a monoid does not need them. In particular, $mathbb R$ is a group, $[0, infty)$ (with addition) is not. If your dynamical system is reversible, you might want to use $mathbb R$, otherwise only $[0,infty)$.
answered Jul 22 at 8:15
Robert Israel
304k22201441
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
The difference is whether the time $t$ can only forward or backward as well. A group has inverses, a monoid does not need them. In particular, $mathbb R$ is a group, $[0, infty)$ (with addition) is not. If your dynamical system is reversible, you might want to use $mathbb R$, otherwise only $[0,infty)$.
answered Jul 22 at 8:15
Robert Israel
304k22201441
add a comment |Â
up vote
0
down vote
accepted
The difference is whether the time $t$ can only forward or backward as well. A group has inverses, a monoid does not need them. In particular, $mathbb R$ is a group, $[0, infty)$ (with addition) is not. If your dynamical system is reversible, you might want to use $mathbb R$, otherwise only $[0,infty)$.
answered Jul 22 at 8:15
Robert Israel
304k22201441
add a comment |Â
up vote
0
down vote
accepted
up vote
0
down vote
accepted
The difference is whether the time $t$ can only forward or backward as well. A group has inverses, a monoid does not need them. In particular, $mathbb R$ is a group, $[0, infty)$ (with addition) is not. If your dynamical system is reversible, you might want to use $mathbb R$, otherwise only $[0,infty)$.
answered Jul 22 at 8:15
Robert Israel
304k22201441
The difference is whether the time $t$ can only forward or backward as well. A group has inverses, a monoid does not need them. In particular, $mathbb R$ is a group, $[0, infty)$ (with addition) is not. If your dynamical system is reversible, you might want to use $mathbb R$, otherwise only $[0,infty)$.
answered Jul 22 at 8:15
Robert Israel
304k22201441
answered Jul 22 at 8:15
Robert Israel
304k22201441
answered Jul 22 at 8:15
Robert Israel
304k22201441
answered Jul 22 at 8:15
Robert Israel
304k22201441
304k22201441
add a comment |Â
add a comment |Â
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1
I am at odds with $T$ being a group. How do you turn back the clock?
– scaaahu
Jul 22 at 7:54
I am confused with group structure too. It is the reason of the question actually. In two trustworthy reference books (in czech) written by experts, there is a groupt $T$ used.
– Jan Filip
Jul 22 at 8:15