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Linear flow on the torus: Difference between revisions

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Revision as of 11:40, 12 June 2008 editPokipsy76 (talk | contribs)Extended confirmed users2,250 edits beginning  Revision as of 11:45, 12 June 2008 edit undoPokipsy76 (talk | contribs)Extended confirmed users2,250 edits correctingNext edit →
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A linear flow on the torus is a ] on the ''n''-dimensional ] A linear flow on the torus is a ] on the ''n''-dimensional ]
:<math>\mathbb{T}^n = \underbrace{S^1 \times S^1 \times \cdots \times S^1}_n</math> :<math>\mathbb{T}^n = \underbrace{S^1 \times S^1 \times \cdots \times S^1}_n</math>
which is represented by a linear differentuial equation with respect to the standard angular coordinates (θ<sub>1</sub>, θ<sub>2</sub>, ..., θ<sub>''n''</sub>), that is which is represented by the following differential equations with respect to the standard angular coordinates (θ<sub>1</sub>, θ<sub>2</sub>, ..., θ<sub>''n''</sub>):
:<math>\dot{\mathbf{\theta}}=A \mathbf{\theta}</math> :<math>\frac{d\theta_1}{dt}=\omega_1, \quad \frac{d\theta_2}{dt}=\omega_2,\quad \cdots, \quad \frac{d\theta_n}{dt}=\omega_n</math>
where ''A'' is a ''n''×''n'' ].

Revision as of 11:45, 12 June 2008

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A linear flow on the torus is a flow on the n-dimensional torus

T n = S 1 × S 1 × ⋯ × S 1 ⏟ n {\displaystyle \mathbb {T} ^{n}=\underbrace {S^{1}\times S^{1}\times \cdots \times S^{1}} _{n}}

which is represented by the following differential equations with respect to the standard angular coordinates (θ1, θ2, ..., θn):

d θ 1 d t = ω 1 , d θ 2 d t = ω 2 , ⋯ , d θ n d t = ω n {\displaystyle {\frac {d\theta _{1}}{dt}}=\omega _{1},\quad {\frac {d\theta _{2}}{dt}}=\omega _{2},\quad \cdots ,\quad {\frac {d\theta _{n}}{dt}}=\omega _{n}}