what does it mean for trajectories to be along a manifold?

It means that the phase space that the trajectory is plotted on is a differentiable manifold.

I don't get the question?

So describe how it arose.

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>>8263959
how do the dimensions of the manifold and the phase space compare? is the former always less than or equal to the latter?

>>8263955
It means that points on the trajectory are locally differentiably connected through a moving subspace which is of a smaller dimension than the full space the manifold lies in, at least for "almost all" points on said trajectory.

>>8263955
Phase spaces are symplectic manifolds.

>>8264180
More specifically for any point on the trajectory we can build an ON coordinate system for which one basis vector is the direction of the trajectory there for at least one (other) basis vector must exist an epsilon >0 so that no other points on the trajectory lies on the line segment [-epsilon,+epsilon] * basis vector.

Note: I'm currently pulling this out of my ass.

>>8264180
This

>>8263979
the phase space IS the manifold. And we want it to be a differentiable manifold so we can do calculus on it (read differential geometry if you don't understand this)

>>8264213
>the phase space IS the manifold
this just made me very confused. i thought the phase space was just [math]\mathbb{R}^{\text{# of state variables your system has}}[/math] and the manifold was the subspace of that where the trajectories actually go.

like the system
[eqn]
\frac{\text{d}}{\text{d}t}

\left(
\begin{array}{c}
x_1 \\
x_2 \\
x_3 \\
\end{array}
\right)

=

\left(
\begin{array}{ccc}
10 & 1 & 1 \\
1 & -1 & 0 \\
1 & 0 & -1 \\
\end{array}
\right)

\left(
\begin{array}{c}
x_1 \\
x_2 \\
x_3 \\
\end{array}
\right)

+

\left(
\begin{array}{c}
1 \\
0 \\
0 \\
\end{array}
\right)

u
[/eqn]
has a three-dimensional phase space (i think?) but the trajectories all lie on a 2d subspace of that specifically the one where y = z

i'm really out of my depth so forgive me if this is retarded.

>>8264248
whoops that 10 should be -10

>>8264248
[math]\left( {{\mathbb{R}^{2n}},\omega } \right)[/math] with [math]\omega = \sum\limits_{i = 1}^n {{{\operatorname{dx} }_i} \wedge \operatorname{d} {p_i}} [/math] is a symplectic manifold.

>>8264293
wish i knew what that meant

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>>8264293
>guy asks babby-tier undergrad question about phase space and manifolds
>give him lagrangian fibrations

>>8264451
That is the most basic example of a symplectic manifold, and generally the only one used in undergrad mechanics/diffeq courses.

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>>8264248
No the trajectory (solution of the equation) is a curve on the manifold.

Yea, the guy that's replying with differential forms, I don't think this poster is at this level yet. And you don't need to give such a complicated answer when the answer is really simple.

>>8264479
>Yea, the guy that's replying with differential forms, I don't think this poster is at this level yet.

He is asking a question about manifolds yet doesn't know what a differential form is?

>>8263959
Never even heard of a phase space before. State space and the state of a dynamical system though. Phase space sounds like something someone obsessed with Fourier transforms would use.

>>8264639
i think that's basically the configuration space

https://en.wikipedia.org/wiki/Configuration_space#Phase_space

this talks about the connection between the two, though the language is unintelligible to myself

>>8264479
what is the proper progression on these topics? i've taken linear algebra, diffeq, and calc II. from here do i go vector calculus -> differential forms -> differential topology?

>>8264663
Do vector calc, Real Analysis, Multilinear algebra, and then Analysis on Manifolds. Maybe some throw some differential geometry in there too. Topology and Topological manifolds wouldn't hurt actually.

>>8263955
>what does it mean for trajectories to be along a manifold?

Math...

Math that MAY be applicable to reality.