So. When the topic of symplectic manifolds first came up, I was at a disadvantage as compared to, for example, temur and Alpha. This was because I learned about these beasts in a purely abstract setting. It seems the following is always true: Supposing \(M\) to be an arbitrary manifold, the co-tangent bundle \(T^{\ast}M\) associated to it is of necessity symplectic (this sort of follows from a theorem of Darboux). Now suppose a "system" of \(n\) particles each with 3 degrees of freedom, then since no 2 particles can occupy the same position, the manifold \(M \subset R^{3n}\) is called "configuration space". Say it has dimension \(d\) This is fine. The co-tangent bundle associated to \(M\) is quite obviously of dimension \(2d\), and again by Darboux is endowed with the symplectic form \(\Omega = dp^i \wedge dq_i\). This follows from the easy demonstration that \(\Omega\) is closed, and the not-so-easy demonstration that it is maximal. Let's assume the truth of all the above (or not, your choice). So the bundle \(T^{\ast}M\) is referred to (by phys jocks) as "phase space". So, of course, this is where I start to unravel, being neither a physicist nor especially bright. The argument that phase space actually IS the co-tangent bundle is what? Is it simply that the Lagragian that sends configuration space to phase space takes vectors as argument at each point \(m \in M\), i.e. is a function on the bundle \(TM\)? I have a slighty less unhinged question about momentum maps, but that will have to wait.
I think you are right. It has to do with the Legendre transformation. I think if you want to go further it becomes simply the fact that Newton's laws are second order differential equations.
I really don't know if this is going to fly (I'm only a systems hacker, after all) but isn't the cotangent space related to the tangent space, in the same way as a cosine to sine relationship in complex transforms, where the cos() is a real and the sin() an imaginary term? The cotangent space represents Re{phase space} and the tangent space is its Im{} dual space...? Or not...
noodler: I'm afraid your post went straight over my head. Yes, the tangent space and the cotangent space are related, since, as vector spaces they are mutually dual. That is, every element in one sends all elements in the other to the real numbers. So what's with the real and imaginary parts? I don't get you. But anyway I was talking about the tangent and co-tangent BUNDLES. Now it is also true that the isomorphism induced by the above duality induces an isomorphism (a BUNDLE isomorphism) between these bundles. But I'm reasonably sure that this doesn't imply duality of bundles in the above sense AND - as far as I am aware, the sine and cosine functions are real-valued functions \(\mathbb{R} \to [-1,1]\). What does it mean for, say sin() to be an imaginary term?
I was talking about complex frequency space, you are talking about complex phase space. I just had this idea that there was a connection to the way the sin and cos terms behave, one or the other vanishes from the transform, conventionally the sin terms in an equation for time dependent waveforms,
noodler is the new account of Vkothii and is doing precisely as he used to do, waffling nonsense and jumbling together the results of Google searches for words in your original post so that it seems like he understands but in fact he doesn't have a clue. I imagine he put in 'cotangent' or 'tangent bundle' into Google and got back a page which talks about having some kind of \(E = TM \oplus T*M\) bundle such that \(X = X^{a}\partial_{a} + iX_{b}dx^{b}\) is a 'vector' in E and failed to understand that that is not a general definition or result but something specific to the article he's reading (or perhaps I've just spend too much time thinking about exactly that generalised bundle...). I can point you to a few examples of him trying and failing to convince people he grasps relativity or group theory. Just ignore him. I am not sure of the direct answer (Guest might, he knows these things) but I'll just ramble and hope something is of use. Given coordinates \(x^{a}\) on M then the induced coordinates on TM will be \(\partial_{a} = \frac{\partial}{\partial x^{a}}\) and on T*M \(dx^{a}\). That's just mathematical results/definitions. Now, suppose I wish to pick a different cotangent basis, ie working in a frame bundle (or co-frame... whatever), which I'm going to call \(dy^{a}\) and likewise for TM. Assume this allows me to build \(\Omega\) as you defined, \(\Omega = dx^{a}\wedge dy_{a}\), which defines, in terms of components etc, a non-degenerate map \(\Omega : TM \times TM \to \mathbb{R}\). Therefore, given some \(X \in TM\) we have that \(\Omega(X,\cdot) : TM \to \mathbb{R}\). Since \(\Omega(X,\cdot)\) takes elements of TM to the Reals it is equivalent to some 1-form (see 'musical isomorphisms' on Wikipedia). Therefore \(\Omega(X,\cdot) \sim \xi \in T*M\). So what happens if I pick my X to be \(\frac{\partial}{\partial x^{a}}\)? Then \(\Omega(\frac{\partial}{\partial x^{a}},\cdot) \sim dy^{a}\) by the component form of \(\Omega\) I've assumed. Therefore I have a manifold coordinate system \(x^{a}\) and a fibre with basis \(dy^{a}\). So what about physics? Given some Hamiltonian H, a function of the position of the objects which the system is built from, \(q^{a}\) and their velocities \(\dot{q}^{a}\), so \(H = H(q,\dot{q})\). The definition of the momentum conjugate to q, p, is \(p_{b} = \frac{\partial}{\partial \dot{q}^{b}}H\), such that \(\{q^{a},p_{b}\} = \delta^{a}_{b}\) where { , } is the Poisson bracket. I suspect, but can't justify much, that due to the way in which a Hamiltonian is defined and how p is defined from it via q that by construction q and p are such that \(dq^{i} \wedge dp_{i}\) is always well defined and by definition p is the momentum associated to q. Therefore, given H in terms of q and q' you can define M and p. Given p you can define \(\Omega\) and given \(\Omega\) you can define your phase space. Is that making much sense?
Comment: I note that Alphanumeric has mentioned some 1-form that "takes elements of TM to the reals". And that I was pointing out that, in the frequency domain, time-dependent transforms take elements to the reals by being complex functions with real and imaginary coefficients. That is all I was doing. I note that he then spends an entire paragraph dissing my opinion and assassinating my character. My observation on that, and most of his previous dissing, is that it's because of a desperate need, whose nature need not concern us, he has.
It is practically the definition of a 1-form. Doing frequency decompositions takes functions to functions, real or complex doesn't come into it. A Fourier transformation or Fourier decomposition is a functional map, given an element of \(\Xi(M)\) you get another element of \(\Xi(M)\). A desperate need to do what? I addressed the original question, I didn't just post to slag you off. Why is it cranks think that when I correct cranks they think its because there's something wrong with me and never consider it might be them? Your reply to the original post was wrong and irrelevant. Your posts in other threads have been nonsense and incorrect. The fact I correct you all the time is not because of some issue I have but because you are wrong all the time! I don't correct people like Rpenner, Ben or Guest, despite them talking about high level physics or mathematics. If I had some issue where I have to put down anyone who talks about complicated stuff surely I'd try to put down them? But I don't. Why? Because, unlike you, they actually know their stuff. When pressed to back up their explanations they can. Infact, I know for a fact the stuff they post here is a great deal simpler than the most complicated stuff they know. You, on the other hand, can't do anything like that and so have to lie and post bull to simply keep up with discussions involving QuarkHead, Ben, myself etc. If you don't want me slagging you off, stop being wrong. Stop being wrong by learning something rather than Google searching.
That's fascinating. What about a function like \( f(t)\; =\; Ae^{\sigma t} e^{j(\omega t+ \alpha)}\;\; \) ?? You're saying real or complex doesn't come into this either? Frequency analysis uses complex functions too, are you saying something else? In which case, what the hell exactly? And lets see you point to a few posts where I demonstrate how little I understand relativity, Mr "I know everything". The desperate need I reference is the one you display so transparently, in your previous post, to convince everyone of something, perhaps how "normal" a person you are... /ha ha ha ha ha ha... etc
Well done on failing to understand what I said. 1-forms are such that they take vectors to a field, ie \(\lambda : TM \to \mathbb{F}\) for a given field. What that field is is not relevant to the general definition of a 1-form. Yes, sometimes \(\mathbb{F} = \mathbb{R}\) and other times \(\mathbb{F} = \mathbb{C}\) but that's just particular cases. A Fourier transformation maps a function, f(x), to a function \(\hat{f}(k)\), ie \(\mathcal{L} : f(x) \to \hat{f}(k)\) by \(\hat{f}(k) = \int_{-\inf}^{+\inf}e^{ikx}f(x) \,dx\). Therefore \(\mathcal{L}\) is a functional operator \(\mathcal{L} : \Xi(M) \to \Xi(M)\) where \(\Xi(M)\) is the space of smooth functions on M (up to particular technicalities due to Schwarz spaces). A Fourier decomposition is similar, in that it provides an infinite set of mode coefficients for a function in terms of a complete basis. Given a function f(t) and a complete basis \(b_{n}(t)\) then a modal decomposition is writing f(t) as \(f(t) = \sum_{n} c_{n}b_{n}(t)\) where \(c_{n}\) is a constant. Typical Fourier decompositions use the basis \(\sin n\pi t\) and \(\cos n\pi t\) such that \(f(t) = \sum \left( c_{n}\sin(n \pi t) + d_{n} \cos(n \pi t)\right)\). Therefore you can view the decomposition as picking out parts of the function in a particular basis, so acts as \(\Xi(M) \to \Xi(M)\) or providing the cooefficnets c_n or d_m which take values in a field, typically \(\mathbb{R}\) or \(\mathbb{C}\) again. As no point have I referred to vectors. The decomposition can be done on vectors too and any other function over a finite interval but that doesn't mean decomposition is related to 1 forms. How about all the posts you made in this thread : http://www.sciforums.com/showthread.php?t=97818 . You didn't even understand Lorentz transformations. Then there's your incoherent rambling here : http://www.sciforums.com/showthread.php?t=97493 And there's two threads in the Feedback forum of you whining about how myself and DH keep saying you're wrong. So me correcting you on something you got wrong is an attempt by me to convince people I'm a 'normal' person? Excellent non-sequitor. Personally I don't care if people think I'm a little off or that I'm a 'normal' person. My corrections of your attempts to BS physics are independent of my personality. I am not trying to be all warm and fuzzy, I'm correcting someone who has gotten something wrong in an area I know a decent amount about. In fact, symplectic geometry is something I have spent the last year doing. I have a great deal of experience in geometries which do combine the tangent and cotangent spaces into a single generalised fibre for space-time. Your first post talks about combining the two of them and so you're talking about something I know a lot about and I can say its irrelevant to QuarkHead's original question. Now I'm more than willing to discuss why I know about such things and to explain how they arise in string theory, either in this thread or a new one (should QH not wish the discussion to be here), as I have nothing to hide. Unlike you when I say "I've read about that" I mean published papers. When I say "I have work in that area" I mean it. So, if you want to dazzle us with your grasp of this area of differential geometry just say and I'll start a new thread and link to a few papers which talk about how its used in string theory in relation to \(TM \oplus T*M\) bundles.
Well, there you go, more incoherent rambling about "not understanding" and "being corrected", where there is no actual evidence of either. You appear to be hallucinating. Yes, I kept saying that too, maybe you and DH are wrong, and my insisting you were both mistaken about some detail (that need not concern us, etc) means I can assume, in the face of no contradicting evidence that your claims are based on more hallucinations. Oh goody. Lets go up to the lab, and see what's on the slab, shall we, just then?
Oh dear! I opened a thread with an honest question about a subject that has been confusing me for a while. What all do I get? Historical wrangles about which I know nothing, and care less. Actually that's not completely true. Alpha gave me a new way to think about this, and I thank him for that. So, as a result, I think I see my way ahead clear, at least until I get to the momentum map, but that will will have to wait.
The many pseudo'd threads of your previous account 'Vkothii' is plenty of evidence. Your Rubik's cube thread is just you rambling. Can you point me to a single post of yours where you show a working understanding of relativity? Another non-sequitor. Unfortunately your inability to post many coherent sentences means I am unable to tell if you accept my offer to discuss the topic of generalised tangent bundles, which your original post eluded to, or you're just trying another evasion. Funny how you completely ignored all parts of my post where actually did some mathematics, where I showed a working understanding. I explained why Fourier methods are not related to what QH asked about, illustrating it with examples and you responded to none of that. Always happens, I put my maths where my mouth is and you ignore it because you can't put your maths where your mouth is. Tell you what, I'll start a thread on the differential geometry used in string theory some time in the next day or so and you can then dazzle us all with the insightful comments you're sure to post.
Funny? I thought it was more like absolutely hilarious. And for your ignorant information, I was stating that in the complex frequency domain you use the real and imaginary parts of a time-dependent waveform--of course the waveform doesn't have imaginary parts, it has real physical parts. And who would bother to even think that a complex phase space of momentum products was remotely connected to group and phase velocity in waveforms?? I mean, jeez louise.... /resorts to incoherent babbling
Yeah, and I'll tell Alpha and noodler/ Vkothii what....... I asked a reasonable question in good faith. temur gave me a steer, bless him, and Alpha helped me get my head straight, for which I am grateful. noodler confused me, but I tried to respond constructively. But, due to these uninteresting spats between noodler and Alpha, I'm afraid this is a dead thread as far as I am concerned. Shame, really, as it is a delicate (i.e interesting) subject in differential geometry as applied to physics. For the record, not that anyone would be interested, I think I have answered all my own original queries. Sorry, but I feel somewhat aggrieved.
Don't get me wrong, I think the interactions are fascinating, just fascinating. That you ask a question about the subject of a phase space, then get confused when someone suggests that frequency is a physical phenomenon, which is a prioiri equipped with phase. Of course, this is probably just a random cosmic variable, we can ignore the fact that there is a phase velocity or a group velocity because frequency and complex amplitudes aren't related to systems of particles in motion, except in a random, incoherent sort of way...
Perhaps I'm too late, but I wanted to say something on the off chance you're still interested. The way I think about the question you raised is by a simple example. If I imagine a particle moving on a manifold with metric (M,g) as a function of time t, then the simplest Lagrangian which describes otherwise free motion (except for being confined to M) is \( L(x,\dot{x}) = \frac{m}{2} g(x)_{i j} \dot{x}^i \dot{x}^j \) where g is the metric on M, m is the particle mass, and \( \dot{x}^i = \partial_t x^i \). This is the generalization of the free particle particle Lagrangian in flat space. Physics people define the canonical momentum to be \( p_i = \frac{\partial L}{\partial \dot{x}^i} \) and we have \( p_i = g_{i j} \dot{x}^j \) in this case. This canonical momentum is the generalization of ordinary momentum to curved manifolds or non-cartesian coordinates. I've chosen the subscript appropriately, and the point is that this object p lives in the cotangent space at x. In fact, this is what physicists usually think of as lowering the index in relativity. But all we're doing is using a fixed special 2-tensor, namely the metric, to define a mapping from tangent vectors \( \dot{x}^i \)to cotangent vectors \( p_i \). This special tensor is encoded in the choice of a special function, the Lagrangian, on the tangent bundle. The point is that the choice of Lagrangian provides a natural way to pass from the tangent bundle to the cotangent bundle. When the Lagrangian describes ordinary motion on a manifold with metric, then the pairing is nothing but the usual map from \(TM \) to \( T^* M \) provided by the metric. The configuration space, the space of \( x \) and \(\dot{x} \) is \( TM \). Phase space, where we use \( x \) and \( p \), is the cotangent bundle \( T^* M \) because \( p \) is naturally a covector. In general, the Lagrangian provides the necessary mapping from \( TM \) to \( T^* M\) (assuming it is non-degenerate and nice).
