User giuseppe tortorella - MathOverflow

Is there a preferable convention for defining the wedge product?

2011-02-04T18:17:41Z

There are different conventions for defininig the wedge product $\wedge$.

In Kobayashi-Nomizu, there is $\alpha\wedge\beta:=Alt(\alpha\otimes\beta)$, in Spivak, we find $\alpha\wedge\beta:=\frac{k!l!}{(k+l)!}Alt(\alpha\otimes\beta)$, where $\alpha$ and $\beta$ are any forms of degree $k$ and $l$ respectively, and $Alt(\cdot)$ take the alternating part of the tensor.

But, is there a rationale to prefer one of them among the others?

If not, what do you prefer? and for what reason?

What foliations are symplectic foliations?

2013-01-22T17:41:18Z

On a manifold $M$, let $\mathcal F$ be a foliation having even-dimensional orientable leaves. I was wondering under what hypothesis I can state that $\mathcal F$ is the symplectic foliation of a Poisson structure on $M$?

Sophus Lie on the symplectic foliation theorem

2013-01-07T16:04:12Z

Given a Poisson manifold $(P,\{\cdot,\cdot\})$ , its characteristic distribution $\mathcal C$ is the singular tangent distribution on $M$ generated by the Hamiltonian vector fields,i.e. $$\mathcal C=\textrm{span}\{X_f:=\{f,\cdot\}\mid f\in C^\infty(M)\}.$$

The Symplectic Foliation Theorem states that:

$\mathcal C$ is completely integrable à la Sussmann-Stefan
each integral manifold of $\mathcal C$ brings a unique symplectic Poisson structure such that the immersion map is a Poisson morphism.

In "The Local Structure of Poisson Manifolds" A. Weinstein attributes the Symplectic Foliation Theorem to Sophus Lie for Poisson manifolds having constant rank (citing precisely the Second Section of "Theorie der Transformationsgruppen") and to A.A.Kirillov and R.Hermann in the general case.

I would like to learn more how this problem is tackled by Lie, and if there are some modern expositions (preferibly in English) of Lie's approach to this theorem?

Answer by Giuseppe Tortorella for Old books still used

2013-01-07T10:47:14Z

Many systematic introductions to the foundations of the edifice of Differential Geometry appeared in the sixties, and they are useful references even today. Some of them are:

Lang, Introduction to Differentiable Manifolds, 1962;
Helgason, Differential Geometry and Symmetric Spaces, 1962;
Kobayashi, Nomizu, Foundations of Differential Geometry, 1st Vol 1963, 2nd Vol 1969;
Sternberg, Lectures on Differential Geometry, 1964;
Bishop, Crittenden, Geometry of Manifolds, 1964;

Answer by Giuseppe Tortorella for A good primer for geometric quantization.

2012-09-10T12:10:38Z

There is "Geometric Quantization: A Crash Course" by Eugene Lerman.

Given a vector field all of whose integral curves are closed, is the period a smooth function?

2012-08-06T07:19:29Z

Disclaimer: The original question consisted of two parts. The first one has been answered negatively (see below the answers of Sam Lisi and Alejandro). It remains the second one.

Background
I am reading about the energy-period relation for Hamiltonian Systems.
In Weinstein's formulation (cf. Abraham, Marsden, Foundations of Mechanics 2nd Ed, page 198) this relation amounts to:

$(\ast)$ Given an Hamiltonian system $(M,\omega, X_H)$, let be $\Phi$ the flow of $X_H$ and $\text{per}:=\{(t,x)\in\mathbb R\times M\mid\Phi(t,x)=x\}.$
If $N$ is a smooth submanifold contained in $\text{per},$ then $\left.dt\wedge dH\right|_N=0,$ i.e. $t=t(H)$ on $N,$ (the period depends only on the energy.)

Question
In Guillemin, Stenberg, Geometric Asymptotics, between pages 170-171, I have additionally found that, when all integral curves of $X_H$ are periodic, we can take $N=\text{per}$ in $(\ast),$ which should mean that in such a case $\text{per}$ is a smooth submanifold of $\mathbb R\times M.$

In order to justify this last point I was wondering myself:

If $X$ is a non singular vector field on $M,$ all of whose integral curves are periodic, and $\tau(p)$ denotes the period of the orbit through $p,$ then $\tau:M\to\mathbb R$ is smooth?

otherwise, how to prove that in such a case $\text{per}$ is a submanifold?

What I have tried about point 2
Probably I am missing something because my guess is that if there were a principal bundle structure $(M,p,X,\mathbb S^1)$ such that the $\mathbb S^1$-orbits are the trajectories of $X$ then the period $\tau:M\to\mathbb R$ should be smooth because of the relation $\zeta=\tau X_H,$ where $\zeta$ is the infinitesimal generator of the action.
But I don't know how to proceed without this additional hypothesis.

Edit1 (After Sebastian's answer about point 1): As illustration of my difficulties with point 1, I imagine that $M$ is the Moebius band $[0,1]\times\mathbb R/\sim$ and $X=\frac{\partial}{\partial x}$ then the period is $$\tau([(x,y)]_{\sim})=\begin{cases}1&\text{if }y=0\\2&\text{if }y\neq 0\end{cases}$$

About a letter by Richard Palais of 1965.

2012-08-15T08:53:24Z

In Cushman and Bates, Global Aspects of Classical Integrable Systems, 1997, I have read

In a widely circulated but unpublished letter in 1965, Palais explained the symplectic formulation of Hamiltonian mechanics.

I would like to know if, in the meanwhile, this letter was made available.

On the smooth structure of the spaces of $k$-jets

2011-07-21T13:52:35Z

I was asking myself, if the following list of conditions is sufficient to determine the usual smooth structure on the spaces of $k$-jets.

the map $j^k f:M\ni x\to j_x^k f\in J^k(M,N)$ is smooth, for any $f\in C^{\infty}(M,N)$;
the map $(\alpha,\beta):J^k(M,N)\to M\times N$, defined by $(j^k_x f)\mapsto (x,f(x))$, is a smooth summersion, for any smooth manifolds $M$ and $N$;
the composition map $\gamma:J^k(N,O)\times_{N} J^k(M,N)\to J^k(M,O)$, defined by $(j^k_{f(x)} g,j^k_x f)\mapsto j^k_x(g\circ f)$, is smooth, for any smooth manifolds $M$,$N$ and $O$, (here $J^k(M,N)\times_{N} J^k(M,N)$ is the fiber product of $\beta:J^k(M,N)\to N$ and $\alpha:J^k(N,O)\to N$);
the map $(\alpha,\beta)^{-1}(U,V)\to J^k(U,V)$ defined by $j^k_x f\mapsto j^k_x(f|_{U\cap f^{-1}(V)})$ is a smooth isomorphism, for any open subsets $U\subset M$, $V\subset N$;
for any open subsets $U\subset \mathbb{R}^m$ and $V\subset \mathbb{R}^n$, the map $J^k (U,V)\to U\times V\times \bigoplus_{i=1}^k{L^i_{sym}(m,n)}$, given by $j^k_x f\mapsto (x,f(x),Df(x),\ldots,(D^kf)(x))$, is a smooth isomorphism, (here $L^i_{sym}(m,n)$ is the vector space of the $\mathbb{R}^n$-valued symmetric $k$-multinear maps on $\mathbb{R}^m$).

Probably it is not sufficient, or redundant, but, in such a case, I would know if there is in the literature such a kind of characterization.

My question is: Once prescribed the usual smooth structure on the $J^k(U,V)$, for arbitrary open sets in euclidean spaces $U$ and $V$ (as in point 5), what kind of conditions are sufficient to uniquely determine the usual smooth structure on $J^k(M,N)$ for all other smooth manifolds $M$ and $N$?

Answer by Giuseppe Tortorella for Generating functions and Lagrangian submanifolds

2012-06-13T20:01:50Z

When $(M_1,\omega_1)$ and $(M_2,\omega_2)$ are symplectic manifolds then we endow $M_1\times M_2$ with the symplectic form $\omega_1\ominus\omega_2:=\pi_1^\ast\omega_1-\pi_2^\ast\omega_2,$ (where $\pi_i:M_1\times M_2\to M_i$ denotes the projection on the $i$-th factor.)

Let $\Lambda$ be an arbitrary Lagrangian submanifold of $M_1\times M_2,\omega_1\ominus\omega_2),$ and $i:\Lambda\to M_1\times M_2$ the inclusion map.
Fixed $p\in\Lambda$, for any primitive $\theta$ of $\omega_1\ominus\omega_2$ in a neighborhood of $p,$ there exists a smooth local function $S$ around $p$ such that $dS=i^\ast\theta.$ (Because of Poincaré Lemma and $0=i^\ast(\omega_1\ominus\omega_2)=i^\ast d\theta=d(i^\ast\theta).$)
Such a function $S$ is called generating function for $\Lambda,$ and, being only locally defined, it depends on the choice of $\theta.$

To be more specific:
In your context $\omega_i=d\xi_i\wedge dx_i$ is the canonical $2$-form on $M_i=\mathbb{R}^{2n},$ for $i=1,2,$ and $\Lambda$ is the image of $i:(u,f(u,v),g(u,v),v)\in\mathbb{R}^{2n}\to(u,v)\in\mathbb{R}^{4n}.$
As remarked above the local generating functions for $\Lambda$ are sensitive to our choice of the local primitive $\theta_i$ of $\omega_i.$

Let us choose $\theta_1=\xi_1 dx_1$ and $\theta_2=-x_2d\xi_2.$
Then the corresponding generating function is determined (up to a constant) by $dS=i^\ast(\theta_1\ominus\theta_2)=fdu+gdv,$ i.e.: $f=\frac{\partial S}{\partial u},\ g=\frac{\partial S}{\partial v}.$

Answer by Giuseppe Tortorella for References for the Poincaré-Cartan forms

2012-05-15T16:21:07Z

I think that you could appreciate "Methods of Differential Geometry in Analytical Mechanics" by P.Rodriguez and M.deLeon (this is a link)

Apart from its intrinsec interest as reference for both the constructions of differential geometry and the geometrization of Lagrangian/Hamiltonian mechanics, in particular, if you are looking for the role played by the Poincaré-Cartan forms in mechanics, you could find it in their Chapters 2 (for the canonical almost-tangent structure on $TQ$) and 9 (for the Lagrangian Mechanics).

There you would find that to any (Lagrangian) function $L$ on $TQ,$ there are associated the Poincaré-Cartan forms $\theta_L:=J^\ast dL$ and $\omega_L:=d\theta_L,$ where $J:T(TQ)\to T(TQ)$ is the vertical endomorphism associated to the canonical almost tangent structure on $TQ.$

When the Lagrangian is not degenerate then $\omega_L$ is non degenerate, and the Euler-Lagrange vector field $\xi_L$ is the hamiltonian vector field w.r.t. $\omega_L$ with Hamilton function $E_L:=\mathcal{L}_{\Delta}L-L.$ (Here $\Delta\in\mathfrak{X}(TQ)$ is the Liouville vector field, i.e. the infinitesimal generator of the action of $\mathbb{R}$ throgh fiber-wise homotopies.)

About the edited question: Truly my historical knowledge is very limited, but probably the origin of these concepts could be in the works by Poincarè on the celestial mechanics, (when the concept itself of differential forms was germinating,) and therefore at the origins of the differential topology as we know it nowadays. You could look at the EoM entry on integral invariants and the references therein. (This is a link.)

Answer by Giuseppe Tortorella for Transversal intersection of a symplectic manifold with a plane

2012-05-04T17:31:25Z

I am assuming you write perpendicular in body of the question to mean transversal as in its title. Now I read your question as searching for local functions $\hat{x}_2=\hat{x}_2(x_2,y_2),\hat{y}_2=\hat{y}_2(x_2,y_2)$ such that $dx_2\wedge dy_2=d\hat{x}_2\wedge d\hat{y}_2$ and $\mathcal{M}=\hat{x}_2^{-1}(0)\cap\hat{y}_2^{-1}(0)$ locally around $0$.

If this reading is correct then in general the answer is no.
Infact if there were local symplectic coordinates $(\hat{x}_2,\hat{y}_2)$ on the $(x_2,y_2)$ plane such that $\mathcal{M}\subseteq\hat{x}^{-1}_2(0)\cap\hat{y}^{-1}_2(0)$ in a neighborhood of $0$, then $\mathcal{M}$ should be locally included in the $(x_1,y_1)$ plane (because $\hat{x}^{-1}_2(0)\cap\hat{y}^{-1}_2(0)=x^{-1}_2(0)\cap y^{-1}_2(0)$.)

But one can construct a counter-example already taking symplectic vector subspaces complementary to the $(x_2,y_2)$-plane in the constant symplectic space $(\mathbb{R}^4,dx_1\wedge dy_1+dx_2\wedge dy_2)$.

Answer by Giuseppe Tortorella for Symplectic submanifolds and first integrals

2012-05-04T14:31:59Z

I am posting here the answer that I gave to the same question when it was posted yesterday on MSE.

Let $f_1$ and $f_2$ be independent functions on a symplectic manifold $(M,\omega).$
Let us denote by $\Sigma$ the submanifold $f_1^{-1}(0)\cap f_2^{-1}(0)$ of codimension $2$ in $M$.
The tangent bundle of $\Sigma$ is $$T \Sigma= (\ker df_1\cap \ker df_2) |_\Sigma=(\text{span}\{X_{f_1},X_{f_2}\})^\perp|_\Sigma.\tag{1}$$
So in the symplectic vector bundle $(T_{\Sigma} M,\omega |_\Sigma)$ the vector sub-bundle $T\Sigma$ has orthogonal complement $$(T\Sigma)^\perp=\operatorname{span}\{X_{f_1},X_{f_2}\}|_\Sigma.\tag{2}$$

By definition, $\Sigma$ is symplectic in $(M,\omega)$ if and only $T\Sigma\cap(T\Sigma)^\perp=0 (\leftarrow\text{the zero section of }\Sigma).$
By (1) and (2), this means : in any point of $\Sigma$ the linear system $$\left\{\begin{array}{c}0=\langle df_1,c_1X_{f_1}+c_2X_{f_2}\rangle=\{f_1,f_2\}c_2, \\0=\langle df_2,c_1X_{f_1}+c_2X_{f_2}\rangle=-\{f_1,f_2\}c_1\end{array}\right.$$ has only the trivial solution $c_1=c_2=0.$
Therefore $\Sigma$ is symplectic iff $\{f_1,f_2\}$ has no zeroes on $\Sigma.$

Answer by Giuseppe Tortorella for Characterization of the Lie derivative

2012-03-15T18:07:50Z

As a possible reference, I would bring your attention on a paper T.J. Willmore, The definition of Lie derivative, Proc. Edinb. Math. Soc. (Ser.2) 1960, 12, 27-29.

It is freely available here.

Answer by Giuseppe Tortorella for Doubly covering an even lattice

2012-03-14T21:14:30Z

Try to take a look to the references in: Yongchang Zhu "Modular invariance of characters of vertex operator algebras" J. Amer. Math. Soc. 9 (1996), 237-302.

Answer by Giuseppe Tortorella for What is soliton

2012-03-05T19:12:27Z

Dear Pradip
There is the historical and theoretical survey "The Symmetry of Solitons" by Richard Palais.
I have liked it very much, so I hope you can find it useful.

Bye.

Is there a coordinate-free proof of the hamiltonian character of the geodesic flow?

2012-02-16T13:03:47Z

I do not know if this question is appropriate for this site, but I posted here without having answers, so now I make this attempt.

Let be $(M,g)$ a pseudoriemannian manifold. Let us identify the tangent and the cotangent bundles through the musical isomorphism $g^\flat:u\in TM\to g(u,\cdot)\in T^\ast M.$

It is well known that:

The geodesics of $(M,g),$ i.e. the solutions of $\frac{D}{dt}\gamma=0,$ are integral curves for the hamiltonian vector field of $K:u\in TM\to \tfrac{1}{2}g(u,u)\in\mathbb{R}$ w.r.t. the canonical symplectic form.

Question Knowing how to show it using coordinates and Christoffell symbols, I am wondering how to prove it in an intrinsic way.

How the Jacobi metrics may be useful in mechanics with or without constraints?

2012-02-06T19:38:20Z

A mechanical system $(Q,K,V)$ is specified by the configuration space $Q,$ the potential energy $V\in C^\infty(Q),$ and the kinetic energy $K=K_g$ given by a Riemannian metric $g$ on $Q.$

If $V{<}e$ then $g_J=(e-V)g$ is the Jacobi metric on $Q,$ and Maupertuis' principle states that, up to reparametrization, the trajectories of motion of $(Q,K,V)$ with energy $e$ are exactly the geodesics of $g_J.$ (which by the way are the trajectory of motion of $(Q,K_{g_J},0).$

My question is motivated by what I read in Jair Koiller's paper "Reduction of Some Classical Non-Holonomic Systems with Symmetry", Arch. Rational Mech. Anal. 118 (1992).

The Jacobi metric is useful for constrained systems as well, an observation which seems not to have been sufficiently explored in the theory of nonholonomic systems.

Koiller goes on saying that, if $D,$ a tangent distribution on $Q,$ represents a linear constraint on the velocities, then the trajectories of motion of the constrained system $(Q,K,V;D)$ with energy $e$ are the same as the ones of $(Q,K_{g_J},0;D),$ with energy $1,$ up to a reparametrization.

Edited Question:

What results of riemannian geometry can be transferred into implications on the dynamics of mechancal systems by way of Jacobi metric?
Furthermore, in the light of the quotation, I would know: what are examples of the usefulness of Jacobi metrics in nonholonomic mechanics?

Obviously any feedback is welcome. Thank you.

Answer by Giuseppe Tortorella for space of geodesics

2012-02-07T10:08:56Z

To expand Ryan Budney's comment, the geodesics of $(M,g)$ are the projection on the base $M$ of the integral curves for the vector field $S_g$ on $TM.$

$S_g$ is the unique vector field on $TM$ which is at the same time:

special (i.e. it represents a $2^\textrm{nd}$-order edo on $M$, or equivalently its integral curves are the tangential lifting of their projection on the base) and

horizontal (i.e. it is a section of the horizontal distribution on $TM$.)

Because $S_g$ is a spray (i.e. $\mathcal{L}_Z S_g=S_g,$ where $Z$ is the Euler vector filed), it is called the geodesic spray of $(M,g).$

It can be realized even that $S_g$ is the hamiltonian vector field of the kinetic energy $K_g:v\in TM\to\tfrac{1}{2}g(v,v)\in\mathbb{R}$ with respect to the pull-back through $g^\flat$ of the canonical symplectic form on $T^\ast M.$

So $S_g$ preserves the sphere bundles, and we get that if $M$ is compact then $S_g$ is complete.

Through the flow of $S_g,$ its integral curves (which are the tangential lifting of their projection on the base) are indentified with $TM$.

In search for a more geometric proof of a result of van der Schaft and Maschke on nonholonomic mechanics.

2011-12-23T17:37:44Z

Edit: Now I have found something that appears to answer my own question. It is section 2 in the paper "On Submanifolds and Quotients of Poisson and Jacobi Manifolds" by Ch.-M. Marle. (There, he transfer all on the cotangent bundle via the Legendre map and uses a different, but really equivalent, construction of the nonholonomic bracket.) Thank you.

In "On The Hamiltonian Formulation of Nonholonomic Mechanical Systems" of Van der Schaft and Maschke (this is a link), a mechanical system subject to linear constraints on the velocities is provided with a bracket $\lbrace\cdot,\cdot\rbrace_{nh}$, which is almost Poisson because it lacks only the Jacobi identity.
Notwithstanding this loss, it equally describes the time evolution of the system according with d'Alembert's principle, that is $\frac{d}{dt}f=\lbrace f,E\rbrace_{nh}$, where $E$ is the total energy of the system.

My interest was attracted above all by the following result:

$\lbrace\cdot,\cdot\rbrace_{nh}$ measures the holonomic character of the linear constraint, that is the bracket satisfies the Jacobi identity iff the linear constraint is an integrable distribution.

I am writing this question for the following reason:

Being their presentation deeply depending on coordinates, I was in search for a more geometric, invariant demonstration of this last result.

In order to convey more information, below I briefly sketched the geometric context as far as I have understood until now. Thank you.

Let us impose on mechanical system having configuration space $Q$ and Lagrangian $L$ a constraint linear on the velocities represented by $D\subset TQ$, a tangent distribution on $Q$.

Let us introduce $J:TTQ\to TTQ$ the endomorphism of $\tau_{TQ}:TTQ\to TQ$ defined locally by $$J(\frac{\partial}{\partial x_i})=\frac{\partial}{\partial v_i},\ J(\frac{\partial}{\partial v_i})=0$$ where (x,v) are the standard coordinates on $TQ$ associated to local coordinates $x$ on $Q$. Below $J^\ast$ will denote the transpose of $J$.

Let $\omega$ be the symplectic form on $TQ$ given by the pull-back through $\mathbb{F}L$ of the canonical symplectic form on $T^\ast Q$. Here $\mathbb{F}L:TQ\to T^|ast Q$ is the Legendre trasformation associated to $L$ which locally acts as $(x,v)\mapsto (x,\frac{\partial L}{\partial v})$. Below $\flat$ and $\sharp$ will denote the musical isomorphisms corresponding to $\omega$.

The Chetaev bundle $F$ of the reaction forces is the tangent distribution on $TQ$ such that $\flat(F)=J^\ast((TD)^0)$. Here $(TD)^0$ denotes the annihilator of $TD$.

If the Lagrangian is natural then $H:=F^\perp\cap TD$ is a symplectic distribution on $(TQ,\omega)$ , and we can define an endomorphism $P:T_DTQ\to T_DTQ$ by projecting fiberwise $T_DTQ=H^\perp\oplus H$ on $H$ along $H^\perp$.

For any pair $f,g$ of smooth functions on $D$, their bracket $\lbrace f,g\rbrace_{nh}$ is defined as $\omega(P\circ\sharp(d\tilde{f}),P\circ\sharp(d\tilde{g}))$ where $\tilde{f}$ and $\tilde{g}$ are arbitrary smooth extensions on $TQ$ of $f$ and $g$ respectively.
Such a bracket is skew-simmetric, bilinear and satisfies the Leibnitz rule in each one of its argument, but couldn't be a Poisson bracket on $D$ because of the lack of Jacobi identity.

The Lagrangian formulation of mechanics without going through variational principles.

2011-11-24T13:06:51Z

In some texts on classical mechanics and not only, the Euler--Lagrange equations of motion are directly obtained as solution of variational problems.
On the other side, sometimes reading about hamiltonian mechanics, one find the expression that this latter formulation is preferred to the lagrangian one because of it does completely avoid the appeal to variational principles.

This observation suggested to myself the following question:

Is the variational approach to the Euler--Lagrange equations the only one viable?
If not, is there some reason that explain why the geometry of the Euler-Lagrange eqns is much more hidden than the geometry of the Hamilton eqns?

I was searching for suggestion of reading for best tackle this question.

As usual any feedback is welcome.

What happens when Appell-Chetaev's rule for constrained mechanical systems is not applicable?

2011-10-11T10:46:45Z

Background: Let be given a mechanical system whose configuration space is a manifold $Q$, and the kinetic energy is a metric $K$ on $Q$, in presence of a potential function $V$.
Let us identify the tangent bundle $TQ$ with the phase space $T^\ast Q$ through the Legendre map $\mathcal{L}_K$. Let $X_L$ be the second order differential equation on $M$ corresponding to the lagrangian $L=K+\pi^\ast V$.

If the system is subject to a kinematical constraint represented by a submanifold $M$ of $TQ$, then we need to find the vector field $X$ on $M$ which determines the dynamics of such a constrained system, the difference $X_C=X-X_L$ being the contribution of the constraint force.

Under mild assumption, a prescription to find $X_C$ is the Appell-Chetaev rule, which extends the Lagrange-D'Alembert principle in the context of constraints not necessarily linear on the velocities. This rule imposes that $X_C$ has to lie on $(TM)^{0}$, the annihilator of $TM$.

Question: Are there constrained mechanical systems whose dynamics is not in agree with the previsions based on Appell-Chetaev?
And in such a case, what rules are the alternatives in prescribing the constraint forces? and what are the domains of applicability of such other rules?

As usual any feedback is welcome, thank you.

Answer by Giuseppe Tortorella for Mathematicians failing to solve problems despite having all methods required

2011-10-03T17:39:45Z

The history of the first version of Poincaré's essay submitted to the competition sponsored by King Oscar II of Sweden, could be representative of this situation but at the same time of the capacity of overcome previous errors.

The problem of the stability of a planetary system was central from the dawn of newtonian mechanics. A father of Analytical mechanics such as Dirichlet thought to have proved the stability for the n-body problem, but he died suddenly before to write it.

The prize of King Oscar was aimed to obtain such a proof of the stability and infact in the first version of his essay, Poincaré claimed the stability of the restricted 3-body problem. This essay won the prize, but just after the publication of the paper in the Acta he realized the presence of a serious error for the presence of homoclinic orbits. Consequently the published issues were recalled and the second version of the essay was printed.

The existence of a first version of Poincaré's essay has been discovered only in 1994 by June Green-Barrow. The second version is known as the starting point of the qualitative geometric methods in mechanics.

For more information a possible source is Diacu, F., The solution of the n-body problem, Math. Intelligencer 18 (3) 66-70, 1996.

Answer by Giuseppe Tortorella for reference for the slice theorem for Banach Lie group actions on Banach manifolds

2011-09-28T13:16:49Z

Dear Orbicular the theorem on the existence of slices is stated without proof as Theorem 5.2.6 in Critical Point Theory and Submanifold Geometry, LNM 1353, of Palais and Terng (for example see here).
The proof should be adapted without difficulty from that in the finite-dimensional case.
For this case you can look at ``On the existence of slices for actions of non-compact groups'' by Palais (for example see here).

Answer by Giuseppe Tortorella for Navier-Stokes equations in riemannian geometry

2011-09-25T14:50:32Z

You could look at the paper: Groups of Diffeomorphisms and the motion of an incompressible fluid, by Ebin and Marsden.

About two centuries after Euler, in 1966 Arnold gave a geometric reformulation of the classical equations for an imcompressible fluid in terms of the geodesic spray of left invariant metric on an infinite dimensional Lie Group.

Ebin and Marsden promptly employed this reformulation to obtain existence and uniqueness results for these equations on compact oriented riemannian manifolds.

This circle of ideas is one of the first important application of infinite dimensional manifolds as remarked by Stephen Smale.

By the way, should not the equation contain the time derivative of the unknown $u$?

What are elementary applications of the Frobenius'Theorem in the Classical Differential Geometry?

2011-09-08T13:28:24Z

Usually in a first course on differential geometry we learn some classical results on the geometry of curves and surfaces in the ordinary euclidean space, and just later in more advanced courses we learn systematically the concepts and the tools of the analysis on manifolds, one of whose pillars is the Frobenius' Theorem.

In order to remark the continuity between the two stages, it would be nice, for example, to present the Frobenius' Theorem together with some of its application in the realm of classical differential geometry.

Adressing to someone who has had already an introductory course on the differential geometry, and now is taking a course on smooth manifolds, what are results from classical differential geometry of curves and surfaces that I could present as good illustrations of Frobenius' Theorem?

Any suggestion is welcome.

Answer by Giuseppe Tortorella for A book on Banach Manifold for a Dynamicist

2011-09-02T07:45:00Z

The first textbook I thought is: Palais, The Foundations of Global Non-linear Analysis, Benjamin-Cummings, 1968.

There is also: Marsden, Applications of Global Analysis in Mathematical Physics, Publish or Perish, 1974.

Finally you could look here at Graff's review of ``The Metric theory of Banach Manifolds'' by Ethan Atkin, for many other references.

On a remark in Foundations of mechanics, 2nd Edition, by Abraham and Marsden

2011-08-28T20:35:26Z

I don't know if this question is appropriate to this site, but I posted here without an answer, so I tried this alternative.

Given a $2$-form $\omega$ on a manifold $M$, let us denote by $N$ the kernel of $\omega$, i.e. $N:=\{u\in TM : \omega(u,\cdot)=0\}$. Their Proposition 5.1.2 shows that if $\omega$ has constant rank (and is closed) then $N$ is a tangent distribution on $M$ (and completely integrable).

In the following remark they say that ``the reader can easily prove the converse of the previous conclusion''. While I understand that $N$ is a tangent distribution if and only if $\omega$ has constant rank. Instead I think that, for $\omega$ of constant rank, $N$ can be completely integrable even if $\omega$ is not closed, (e.g. $\omega=e^z dx\wedge dy$).

Starting from this consideration I have asked myself a question:
Given a $\Omega\in\mathcal{A}^p(M)$, with $p>1$, whose rank is constant, let us define its kernel $N$ as above. Evidently $N$ is a tangent distribution on $M$, and I find it is completely integrable at least when there exists a $1$-form $\phi$ such that $d\Omega=\phi\wedge\Omega$. Clearly, if $\Omega$ is decomposable then the last condition is even necessary.

My question (edited after the comment of Willie Wong):

Is this last condition (the ``divisibility'' of $d\Omega$ by $\Omega$) necessary for the complete integrability of $N$ even when $\Omega$ is not decomposable? (Using Frobenius' Theorem I understand the case $p=1$, but what about the case $p>1$?.)

Any suggestion and\or counterexample are welcome.

Answer by Giuseppe Tortorella for when can we lift an action of Lie algebra?

2011-08-24T14:46:27Z

Let be $G$ a simply connected Lie group, $\mathfrak{g}$ its Lie algebra and $M$ an arbitrary smooth manifold. Let be $\zeta$ a smooth action of $\mathfrak{g}$ on a $M$, i.e. $\zeta:X\in\mathfrak{g}\to\mathfrak{X}(M)$ is a Lie algebra homomorphism.

Then there exists a local left action $\Phi$ of $G$ on $M$ such that, for any $X\in\mathfrak{g}$, the t-time local flow of $\zeta(X)$ is given by $m\mapsto\Phi(e^{-t.X},m)$

In general the action of $\mathfrak{g}$ on $M$ can only be lifted to a local left action $\Phi$ of $G$ on $M$, i.e. defined only on a neighborhood of $\{e\}\times M$ in $G\times M$.

But, if $\zeta(X)$ is a complete vector on $M$ for any $X\in\mathfrak{g}$, then $\zeta$ can be lifted to a global left action of $G$ on $M$.

These results should be found in the work of Richard Palais on the Lie theory of transformation groups.

Answer by Giuseppe Tortorella for Most memorable titles

2011-08-23T08:31:48Z

Larry Bates, "You can't get there from here", Differential Geometry and its Applications 8.3 (1998): 273-274

What is the role of contact geometry in the hamiltonian mechanics?

2011-08-09T18:32:39Z

Let us assume someone is interested in the study of Hamiltonian mechanics.
What are good examples to illustrate him of the usefulness of contact geometry in this context?

On one hand the Hamiltonian mechanics was time ago expressed in the language of symplectic geometry, but, on the other hand, the contact geometry is often presented like the brother of the symplectic geometry.

My question is:

In the hamiltonian mechanics, not necessarily only for Hamiltonian of mechanical type, what is the role played by the contact geometry?

Any kind of suggestion is welcome.

Comment by Giuseppe Tortorella

2013-03-10T17:57:33Z

Cross-posted on MathStackexchange here: math.stackexchange.com/questions/326503/….

Comment by Giuseppe Tortorella

2013-03-10T12:25:36Z

From "How to write math" (look on the right $\rightarrow$): if you're having problems with the preview (or the post looks wrong), put backticks around any math that contains underscores or asterisks. E.g. write $f'_n=g_{n+1}$ .

Comment by Giuseppe Tortorella

2013-01-28T18:21:00Z

Dear Andrea Mori, I have tried to get displayed the matrices, without modify their content.

Comment by Giuseppe Tortorella

2013-01-22T11:46:53Z

Here is: mathoverflow.net/questions/91385/…

Comment by Giuseppe Tortorella

2013-01-22T11:36:24Z

Dear Chandan Singh Dalawat, last year Roger Godement answered a question here on MathOverflow, surely he could give invaluable information.

Comment by Giuseppe Tortorella

2013-01-22T11:21:01Z

Dear Delio Mugnolo, I don't have an answer, but I would just remark that Bourbaki's "Elements of Hystory of Mathematics" recollects the historical sections disseminated in the volumes of the "Elements of Mathematics". By the way, Jean Dieudonné wrote historical works on his own, and directed the work of a group of mathematician to write "Abrégé d'histoire des mathématiques 1700-1900".

Comment by Giuseppe Tortorella

2013-01-19T16:28:37Z

Dear Peter Arndt, I have tried to edit your question without altering its content. Is it ok for you?

Comment by Giuseppe Tortorella

2013-01-07T15:38:21Z

When MathJax seems having problems, the basic solution is to enclose between backtips any TeX code containing underscores or asterisks.

Comment by Giuseppe Tortorella

2013-01-07T15:23:24Z

Dear Italo I have tried to edit your question, it is ok for you?

Comment by Giuseppe Tortorella

2013-01-07T11:38:28Z

The first Russian edition is dated 1953.

Comment by Giuseppe Tortorella

2013-01-07T09:56:22Z

@Qfwfq @Joe Silverman the first edition of Lang's Algebra appeared in 1965.

Comment by Giuseppe Tortorella

2013-01-07T09:49:08Z

The four volumes were published between 1887 and 1896.

Comment by Giuseppe Tortorella

2012-12-05T20:18:07Z

Dear Liviu Nicolaescu, if you do not mind, I have tried to adjust the references you included.

Comment by Giuseppe Tortorella

2012-09-05T17:00:55Z

@Squid, Do you know "Mathematics: its content, methods, and meaning" by Aleksandrov, Kolmogorov, Lavrent'ev?

Comment by Giuseppe Tortorella

2012-08-30T06:44:22Z

Thanks for the answer, Alejandro.