Actually the first case in history of a symplectic manifold wasn't a cotangent space. It was the space of Keplerian motions of a planet, represented locally by its Keplerian elements. The Lagrange symplectic structure on this space is defined by the so-called Lagrange parenthesis he introduced at this time (three papers in 1808/09/10)(*). That manifold is actually even non Hausdorff, but its greatest hausdorff quotient is a still a manifold (this is known as the "regularization theorem"). This manifold is symplectic but not a cotangent (but however contractible to the sphere $S^3$). Extended with the repulsive motions, it is an algebraic manifold defined by the following equations [Sou]. $$ \left\{ \begin{array}{rcl} ||{A}||^2 -fx^2 & = & 1 \\ y^2 - f ||{B}||^2 & = & 1 \end{array} \right. \quad \& \quad \left\{ \begin{array}{rcl} A \cdot B - xy &=& 0 \\ xy - f\tau &=& 0, \end{array} \right. $$ where $A,B \in {\bf R}^3$ and $x,y,f,\tau$ belong to $\bf R$. Actually this manifold is the result of the gluing of $TS^3$ and $TH^3$ along $TS^2\times {\bf R}$ (where $H^3$ is the 3 dimensional pseudo-sphere). I made the following picture, for $A$ and $B$ in $\bf R$, to get a visual idea of the manifold. The bottom represents the $TS^3$ part, the top represents $TH^3$ and it is glued along two lines representing $TS^2\times {\bf R} \simeq S^2 \times {\bf R}^3$.

Remark. There exists also the examples of compact symplectic manifolds representing internal degrees of symmetries, as mentioned in Tobias answer. In the same spirit there is the Grassmannian manifolds ${\rm Gr}(2,n+1)$ of $2$-planes in ${\bf R}^{n+1}$, representing the space of un-parametrized geodesics on the sphere $S^n$. We can regard this space of geodesics as the space of light rays on the Euclidean sphere where the speed of light would be infinite.

---------- Edit March 28, 2017

On a conceptual point of view, I just finished to write a paper:

Universal Structure Of Symplectic Manifolds
http://math.huji.ac.il/~piz/documents/ESMIACO.pdf,

that proposes a way, based on diffeology, to understand the statement: "Every symplectic manifold is a coadjoint orbit".

---------- Notes

(*) I published this paper on the origins of symplectic geometry, but in french, where Lagrange's construction is explained.

---------- Reference

[Sou] Jean-Marie Souriau. Géométrie globale du problème à deux corps. In Modern Developments in Analytical Mechanics, pp. 369-418. Accademia della Scienza di Torino, 1983.

Actually the first case in history of a symplectic manifold wasn't a cotangent space. It was the space of Keplerian motions of a planet, represented locally by its Keplerian elements. The Lagrange symplectic structure on this space is defined by the so-called Lagrange parenthesis he introduced at this time (three papers in 1808/09/10)(*). That manifold is actually even non Hausdorff, but its greatest hausdorff quotient is a still a manifold (this is known as the "regularization theorem"). This manifold is symplectic but not a cotangent (but however contractible to the sphere $S^3$). Extended with the repulsive motions, it is an algebraic manifold defined by the following equations [Sou]. $$ \left\{ \begin{array}{rcl} ||{A}||^2 -fx^2 & = & 1 \\ y^2 - f ||{B}||^2 & = & 1 \end{array} \right. \quad \& \quad \left\{ \begin{array}{rcl} A \cdot B - xy &=& 0 \\ xy - f\tau &=& 0, \end{array} \right. $$ where $A,B \in {\bf R}^3$ and $x,y,f,\tau$ belong to $\bf R$. Actually this manifold is the result of the gluing of $TS^3$ and $TH^3$ along $TS^2\times {\bf R}$ (where $H^3$ is the 3 dimensional pseudo-sphere). I made the following picture, for $A$ and $B$ in $\bf R$, to get a visual idea of the manifold. The bottom represents the $TS^3$ part, the top represents $TH^3$ and it is glued along two lines representing $TS^2\times {\bf R} \simeq S^2 \times {\bf R}^3$.

Remark. There exists also the examples of compact symplectic manifolds representing internal degrees of symmetries, as mentioned in Tobias answer. In the same spirit there is the Grassmannian manifolds ${\rm Gr}(2,n+1)$ of $2$-planes in ${\bf R}^{n+1}$, representing the space of un-parametrized geodesics on the sphere $S^n$. We can regard this space of geodesics as the space of light rays on the Euclidean sphere where the speed of light would be infinite.

---------- Edit March 28, 2017

On a conceptual point of view, I just finished to write a paper:

Universal Structure Of Symplectic Manifolds
http://math.huji.ac.il/~piz/documents/ESMIACO.pdf,

---------- Edit November 15, 2019

http://math.huji.ac.il/~piz/documents/ESMIALCO.pdf,

This paper has been enhanced to make the symplectic manifold an orbit of the linear coadjoint action of a central extension of the group of Hamiltonian diffeomorphisms, independently of the group of periods. That is, even if the symplectic form is not integral.

that proposes a way, based on diffeology, to understand the statement: "Every symplectic manifold is a coadjoint orbit".

---------- Notes

(*) I published this paper on the origins of symplectic geometry, but in french, where Lagrange's construction is explained.

---------- Reference

[Sou] Jean-Marie Souriau. Géométrie globale du problème à deux corps. In Modern Developments in Analytical Mechanics, pp. 369-418. Accademia della Scienza di Torino, 1983.

Actually the first case in history of a symplectic manifold wasn't a cotangent space. It was the space of Keplerian motions of a planet, represented locally by its Keplerian elements. The Lagrange symplectic structure on this space is defined by the so-called Lagrange parenthesis he introduced at this time (three papers in 1808/09/10)(*). That manifold is actually even non Hausdorff, but its greatest hausdorff quotient is a still a manifold (this is known as the "regularization theorem"). This manifold is symplectic but not a cotangent (but however contractible to the sphere $S^3$). Extended with the repulsive motions, it is an algebraic manifold defined by the following equations [Sou]. $$ \left\{ \begin{array}{rcl} ||{A}||^2 -fx^2 & = & 1 \\ y^2 - f ||{B}||^2 & = & 1 \end{array} \right. \quad \& \quad \left\{ \begin{array}{rcl} A \cdot B - xy &=& 0 \\ xy - f\tau &=& 0, \end{array} \right. $$ where $A,B \in {\bf R}^3$ and $x,y,f,\tau$ belong to $\bf R$. Actually this manifold is the result of the gluing of $TS^3$ and $TH^3$ along $TS^2\times {\bf R}$ (where $H^3$ is the 3 dimensional pseudo-sphere). I made the following picture, for $A$ and $B$ in $\bf R$, to get a visual idea of the manifold. The bottom represents the $TS^3$ part, the top represents $TH^3$ and it is glued along two lines representing $TS^2\times {\bf R} \simeq S^2 \times {\bf R}^3$.

Remark. There exists also the examples of compact symplectic manifolds representing internal degrees of symmetries, as mentioned in Tobias answer. In the same spirit there is the Grassmannian manifolds ${\rm Gr}(2,n+1)$ of $2$-planes in ${\bf R}^{n+1}$, representing the space of un-parametrized geodesics on the sphere $S^n$. We can regard this space of geodesics as the space of light rays on the Euclidean sphere where the speed of light would be infinite.

(*) I published this paper on the origins of symplectic geometry, but in french, where Lagrange's construction is explained.

Reference

[Sou] Jean-Marie Souriau. Géométrie globale du problème à deux corps. In Modern Developments in Analytical Mechanics, pp. 369-418. Accademia della Scienza di Torino, 1983.

Actually the first case in history of a symplectic manifold wasn't a cotangent space. It was the space of Keplerian motions of a planet, represented locally by its Keplerian elements. The Lagrange symplectic structure on this space is defined by the so-called Lagrange parenthesis he introduced at this time (three papers in 1808/09/10)(*). That manifold is actually even non Hausdorff, but its greatest hausdorff quotient is a still a manifold (this is known as the "regularization theorem"). This manifold is symplectic but not a cotangent (but however contractible to the sphere $S^3$). Extended with the repulsive motions, it is an algebraic manifold defined by the following equations [Sou]. $$ \left\{ \begin{array}{rcl} ||{A}||^2 -fx^2 & = & 1 \\ y^2 - f ||{B}||^2 & = & 1 \end{array} \right. \quad \& \quad \left\{ \begin{array}{rcl} A \cdot B - xy &=& 0 \\ xy - f\tau &=& 0, \end{array} \right. $$ where $A,B \in {\bf R}^3$ and $x,y,f,\tau$ belong to $\bf R$. Actually this manifold is the result of the gluing of $TS^3$ and $TH^3$ along $TS^2\times {\bf R}$ (where $H^3$ is the 3 dimensional pseudo-sphere). I made the following picture, for $A$ and $B$ in $\bf R$, to get a visual idea of the manifold. The bottom represents the $TS^3$ part, the top represents $TH^3$ and it is glued along two lines representing $TS^2\times {\bf R} \simeq S^2 \times {\bf R}^3$.

Remark. There exists also the examples of compact symplectic manifolds representing internal degrees of symmetries, as mentioned in Tobias answer. In the same spirit there is the Grassmannian manifolds ${\rm Gr}(2,n+1)$ of $2$-planes in ${\bf R}^{n+1}$, representing the space of un-parametrized geodesics on the sphere $S^n$. We can regard this space of geodesics as the space of light rays on the Euclidean sphere where the speed of light would be infinite.

---------- Edit March 28, 2017

On a conceptual point of view, I just finished to write a paper:

Universal Structure Of Symplectic Manifolds
http://math.huji.ac.il/~piz/documents/ESMIACO.pdf,

that proposes a way, based on diffeology, to understand the statement: "Every symplectic manifold is a coadjoint orbit".

---------- Notes

(*) I published this paper on the origins of symplectic geometry, but in french, where Lagrange's construction is explained.

---------- Reference

[Sou] Jean-Marie Souriau. Géométrie globale du problème à deux corps. In Modern Developments in Analytical Mechanics, pp. 369-418. Accademia della Scienza di Torino, 1983.