I was just recently trying to understand a similar lecture by a colleague at my university. He's using Folland's "Quantum field theory" as source; you may find it helpful.

The symplectic structure on the cotangent bundle to $M$ is given by the differential $\Omega$ of the 1-form $\omega_{v^*}(X, X^*) = v^*(X)$, where $v^* \in T^*(M)$ lies over $p \in M$ and we have identified $T_{v^*}(T^*(M)) \cong T_p(M) \times T_p^*(M)$. (I couldn't tell from your question whether or not this was already clear. I also originally wrote something a little nonsensical; thanks to Orbicular for pointing it out.)

The following is a direct quote from Folland:

… we start with a configuration space $N$, which is taken to be a manifold and is supposed to be a description of the possible “positions” of the system. There is quite a lot of flexibility here. For example, if the system consists of $k$ particles moving in $\mathbb R^3$ …, $N$ will be $\mathbb R^{3k}$, or perhaps $\{(\mathbf x_1, \ldots, \mathbf x_k) \in \mathbb R^{3k} : \mathbf x_i \ne \mathbf x_j\text{ for }i \ne j\}$. If the motion is subject to some constraints, $N$ could be a submanifold of $\mathbb R^{3k}$ instead. For an asymmetric rigid body, however, the appropriate configuration space is $\mathbb R^3 \times \operatorname{SO}(3)$: three linear coordinates to give the location of the center of mass (or some other convenient point in the body), and three angular coordinates to give the body's orientation in space.

Velocities are taken to be tangent vectors to the configuration space, so the position–velocity state space is $T N$. On the other hand, the appearance of Poisson brackets in the Hamiltonian formalism [earlier] leads us to take the position–momentum state space to be the symplectic manifold $T^*N$ …. [T]he mass matrix $m$ should be interpreted as a Riemannian metric on $N$ that mediates between vectors and covectors …. $T^*N$ is traditionally called the phase space of the system. (The origin of the name lies in statistical mechanics.)

Thus, it seems to me that it is the configuration, not phase, space that is meant to be physically obvious; and that the designation of the cotangent bundle as the phase space is mostly a matter of terminology.

Once we have the symplectic structure, we use it to equip, not $N$ or $T^*N$, but $C^\infty(T^*N)$, with a Poisson bracket, with respect to which it is a Lie algebra. However, this is a really immense (hugely infinite dimensional) Lie algebra, so it is hard to believe that it has anything to do with the much smaller Lie algebra $T_1G$ when $N = G$ is a Lie group.

I was just recently trying to understand a similar lecture by a colleague at my university. He's using Folland's "Quantum field theory" as source; you may find it helpful.

The symplectic structure on the cotangent bundle to $M$ is given by the differential $\Omega$ of the 1-form $\omega$ defined by $\omega_{v^*}(v) = v^*(\pi_*(v))$ for $v^* \in T^*M$ and $v \in T_{v^*}(T^*M)$. (I couldn't tell from your question whether or not this was already clear. Thanks to Orbicular for pointing out that my original version depended on a choice of coördinates.)

The following is a direct quote from Folland:

… we start with a configuration space $N$, which is taken to be a manifold and is supposed to be a description of the possible “positions” of the system. There is quite a lot of flexibility here. For example, if the system consists of $k$ particles moving in $\mathbb R^3$ …, $N$ will be $\mathbb R^{3k}$, or perhaps $\{(\mathbf x_1, \ldots, \mathbf x_k) \in \mathbb R^{3k} : \mathbf x_i \ne \mathbf x_j\text{ for }i \ne j\}$. If the motion is subject to some constraints, $N$ could be a submanifold of $\mathbb R^{3k}$ instead. For an asymmetric rigid body, however, the appropriate configuration space is $\mathbb R^3 \times \operatorname{SO}(3)$: three linear coordinates to give the location of the center of mass (or some other convenient point in the body), and three angular coordinates to give the body's orientation in space.

Velocities are taken to be tangent vectors to the configuration space, so the position–velocity state space is $T N$. On the other hand, the appearance of Poisson brackets in the Hamiltonian formalism [earlier] leads us to take the position–momentum state space to be the symplectic manifold $T^*N$ …. [T]he mass matrix $m$ should be interpreted as a Riemannian metric on $N$ that mediates between vectors and covectors …. $T^*N$ is traditionally called the phase space of the system. (The origin of the name lies in statistical mechanics.)

Thus, it seems to me that it is the configuration, not phase, space that is meant to be physically obvious; and that the designation of the cotangent bundle as the phase space is mostly a matter of terminology.

Once we have the symplectic structure, we use it to equip, not $N$ or $T^*N$, but $C^\infty(T^*N)$, with a Poisson bracket, with respect to which it is a Lie algebra. However, this is a really immense (hugely infinite dimensional) Lie algebra, so it is hard to believe that it has anything to do with the much smaller Lie algebra $T_1G$ when $N = G$ is a Lie group.

I was just recently trying to understand a similar lecture by a colleague at my university. He's using Folland's "Quantum field theory" as source; you may find it helpful.

The symplectic structure on the cotangent bundle to $M$ is given by the differential $\Omega$ of the 1-form $\omega_{(p, v^*)}(X, X^*) = v^*(X)$, where we have identified $T_{(p, v^*)}(T^*(M)) \cong T_p(M) \times T_p^*(M)$. (I couldn't tell from your question whether or not this was already clear.)

The following is a direct quote from Folland:

… we start with a configuration space $N$, which is taken to be a manifold and is supposed to be a description of the possible “positions” of the system. There is quite a lot of flexibility here. For example, if the system consists of $k$ particles moving in $\mathbb R^3$ …, $N$ will be $\mathbb R^{3k}$, or perhaps $\{(\mathbf x_1, \ldots, \mathbf x_k) \in \mathbb R^{3k} : \mathbf x_i \ne \mathbf x_j\text{ for }i \ne j\}$. If the motion is subject to some constraints, $N$ could be a submanifold of $\mathbb R^{3k}$ instead. For an asymmetric rigid body, however, the appropriate configuration space is $\mathbb R^3 \times \operatorname{SO}(3)$: three linear coordinates to give the location of the center of mass (or some other convenient point in the body), and three angular coordinates to give the body's orientation in space.

Velocities are taken to be tangent vectors to the configuration space, so the position–velocity state space is $T N$. On the other hand, the appearance of Poisson brackets in the Hamiltonian formalism [earlier] leads us to take the position–momentum state space to be the symplectic manifold $T^*N$ …. [T]he mass matrix $m$ should be interpreted as a Riemannian metric on $N$ that mediates between vectors and covectors …. $T^*N$ is traditionally called the phase space of the system. (The origin of the name lies in statistical mechanics.)

Thus, it seems to me that it is the configuration, not phase, space that is meant to be physically obvious; and that the designation of the cotangent bundle as the phase space is mostly a matter of terminology.

Once we have the symplectic structure, we use it to equip, not $N$ or $T^*N$, but $C^\infty(T^*N)$, with a Poisson bracket, with respect to which it is a Lie algebra. However, this is a really immense (hugely infinite dimensional) Lie algebra, so it is hard to believe that it has anything to do with the much smaller Lie algebra $T_1G$ when $N = G$ is a Lie group.

I was just recently trying to understand a similar lecture by a colleague at my university. He's using Folland's "Quantum field theory" as source; you may find it helpful.

The symplectic structure on the cotangent bundle to $M$ is given by the differential $\Omega$ of the 1-form $\omega_{v^*}(X, X^*) = v^*(X)$, where $v^* \in T^*(M)$ lies over $p \in M$ and we have identified $T_{v^*}(T^*(M)) \cong T_p(M) \times T_p^*(M)$. (I couldn't tell from your question whether or not this was already clear. I also originally wrote something a little nonsensical; thanks to Orbicular for pointing it out.)

The following is a direct quote from Folland:

… we start with a configuration space $N$, which is taken to be a manifold and is supposed to be a description of the possible “positions” of the system. There is quite a lot of flexibility here. For example, if the system consists of $k$ particles moving in $\mathbb R^3$ …, $N$ will be $\mathbb R^{3k}$, or perhaps $\{(\mathbf x_1, \ldots, \mathbf x_k) \in \mathbb R^{3k} : \mathbf x_i \ne \mathbf x_j\text{ for }i \ne j\}$. If the motion is subject to some constraints, $N$ could be a submanifold of $\mathbb R^{3k}$ instead. For an asymmetric rigid body, however, the appropriate configuration space is $\mathbb R^3 \times \operatorname{SO}(3)$: three linear coordinates to give the location of the center of mass (or some other convenient point in the body), and three angular coordinates to give the body's orientation in space.

Velocities are taken to be tangent vectors to the configuration space, so the position–velocity state space is $T N$. On the other hand, the appearance of Poisson brackets in the Hamiltonian formalism [earlier] leads us to take the position–momentum state space to be the symplectic manifold $T^*N$ …. [T]he mass matrix $m$ should be interpreted as a Riemannian metric on $N$ that mediates between vectors and covectors …. $T^*N$ is traditionally called the phase space of the system. (The origin of the name lies in statistical mechanics.)

Thus, it seems to me that it is the configuration, not phase, space that is meant to be physically obvious; and that the designation of the cotangent bundle as the phase space is mostly a matter of terminology.

Once we have the symplectic structure, we use it to equip, not $N$ or $T^*N$, but $C^\infty(T^*N)$, with a Poisson bracket, with respect to which it is a Lie algebra. However, this is a really immense (hugely infinite dimensional) Lie algebra, so it is hard to believe that it has anything to do with the much smaller Lie algebra $T_1G$ when $N = G$ is a Lie group.