Apparently one identifies the configuration space in physics often with a manifold $M$. The tangent bundle $TM$ is then the space of all possible positions and velocities.

Furthermore, many sources seem to claim that $T^*M$ can be regarded as the phase space, where $(q,p) \in T^*M$ satisfies by definition that $p \in T_p^*M.$

Again by definition this means that $p:=\partial_2L$ takes velocities as arguments and is linear(!) in them. Unfortunately, I don't see from the definition of the momentum by the Lagrangian why this should be a linear functional. So something is confusing me here.

Apparently one identifies the configuration space in physics often with a manifold $M$. The tangent bundle $TM$ is then the space of all possible positions and velocities.

Furthermore, many sources seem to claim that $T^*M$ can be regarded as the phase space, where $(q,p) \in T^*M$ satisfies by definition that $p \in T_q^*M.$

Again by definition this means that $p:=\partial_2L$ takes velocities as arguments and is linear(!) in them. Unfortunately, I don't see from the definition of the momentum by the Lagrangian why this should be a linear functional. So something is confusing me here.