'body-fixed coordinates' of . Initially is also the space-fixed coordinate vector of . Upon rotation of the body, the body-fixed coordinates of do not change, but the space-fixed coordinate vector of becomes, In particular, if is initially on the space-fixed Z-axis, it has the space-fixed coordinates which shows the correspondence with the spherical polar coordinates (in the physical convention). Knowledge of the Euler angles as function of time t and the initial coordinates determine the kinematics of the rigid rotor. Classical kinetic energy The following text forms a generalization of the well-known special case of the rotational energy of an object that rotates around one axis. It will be assumed from here on that the body-fixed frame is a principal axes frame; it diagonalizes the instantaneous inertia tensor (expressed with respect to the space-fixed frame), i.e., where the Euler angles are time-dependent and in fact determine the time dependence of by the inverse of this equation. This notation implies that at the Euler angles are zero, so that at the body-fixed frame coincides with the space-fixed frame. The classical kinetic energy T of the rigid rotor can be expressed in different ways: as a function of angular velocity in Lagrangian form as a function of angular momentum in Hamiltonian form. Since each of these forms has its use and can be found in textbooks we will present all of them. Angular velocity form As a function of angular velocity T reads, with The vector on the left hand side contains the components of the angular velocity of the rotor expressed with respect to the body-fixed frame. The angular velocity satisfies equations of motion known as Euler's equations (with zero applied torque, since by assumption the rotor is in field-free space). It can be shown that is not the time derivative of any vector, in contrast to the usual definition of velocity. The dots over the time-dependent Euler angles on the right hand side indicate time derivatives. Note that a different rotation matrix would result from a different choice of Euler angle convention used. Lagrange form Backsubstitution of the expression of into T gives the kinetic energy in Lagrange form (as a function of the time derivatives of the Euler angles). In matrix-vector notation, where is the metric tensor expressed in Euler angles—a non-orthogonal system of curvilinear coordinates— Angular momentum form Often the kinetic energy is written as a function of