Scleronomous

In 3-D space, a particle with mass m\,\!, velocity \mathbf{v} has kinetic energy T T =\frac{1}{2}m v^2 . Velocity is the derivative of position r with respect to time t\,\!. Use chain rule for several variables: \mathbf{v} = \frac{d\mathbf{r}}{dt} = \sum_i\ \frac{\partial \mathbf{r}}{\partial q_i} \dot{q}_i + \frac{\partial \mathbf{r}}{\partial t} . where q_i are generalized coordinates. Therefore, T = \frac{1}{2} m \left(\sum_i\ \frac{\partial \mathbf{r}}{\partial q_i}\dot{q}_i+\frac{\partial \mathbf{r}}{\partial t}\right)^2 . Rearranging the terms carefully, \begin{align} T &= T_0 + T_1 + T_2 : \\[1ex] T_0 &= \frac{1}{2} m \left(\frac{\partial \mathbf{r}}{\partial t}\right)^2 , \\ T_1 &= \sum_i\ m\frac{\partial \mathbf{r}}{\partial t}\cdot \frac{\partial \mathbf{r}}{\partial q_i}\dot{q}_i\,\!, \\ T_2 &= \sum_{i,j}\ \frac{1}{2}m\frac{\partial \mathbf{r}}{\partial q_i}\cdot \frac{\partial \mathbf{r}}{\partial q_j}\dot{q}_i\dot{q}_j, \end{align} where T_0\,\!, T_1\,\!, T_2 are respectively homogeneous functions of degree 0, 1, and 2 in generalized velocities. If this system is scleronomous, then the position does not depend explicitly with time: \frac{\partial \mathbf{r}}{\partial t}=0. Therefore, only term T_2 does not vanish: T = T_2. Kinetic energy is a homogeneous function of degree 2 in generalized velocities. ==Example: pendulum==

As shown at right, a simple pendulum is a system composed of a weight and a string. The string is attached at the top end to a pivot and at the bottom end to a weight. Being inextensible, the string’s length is a constant. Therefore, this system is scleronomous; it obeys scleronomic constraint \sqrt{x^2+y^2} - L = 0, where (x,y) is the position of the weight and L is length of the string. Take a more complicated example. Refer to the next figure at right, Assume the top end of the string is attached to a pivot point undergoing a simple harmonic motion x_t=x_0\cos\omega t , where x_0 is amplitude, \omega is angular frequency, and t is time. Although the top end of the string is not fixed, the length of this inextensible string is still a constant. The distance between the top end and the weight must stay the same. Therefore, this system is rheonomous as it obeys constraint explicitly dependent on time \sqrt{(x - x_0\cos\omega t)^2+y^2} - L = 0. ==See also==