McCarthy Formalism

A mechanical system is scleronomous if the equations of constraints do not contain the time as an explicit variable. Such constraints are called scleronomic constraints.

Application

Main article：Generalized velocity

In 3-D space, a particle with mass $m$, velocity $𝐯$ has kinetic energy

$T=\frac{1}{2}m{v}^2.$

Velocity is the derivative of position with respect time. Use chain rule for several variables:

$𝐯=\frac{d𝐫}{dt}=\sum _i\frac{\partial 𝐫}{\partial q_i}{\dot{q}}_i+\frac{\partial 𝐫}{\partial t}.$

Therefore,

$T=\frac{1}{2}m{(\sum _i\frac{\partial 𝐫}{\partial q_i}{\dot{q}}_i+\frac{\partial 𝐫}{\partial t})}^2.$

Rearranging the terms carefully,

$T=T_0+T_1+T_2:$

$T_0=\frac{1}{2}m{\left(\frac{\partial 𝐫}{\partial t}\right)}^2,$

$T_1=\sum _{i}m\frac{\partial 𝐫}{\partial t}\cdot \frac{\partial 𝐫}{\partial q_i}{\dot{q}}_i,$

$T_2=\sum _{i,j}\frac{1}{2}m\frac{\partial 𝐫}{\partial q_i}\cdot \frac{\partial 𝐫}{\partial q_j}{\dot{q}}_i{\dot{q}}_j,$

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:

$\frac{\partial 𝐫}{\partial t}=0.$

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

File:SimplePendulum01.JPG

A simple 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.

A simple pendulum with oscillating pivot point

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; it obeys rheonomic constraint

$\sqrt{(x-x_0\cos \omega t{)}^2+y^2}-L=0.$

See also

• Lagrangian mechanics
• Holonomic system
• Nonholonomic system
• Rheonomous