In 3-D space, a particle with mass ${\displaystyle m\,\!}$ , velocity ${\displaystyle \mathbf {v} \,\!}$  has kinetic energy ${\displaystyle T\,\!}$ 

${\displaystyle T={\frac {1}{2}}mv^{2}\,\!.}$ 

Velocity is the derivative of position ${\displaystyle r\,\!}$  with respect to time ${\displaystyle t\,\!}$ . Use chain rule for several variables:

${\displaystyle \mathbf {v} ={\frac {d\mathbf {r} }{dt}}=\sum _{i}\ {\frac {\partial \mathbf {r} }{\partial q_{i}}}{\dot {q}}_{i}+{\frac {\partial \mathbf {r} }{\partial t}}\,\!.}$ 

where ${\displaystyle q_{i}\,\!}$  are generalized coordinates.

Therefore,

${\displaystyle T={\frac {1}{2}}m\left(\sum _{i}\ {\frac {\partial \mathbf {r} }{\partial q_{i}}}{\dot {q}}_{i}+{\frac {\partial \mathbf {r} }{\partial t}}\right)^{2}\,\!.}$ 

Rearranging the terms carefully,^[1]

${\displaystyle T=T_{0}+T_{1}+T_{2}\,\!:}$ 

${\displaystyle T_{0}={\frac {1}{2}}m\left({\frac {\partial \mathbf {r} }{\partial t}}\right)^{2}\,\!,}$ 

${\displaystyle T_{1}=\sum _{i}\ m{\frac {\partial \mathbf {r} }{\partial t}}\cdot {\frac {\partial \mathbf {r} }{\partial q_{i}}}{\dot {q}}_{i}\,\!,}$ 

${\displaystyle T_{2}=\sum _{i,j}\ {\frac {1}{2}}m{\frac {\partial \mathbf {r} }{\partial q_{i}}}\cdot {\frac {\partial \mathbf {r} }{\partial q_{j}}}{\dot {q}}_{i}{\dot {q}}_{j}\,\!,}$ 

where ${\displaystyle T_{0}\,\!}$ , ${\displaystyle T_{1}\,\!}$ , ${\displaystyle T_{2}\,\!}$  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:

${\displaystyle {\frac {\partial \mathbf {r} }{\partial t}}=0\,\!.}$ 

Therefore, only term ${\displaystyle T_{2}\,\!}$  does not vanish:

${\displaystyle T=T_{2}\,\!.}$ 

Kinetic energy is a homogeneous function of degree 2 in generalized velocities .

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

${\displaystyle {\sqrt {x^{2}+y^{2}}}-L=0\,\!,}$ 

where ${\displaystyle (x,y)\,\!}$  is the position of the weight and ${\displaystyle L\,\!}$  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

${\displaystyle x_{t}=x_{0}\cos \omega t\,\!,}$ 

where ${\displaystyle x_{0}\,\!}$  is amplitude, ${\displaystyle \omega \,\!}$  is angular frequency, and ${\displaystyle t\,\!}$  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

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

• Lagrangian mechanics
• Holonomic system
• Nonholonomic system
• Rheonomous
• Mass matrix