# Kinetic energy in Lagragian mechanics [duplicate]

In my classical mechanics class they asked why the kinetic energy for an holonomic mechanical system has the homogeneous quadratic form. Of course for a autonomous standard system( system that is holonomic and the constraint do no virtual work) you can just calculate that $$T = \frac { 1 } { 2 } \sum _ { i = 1 } ^ { M } m _ { i } \left( v _ { i } \cdot v _ { i } \right) = \sum _ { j = 1 } ^ { n } \sum _ { k = 1 } ^ { n } a _ { j k } ( q ) \dot { q } _ { j } \dot { q } _ { k }$$ with $$a _ { j k } ( \boldsymbol { q } ) = \frac { 1 } { 2 } \sum _ { i = 1 } ^ { N } m _ { i } \left( \frac { \partial \boldsymbol { r } _ { i } } { \partial q _ { j } } \cdot \frac { \partial r _ { i } } { \partial q _ { k } } \right)$$ So you can write $$T = \dot { q } ^ { T } \cdot A \cdot \dot { q }$$ But what is the physical reason?

That the kinetic term $$T(q,\dot{q},t)$$ is a second-order polynomial in the generalized velocities $$\dot{q}$$ can be viewed as:

1. a consequence of

• the holonomic constraint that the positions $${\bf r}_i(q,t)$$ do not depend on the generalized velocities $$\dot{q}$$,

• and that the non-relativistic kinetic energy is $$T=\sum_{i=1}^N\frac{m_i}{2}\dot{\bf r}_i^2$$,