Kinetic Energy

The kinetic energy of a Punctual Mass whose position with respect to a fixed (inertial) reference frame is given by $p\in {\mathbb{R}}^D$ (where the number of dimensions can be $D=1,2,3$) reads as:

$$T=\frac{1}{2}m||\dot{p}|{|}^2=\frac{1}{2}m{\dot{p}}^T\dot{p}$$

Where $\dot{p}$ is the mass velocity. We will often use the $W$ matrix

$$\begin{array}{rl}T& =\frac{1}{2}{\dot{q}}^T\left(\sum _{i=1}^N{m}_i{\left(\frac{\partial p_i}{\partial q}\right)}^T\frac{\partial p_i}{\partial q}\right)\dot{q}\\ T& =\frac{1}{2}{\dot{q}}^{T}W\dot{q}\end{array}$$

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Some Observations

Given that q is the generalized coordinates in the Configuration Space, then one could more formally write the mass velocity as

$$\dot{p}=\frac{\partial p}{\partial q}\dot{q}$$

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If the mechanical systems is made of $N$ punctual masses of position $p_i$, determined by the generalized coordinates $q$

$$T=\frac{1}{2}\sum _{i=1}^N{m}_i||{\dot{p}}_i|{|}^2=\frac{1}{2}\sum _{i=1}^N{m}_i{\dot{p}}_i^T{\dot{p}}_i$$

Where each velocity isa given by the chain rule

$${\dot{p}}_i=\frac{\partial p_i}{\partial q}\dot{q}$$

Combination of these gives

$$T=\frac{1}{2}{\dot{q}}^T\left(\sum _{i=1}^N{m}_i(\frac{\partial p_i}{\partial q}{)}^T\frac{\partial p_i}{\partial q}\right)\dot{q}$$

Where we defined $W$, which is a square symmetric, (semi)positive-definite matrix of size ${\mathbb{R}}^{n_q\times n_q}$.

$$W=\sum _{i=1}^N{m}_i(\frac{\partial p_i}{\partial q}{)}^T\frac{\partial p_i}{\partial q}$$