Kinetic Energy of Classical Particle

Theorem

Let $\MM$ be an $n$-dimensional Euclidean manifold.

Let $P$ be a particle with an inertial mass $m_i$.

Let $t$ be the time variable of $P$.

Suppose the position of $P$ is a real differentiable $n$-dimensional vector-valued mapping $\mathbf x = \map {\mathbf x} t$.

Then the kinetic energy of a classical particle $P$ is:

$T = \dfrac {m_i} 2 \paren {\dfrac {\d \mathbf x} {\d t} }^2$

where $\paren {\dfrac {\d \mathbf x} {\d t} }^2$ is the dot product of the vector $\dfrac {\d \mathbf x} {\d t}$ with itself.

Proof

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Sources

• 1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 4.21$: The Principle of Least Action