Action in terms of Kinetic Energy

Definition

Let $A$ be the action for a given segment from point $A$ to point $B$ on the trajectory of a system is defined by the line integral:

$\ds \sum_i \int_A^B p_i \rd q_i$

Let $A$ act over a time interval of length $t$.

Then:

$A = 2 K t$

where $K$ denotes the mean kinetic energy of the system over $t$.

Proof

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Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): action: 1.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): action: 1.