Euler-Lagrange Equation for Conservative System

Theorem

Let $P$ be a conservative system composed of $n \in \N$ particles.

The Euler-Lagrange equation for $P$ is given by:

$\map {\dfrac \d {\d t} } {\dfrac {\partial L} {\partial \dot q_j} } - \dfrac {\partial L} {\partial q_j} = 0$

where:

$L$ denotes the Lagrangian of $P$

$q_j$ denotes the positions of the $n$ particles composing $P$

$t$ denotes time.

Proof

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Sources

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