Partial Derivative of Hamiltonian with respect to Time

Theorem

Let $H$ be a Hamiltonian of a system:

$\ds H = -F + \sum_{i \mathop = 1}^n y_i' F_{y_i'}$

where:

$F$ is the Lagrangian of the system

$y_i$ are the generalized coordinates

$y_i'$ is the first derivative of $q_i$ with respect to time

$p_i$ is the momentum of the system in those generalized coordinates:

$p_i = \dfrac {\partial F} {\partial q_i}$

Then the partial derivative of $H$ with respect to time is given by:

$\dfrac {\partial H} {\partial t} = \dfrac {\partial F} {\partial t}$

Proof

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Sources

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