Partial Derivative of Hamiltonian with respect to Displacement
Jump to navigation
Jump to search
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 momentum is given by:
- $\dfrac {\partial H} {\partial y_i} = -p_i'$
Proof
This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
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