Hamiltonian of Standard Lagrangian is Total Energy
Jump to navigation
Jump to search
Theorem
Let $P$ be a physical system of classical particles.
Let $L$ be a standard Lagrangian associated with $P$.
Then the Hamiltonian of $P$ is the total energy of $P$.
Proof
\(\ds H\) | \(=\) | \(\ds -L + \sum_{i \mathop = 1}^n \dot {x_i} L_{\dot {x_i} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds - \paren{T - V} + \sum_{i \mathop = 1}^n \dot {x_i} \dfrac {\partial} {\partial {\dot x}_i }\paren {\map T {\dot x} - \map V {t, x} }\) | Definition of Standard Lagrangian | |||||||||||
\(\ds \) | \(=\) | \(\ds -\paren{T - V} + \sum_{i \mathop = 1}^n m_i \dot {x_i}^2\) | Kinetic Energy of Classical Particle | |||||||||||
\(\ds \) | \(=\) | \(\ds -T + V + 2 T\) | Kinetic Energy of Classical Particle | |||||||||||
\(\ds \) | \(=\) | \(\ds T + V\) |
By definition, the last expression is the total energy of $P$.
$\blacksquare$
Sources
- 1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 4.21$: The Principle of Least Action
- 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