# 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

\(\displaystyle H\) | \(=\) | \(\displaystyle -L + \sum_{i \mathop = 1}^n \dot {x_i} L_{\dot {x_i} }\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle - \paren{T - V} + \sum_{i \mathop = 1}^n \dot {x_i} \dfrac {\partial} {\partial {\dot x}_i }\paren {\map T {\dot x} - \map V {t, x} }\) | Definition of standard Lagrangian | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle - \paren{T - V} + \sum_{i \mathop = 1}^n m_i \dot {x_i}^2\) | Definition of kinetic energy of classical particles | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle - T + V + 2 T\) | Definition of kinetic energy of classical particles | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle T + V\) |

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