# Hamiltonian of Standard Lagrangian is Total Energy

## 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 $P$.

$\blacksquare$