Hamiltonian of Standard Lagrangian is Total Energy

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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 $P$.


$\blacksquare$

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