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Lagrange and Hamilton Equations for Many-Particle Systems

Lagrange equations (4.8) and Hamilton equations (4.10) are valid also for an ensemble of material points interacting with each other and with some external potentials. In this case

L\left(q_1, \ldots, q_n, \dot{q}_1, \ldots, \dot{q}_n\right...
...= 1}^n
\frac{m_i \dot{q}_i^2}{2} - U(q_1, q_2, \ldots, q_n),
\end{displaymath} (4.8)

where $U(q_1, q_2, \ldots, q_n)$ is the potential energy of the multi-particle system, and

H\left(q_1, \ldots, q_n, p_1, \ldots, p_n\right)
= \sum_{i=1}^n p_i q_i - L
\end{displaymath} (4.9)

Observe how these two functions are defined in terms of coordinates qi and either velocities $\dot{q}_i$ or momenta pi. Hamiltonian is a function defined on the phase space of a system.

Zdzislaw Meglicki