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

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

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

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

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