“Lagrange’s equations”的概念、定义、翻译、参考文献-科学参考

Lagrange’s equations

Mathematics

单词	Lagrange’s equations
释义	Lagrange’s equations Mathematics For a mechanical system with n degrees of freedom in which conservative forces act and all constraints are holonomic, the Lagrangian is L = T−V, where T denotes kinetic energy and V denotes potential energy. For a light pendulum of length l with a mass m at the end and making an angle θ‎ with the downward vertical, $L = \frac{1}{2} m l^{2} {\dot{θ}}^{2} + m g l cos θ .$ For such a system, defined by generalized coordinates q₁, q₂, …, q_n, Lagrange’s equations state that $\frac{d}{d t} (\frac{\partial L}{\partial {\dot{q}}_{i}}) - \frac{\partial L}{\partial q_{i}} = 0 for 1 \leq i \leq n .$ In the example of the pendulum, this gives $0 = \frac{d}{d t} (m l^{2} \dot{θ}) + m g l sin θ = m l^{2} \ddot{θ} + m g l sin θ .$ See also Hamiltonian mechanics.
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