Final answer:
The equations ∂L/∂qi = p˙i and ∂L/∂q˙i = pi, known as the Lagrange's equations, are derived from the principle of stationary action and hold indefinitely for any mechanical system described by a Lagrangian function.
Step-by-step explanation:
The equations ∂L/∂qi = p˙i and ∂L/∂q˙i = pi are known as the Lagrange's equations and are fundamental in Lagrangian mechanics.
These equations are derived from the principle of stationary action, also known as the Euler-Lagrange equations.
They hold indefinitely and are valid for any mechanical system described by a Lagrangian function, which is the difference between the kinetic and potential energies of the system.
What the equations mean is this:
The first equation, ∂L/∂qi = p˙i, states that the derivative of the Lagrangian function L with respect to the coordinates qi gives us the generalized momentum p˙i, which is the time derivative of the generalized coordinate qi.
The second equation, ∂L/∂q˙i = pi, states that the derivative of the Lagrangian function L with respect to the generalized velocities q˙i gives us the generalized force pi, which is the conjugate momentum associated with the generalized coordinate qi.
These equations are invaluable in simplifying the mathematical description of complex physical systems and allow for the derivation of the equations of motion for a wide range of problems.