Hamilton's Principle

We saw, in Chapter 9, that we can specify the instantaneous configuration of a conservative dynamical system with ${\cal F}$ degrees of freedom in terms of ${\cal F}$ independent generalized coordinates $q_i$, for $i=1,{\cal F}$. Let $K(q_1,q_2,\cdots,q_{\cal F},$ $\dot{q}_1,\dot{q}_2,\cdots,\dot{q}_{\cal F},t)$ and $U(q_1,q_2,\cdots,q_{\cal F},t)$ represent the kinetic and potential energies of the system, respectively, expressed in terms of these generalized coordinates. Here, $\dot{~}\equiv d/dt$. The Lagrangian of the system is defined

$$L(q_1,q_2,\cdots,q_{\cal F}, \dot{q}_1,\dot{q}_2,\cdots,\dot{q}_{\cal F},t) = K - U.$$ (711)

Finally, the ${\cal F}$ Lagrangian equations of motion of the system take the form

$$\frac{d}{dt}\!\left(\frac{\partial L}{\partial \dot{q}_i}\right) -\frac{\partial L}{\partial q_i} =0,$$ (712)

for $i=1,{\cal F}$.

Note that the above equations of motion have exactly the same mathematical form as the Euler-Lagrange equations (708). Indeed, it is clear, from Section 10.4, that the ${\cal F}$ Lagrangian equations of motion (712) can all be derived from a single equation: namely,

$$\delta\int_{t_1}^{t_2} L(q_1,q_2,\cdots,q_{\cal F}, \dot{q}_1,\dot{q}_2,\cdots,\dot{q}_{\cal F},t)\,dt = 0.$$ (713)

In other words, the motion of the system in a given time interval is such as to maximize or minimize the time integral of the Lagrangian, which is known as the action integral. Thus, the laws of Newtonian dynamics can be summarized in a single statement:

The motion of a dynamical system in a given time interval is such as to maximize or minimize the action integral.

(In practice, the action integral is almost always minimized.) This statement is known as Hamilton's principle, and was first formulated in 1834 by the Irish mathematician William Hamilton.