Hamilton's Principle

The path of a physical system between two states is the one that minimizes the difference between kinetic and potential energies.

Pasted image 20241014115408.png

Action:

The action is defined as the integral of the Lagrangian $L$ , which is a function of the generalized coordinates $q_{i}$ , their time derivatives ${\dot{q}}_{i}$ , and time $t$ :

S = \int_{t_{1}}^{t_{2}} L (q_{i}, {\dot{q}}_{i}, t) d t

Here, $L (q_{i}, {\dot{q}}_{i}, t)$ is the Lagrangian, typically given by the difference between the kinetic energy $T$ and potential energy $U$ :

L = T - U

Hamilton’s principle can be expressed as:

δ S = 0

This means that the actual path a system takes is the one that makes the variation of the action zero.

To derive the equations of motion from this, we compute the variation $δ S$ and set it to zero:

δ S = δ \int_{t_{1}}^{t_{2}} L (q_{i}, {\dot{q}}_{i}, t) d t = 0

This leads to the Euler-Lagrange equations:

\frac{d}{d t} (\frac{\partial L}{\partial {\dot{q}}_{i}}) - \frac{\partial L}{\partial q_{i}} = 0

The Euler-Lagrange equations are the equations of motion that follow from Hamilton's principle.

lookout Euler-Lagrange Equation .
and Generalized Coordinates.