## Hamilton's principle Hamilton's principle, also known as the "least action principle", proposes a different approach compared to Newton's laws of motion. For Newton, the motion of a system follows $\vec{F_i} = m_i \vec{a_i},$ i.e., Newton's second law. However, Hamilton considers a functional between initial and final states, during the time interval $t_1$ to $t_2$. ![[Drawing 2023-07-22 23.28.53.excalidraw.svg]] If the initial and final time & positions are the same, then we have $I = \int_{t_1}^{t_2} \mathcal{L}(q_1,\ldots,q_n,\dot{q_1},\ldots,\dot{q_n},t) \, \mathrm{d}t = 0,$ i.e., $I$ is constant. Here, $\mathcal{L}$ is the *Lagrangian*, with $\mathcal{L} = T - V$, representing the difference between kinetic energy and potential energy, and $q_i$ denotes generalized coordinates. > [!note] > The main idea of Hamilton's principle is that there exists a quantity, such that the integration of this quantity along the path remains constant. > $\delta I = 0$ signifies that this quantity is a **stable** quantity, and does not need to be minimized. > >[!info] >Read more on the wiki: https://en.wikipedia.org/wiki/Hamilton%27s_principle