Principle of Stationary Action with Standard Lagrangian implies Newton's Laws of Motion

Theorem
Let $\mathcal M$ be an $n$-dimensional Euclidean manifold.

Let $P$ be a physical system composed of classical particles with inertial masses $m_i$ with $i \in \N$.

Let $\mathbf x = \map {\mathbf x} t$ be twice-differentiable vector-valued function embedded in $\mathcal M$.

Suppose ${\mathbf x}_i$ represents the position of the $i$-th particle of $P$.

Suppose, the action of $P$ is of the following form:


 * $\displaystyle S = \int_{t_0}^{t_1} L \rd t$

where $L$ is the standard Lagrangian.

Suppose, all (internal or external) forces ${\mathbf F}_i$ acting upon $P$ are of the form


 * ${\mathbf F}_i = - \dfrac {\partial U}{\partial \mathbf x_i}$

where


 * $\displaystyle U = \map U {t, \set{\mathbf x_i} }$

is a differentiable real function, and $\set{\mathbf x_i}$ denotes dependence on the positions of all the particles.

Then, the stationary point of $S$ implies Newton's Laws of Motion.

Proof
Standard Lagrangian is of the following form:


 * $\displaystyle L = \sum_{i = 1}^n \frac {m_i} 2 \dot {\mathbf x}_i^2 - U$

By the Principle of Stationary Action, equations of motion of $P$ are:


 * $\displaystyle m_i {\ddot {\mathbf x} }_i + \dfrac {\partial U}{\partial \mathbf x_i} = 0 \; \forall i \in \N$

By definition of velocity:


 * $\displaystyle {\dot {\mathbf x} }_i = {\mathbf v}_i$.

By assumption:


 * ${\mathbf F}_i = - \dfrac {\partial U}{\partial {\mathbf x}_i}$

Then it follows that the equations of motion of $P$ can be rewritten as:


 * $ \dfrac {\d}{\d t} \paren {m_i {\mathbf v}_i } = {\mathbf F}_i$

By definition, these equations together represent Newton's second law.