# Does Newtonian $F=ma$ imply the least action principle in mechanics?

I've learned that Newtonian mechanics and Lagrangian mechanics are equivalent, and Newtonian mechanics can be deduced from the least action principle.

Could the least action principle $\min\int L(t,q,q')dt$ in mechanics be deduced from Newtonian $F=ma$?

Sorry if the question sounds beginnerish

Newton's second law implies into Least Action Principle under two assumptions:

1. Virtual Work Principle holds.
2. There are no dissipative forces.

When applied to a system of particles, Newton´s second law can be written as $$\sum_i(\vec F_i-\dot{\vec p}_i)=0.$$ Decomposing this force into a constraint force $\vec F_i^c$ and an applied force $\vec F_i^a$ we obtain $$\sum_i(\vec F_i^a+\vec F_i^c-\dot{\vec p}_i)\cdot\delta\vec r_i=0,$$ where $\delta \vec r_i$ is the virtual displacement of particle $i$. The Principle of Virtual Work says that the total work of constraint forces is zero along virtual displacements, $$\sum_i\vec F_i^c\cdot\delta\vec r_i=0.\tag1$$ The last two equations can be combined into the d'Alembert Principle, $$\sum_i(\vec F_i^a-\dot{\vec p}_i)\cdot\delta\vec r_i=0,\tag2$$ which gives the equations of motion of the system. The last step is to integrate Eqn. (2) from time $t_1$ to $t_2$, $$\int_{t_1}^{t_2}\sum_i(\vec F_i^a-\dot{\vec p}_i)\cdot\delta\vec r_idt=0$$ and assume that the applied forces are derived from a potential (no dissipative forces). The first term gives raise to a potential energy whereas the second leads to a kinetic energy. Moreover, the variation $\delta$ commutes with the integral and the above equation can be written as $$\delta\int_{t_1}^{t_2}(T-V)dt=0,$$ which is the Least Action Principle.

• But the Principle of Least Action and the Principle of Virtual Work are equivalent. Your answer simply redirects the question to "how does Newton's law imply the Principle of Virtual Work"? Unless I am missing something huge, this isn't very useful.
– Max
Jul 21, 2020 at 12:42
• @Max Least Action and Virtual Work are not equivalent. The latter holds even for dissipative forces whereas the former not. I don´t know any proof relating Newton's law and Virtual Work. Both, Newton's law and Virtual Work together, lead to Least Action. Jul 23, 2020 at 0:17

You also need an expression for the Lagrangian, which in classical mechanics is $$L = T - U$$

Where $T$ is the kinetic energy and $U$ is the potential energy.

Provided that you can associate a potential $U$ to the force $\vec{F}$ such that $\vec{F} = - \vec{\nabla} U$ (such a force is said to be conservative), the principle of least action and Newton second's law are equivalent.

The demonstration for a single particle in 1D ($T = m v_x^2 /2$, $F = -dU(x)/dx$) is actually a good exercise.

• Here are some hints: First rewrite Newton's 2nd law in terms of the Lagrangian notation: $F = - dU/dq$ , $a = d\dot{q}/dt$ Jul 8, 2017 at 23:45