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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

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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.

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  • $\begingroup$ Here are some hints: First rewrite Newton's 2nd law in terms of the Lagrangian notation: $F = - dU/dq$ , $a = d\dot{q}/dt$ $\endgroup$ – Alexis Prel Jul 8 '17 at 23:45
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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.

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