Is the principle of least action a tautology ?

[motion_ar]

In classical mechanics, if we consider a force field (uniform or non-uniform) in which the acceleration $\vec{a}_{\scriptscriptstyle \mathrm A}$ of a particle A is constant, then

$$\vec{a}_{\scriptscriptstyle \mathrm A} - \, \vec{a}_{\scriptscriptstyle \mathrm A} = 0$$$${\vphantom{\delta \int_{t_{1}}^{t_{2}}}} \left( \vec{a}_{\scriptscriptstyle \mathrm A} - \, \vec{a}_{\scriptscriptstyle \mathrm A} \right) \cdot \delta \vec{r}_{\scriptscriptstyle \mathrm A} = 0$$$$\int_{t_{1}}^{t_{2}} \left( \vec{a}_{\scriptscriptstyle \mathrm A} - \, \vec{a}_{\scriptscriptstyle \mathrm A} \right) \cdot \delta \vec{r}_{\scriptscriptstyle \mathrm A} \; \, dt = 0$$$$\delta \int_{t_{1}}^{t_{2}} \left( {\textstyle \frac{1}{2}} \; \vec{v}_{\scriptscriptstyle \mathrm A}^{\,2} \, + \, \vec{a}_{\scriptscriptstyle \mathrm A}^{\vphantom{^{\,2}}} \cdot \vec{r}_{\scriptscriptstyle \mathrm A}^{\vphantom{^{\,2}}} \right) \, dt = 0$$$$m_{\scriptscriptstyle \mathrm A}^{\vphantom{^{\,2}}} \; \delta \int_{t_{1}}^{t_{2}} \left( {\textstyle \frac{1}{2}} \; \vec{v}_{\scriptscriptstyle \mathrm A}^{\,2} \, + \, \vec{a}_{\scriptscriptstyle \mathrm A}^{\vphantom{^{\,2}}} \cdot \vec{r}_{\scriptscriptstyle \mathrm A}^{\vphantom{^{\,2}}} \right) \, dt = 0$$$$\delta \int_{t_{1}}^{t_{2}} \left( T_{\scriptscriptstyle \mathrm A} - \, V_{\scriptscriptstyle \mathrm A} \right) \, dt = 0$$$$\delta \int_{t_{1}}^{t_{2}} L_{\scriptscriptstyle \mathrm A} \; \, dt = 0$$where$$T_{\scriptscriptstyle \mathrm A} = {\textstyle \frac{1}{2}} \; m_{\scriptscriptstyle \mathrm A}^{\vphantom{^{\,2}}}\vec{v}_{\scriptscriptstyle \mathrm A}^{\,2}$$$$V_{\scriptscriptstyle \mathrm A} = - \; m_{\scriptscriptstyle \mathrm A}^{\vphantom{^{\,2}}} \; \vec{a}_{\scriptscriptstyle \mathrm A}^{\vphantom{^{\,2}}} \cdot \vec{r}_{\scriptscriptstyle \mathrm A}^{\vphantom{^{\,2}}}$$
If $\vec{a}_{\scriptscriptstyle \mathrm A}$ is not constant but $\vec{a}_{\scriptscriptstyle \mathrm A}$ is function of $\vec{r}_{\scriptscriptstyle \mathrm A}$ then the same result is obtained, even if Newton's second law were not valid.
${\vphantom{aat}}$

 

[eljose79]

Thank you for your post. I am a scientist and I would like to clarify a few points in your statement.

Firstly, it is important to note that the equations you have presented are derived from Newton's second law, which states that the net force acting on an object is equal to its mass multiplied by its acceleration. Therefore, it is not possible to obtain these equations without assuming the validity of Newton's second law.

Secondly, the equations you have presented are specific to classical mechanics and may not hold true in other branches of physics, such as quantum mechanics or relativity. In these branches, the concept of force and acceleration may be different and therefore the equations may also be different.

Lastly, while it is true that the equations hold even if the acceleration is a function of position, it is important to note that this is a simplification and may not always be the case in real-world situations. In more complex systems, the acceleration may also depend on other factors such as time, velocity, or external forces.

In conclusion, while the equations you have presented are valid in classical mechanics, it is important to keep in mind that they are derived from Newton's second law and may not hold true in other branches of physics or in more complex systems. Thank you for your post and I hope this helps clarify any confusion.