Exploring LaGrangian Derivation: Why is It T-V?

[BiGyElLoWhAt]

Last semester I had intermediate mechanics, and we spent a good amount of the class studying the LaGrangian. One thing that I never got an explanation for was why $L = T-V$, as opposed to $T+V$.

The only reason I can think of is the "give and take" relationship that Kinetic and Potential energy have in an isolated system; but is this correct?

Total energy seems more intuitive to me than the difference ($T-V$). I was just hoping someone could shed some light on this for me.

 

[Fredrik]

Is it not sufficient to know that T+V gives us the wrong equation of motion (F=-ma instead of F=ma)?

\begin{align}
&L=\frac{1}{2}m\dot x^2-V(x)\\
&0=\frac{\partial L}{\partial x} -\frac{d}{dt}\left(\frac{\partial L}{\partial\dot x}\right) =-\frac{dV}{dx} -\frac{d}{dt}(m\dot x)=F-m\ddot x.
\end{align}

 

[BiGyElLoWhAt]

Not really, I was looking for a more in depth explanation.

I guess that works. I was really hoping to grab some warm milk and get ready for story time, because that just isn't something that I would think to try, especially since the Chain Rule relationship holds for any definition of L (or it should mathematically), including defining it as T + V. It just so happens that you get a wrong acceleration value from that.

 

[George Jones]

Try

http://www.feynmanlectures.caltech.edu/II_19.html

 

[pmacias]

The Lagrangian formulation of mechanics is a powerful tool for studying the dynamics of a system. It is derived from the principle of least action, which states that the path a system takes between two points in time is the one that minimizes the action, defined as the integral of the Lagrangian over time.

The Lagrangian, denoted by $L$, is defined as the difference between the kinetic energy $T$ and the potential energy $V$ of the system, i.e. $L = T-V$. This may seem counterintuitive at first, as we are used to thinking of energy as a total quantity, represented by the sum of kinetic and potential energy. However, the Lagrangian approach offers several advantages over the traditional Newtonian approach.

Firstly, the Lagrangian formulation allows us to treat all types of forces in a unified manner, without having to explicitly define each one separately. This is because the Lagrangian takes into account the total energy of the system, including all types of forces. In contrast, the Newtonian approach requires us to consider each force separately and apply the principle of superposition.

Secondly, the Lagrangian formulation is independent of the choice of coordinates used to describe the system. This is because the Lagrangian is a scalar quantity, whereas the Newtonian equations of motion are vector equations that depend on the choice of coordinate system. This makes the Lagrangian approach more elegant and easier to work with, especially for complex systems.

Lastly, the Lagrangian formulation also provides a more intuitive understanding of the dynamics of a system. The Lagrangian represents the total energy of the system, and the difference between the kinetic and potential energy represents the energy that is available for the system to perform work. This highlights the "give and take" relationship between kinetic and potential energy that you mentioned.

In summary, the Lagrangian formulation is a powerful and elegant approach to studying the dynamics of a system. The choice of $L = T-V$ is not arbitrary, but rather a result of the principle of least action and offers several advantages over the traditional Newtonian approach. I hope this explanation has helped clarify the reasoning behind $L = T-V$ in the Lagrangian formulation.