Lagrangian Mechanics Question: How do we get the T and V terms?

[McLaren Rulez]

According to my CM text, Lagrangian Mechanics can be used to derive Newton's laws. We define the Lagrangian as L=T-V.

Now, how do we know what T is? Is it defined to be 1/2mv^2? The only way I know how to derive that is using the work energy theorem which feels like 'cheating' since I am sort of using the Newtonian formulation of things to get it.

Similarly, how do we find the potential? The way it is done in regular Newtonian mechanics is by showing that curl of F is zero and hence we can find a potential function from the original force. For Lagrangian Mechanics, is the potential thought of as the fundamental quantity that is always given (like the way the force is given in Newtonian Mechanics)?

Thank you

 

[praharmitra]

Well, yes.

T is defined to be $$\frac{1}{2}m\dot{\textbf{r}}^2$$

V however is defined only for conservative forces. One cannot write a Lagrangian of the form T-V for non-conservative forces. However, there are other forms of the Lagrangian that can be used in such situations (I remember learning this when I did classical mechanics, but am unclear now what exactly this was. I'll read it up and reply back).

 

[A. Neumaier]

According to my CM text, Lagrangian Mechanics can be used to derive Newton's laws. We define the Lagrangian as L=T-V.

Now, how do we know what T is? Is it defined to be 1/2mv^2?

Starting with Newton's mechanics (as undiscussed assumptions), one finds that one can encode it all into a Lagrangian where T is the sum of all kinetic energies of the particles and V the sum of all potential energies.

Conversely, one may start with these formulas for T and V (as undiscussed assumptions) and derive Newton's laws. However, one can then explore what different choices for T and V imply and gets a much richer and more flexible mechanics as Newton's. Thus one learns Newton to get a start, and then learns the much more powerful Lagrangian and Hamiltonian frameworks to do more modern stuff.

 

[Cleonis]

We define the Lagrangian as L=T-V.

Now, how do we know what T is? Is it defined to be 1/2mv^2? The only way I know how to derive that is using the work energy theorem which feels like 'cheating' since I am sort of using the Newtonian formulation of things to get it.

Well, historically the Newtonian formulation came first. The Newtonian formulation is momentum oriented. The concept of kinetic energy did not become a regular part of the physicist's toolbox until much, much later.

As Neumaier mentions, the Lagrangian and Hamiltonian frameworks are more versatile, they have a wider range of deployment.

The cases where the Newtonian formulation is suitable form a subset of the sets of cases where a Lagrangian/Hamiltonian framework is suitable. I suppose that that is why some authors describe that 'Newtonian dynamics can be derived from Lagrangian mechanics'. In the sense that the Newtonian formulation has been eclipsed by the Lagrangian/Hamiltonian formulation that is in itself a true statement.

But historically it was the lessons learned from the Newtonian formulation that prepared the physics community for arriving at Lagrangian/Hamiltonian mechanics.

 

[PJC01]

for your question. In Lagrangian Mechanics, the T and V terms are obtained through a process called "Lagrangian formulation". This formulation involves defining the kinetic energy (T) and potential energy (V) of a system in terms of the generalized coordinates and their time derivatives. The kinetic energy is defined as the sum of the kinetic energies of each individual particle in the system, while the potential energy is defined as the sum of the potential energies of each individual particle in the system.

To find the kinetic energy, we can use the classical definition of kinetic energy, 1/2mv^2, for each particle in the system. However, in more complex systems, this definition may not be applicable and we may need to use other methods to calculate the kinetic energy. For example, in a rotating system, we may use the formula 1/2Iω^2, where I is the moment of inertia and ω is the angular velocity.

Similarly, the potential energy can be calculated using various methods depending on the system. In some cases, the potential energy may be given directly as a function of the generalized coordinates, while in other cases it may need to be derived from the forces acting on the system. In Lagrangian Mechanics, the potential energy is considered to be a fundamental quantity, much like the force in Newtonian Mechanics.

In summary, the T and V terms in the Lagrangian formulation are obtained through various methods depending on the system. They are not simply defined as 1/2mv^2 and the potential function, but rather they are calculated using the principles of classical mechanics and the specific characteristics of the system being studied.