Lagrangian Explained: T-V Meaning & Time Integration

[gulsen]

I've been thinking on it for a while, and can't find a satisfying argument. What would L=T-V mean physically? And what would it's time integral mean? What (physically what) are we minimizing?

Following Feynman, it should be explained at freshmen level if we have understood it.

 

[lalbatros]

Gulsen,

First, what's the meaning of "meaning"?
Do you want to understand how to use it?
Do you want to know why is physics like that?

By analogy, how would you answer the question "what is energy"?
Is energy a conserved quantity: that's not enough to define it.
Is energy the quantity conserved for a time-shift symmetric system: that's enough to define it, but are you happy with this meaning?

Note here that the Lagrangian is not a conserved quantity like energy.
Its meaning doesn't appear in a conservation law.
Rather, it appears in a "variational law" called the "least action principle".
Physics can be described (partly) by conservation principles: energy, momentum, ...
Alternatively, it can be described by (more general) variational principles: just as if nature was based on a kind of 'economical' principle.

I think that the meaning of a concept in physics lies only in its relation to other concepts. There is no "out of the box" understanding that would provide an all-encompassing meaning. There are still irreductible concepts.

Nevertheless a "strong meaning" to the lagragian is given by its relation to the action and the relation of action with quantum mechanics. The action is the quantity that is extremised by the actual path of particule in between two points. This was discovered a long a long time ago and derived from the more 'pictural' basis of classical mechanics (usual forces and Newton laws). It is striking that this was discovered much before quantum mechanics was discovered, a sort of scientific miracle.

You will find easily on the web the link between the least action principle and quantum mechanics. As a matter of fact, the principle of least action results from quantum mechanics approaching classical mechanics when the wavelengths of the wave-functions become 'small' (the object becoming "large"!). We may call that a meaning, since this shows that the Lagragian is related to the phases of the wavefunctions when wavelengths become very small.

But the best reading for that is Feynman ...

But note an even more striking fact: variational principles don't only explain the structure of classical mechanics by its quantum legacy, it appears that variational principles are also within the structure of quantum mechanics itself.

And now a more speculative point of view:

Row Frieden speculated that the appearance of variational principles everywhere in physics has a fundamental reason: the information that can be obtained on a system need an interaction. Therefore, information cannot be considered as independent from physics itself, and this is reflected in the law of physics by the variational principles. It should be no surprise then that Roy Frieden gives a new meaning to the Lagrangian and also to the two terms of the Lagragian. Unfortunaly I got the impression that this new theory doesn't contribute yet to physics, it is more like an interpretation.

But is "meaning" not a synonim for "interpretation" ?

 

[charng]

Lagrangian is a mathematical concept that is used in classical mechanics to describe the motion of a system. It is based on the principle of least action, which states that the path a system takes between two points in time is the one that minimizes the total action (or energy) of the system.

In the Lagrangian formulation, the total energy of a system is expressed as the difference between the kinetic energy (T) and the potential energy (V). This is what the equation L=T-V represents. T represents the energy of motion, while V represents the energy associated with the position of the system.

The time integral of the Lagrangian, also known as the action, has a physical meaning of its own. It represents the total energy of the system over a certain period of time. This means that by minimizing the action, we are essentially finding the path that the system takes that results in the minimum energy expenditure.

In simpler terms, the Lagrangian formulation allows us to understand the motion of a system by considering the energy associated with its position and motion. By minimizing the action, we can determine the path that the system takes that requires the least amount of energy.

To explain this concept to a freshmen level, we can use the analogy of a ball rolling down a hill. The total energy of the ball is made up of its kinetic energy (energy of motion) and its potential energy (energy due to its position on the hill). The path that the ball takes down the hill is the one that requires the least amount of energy, which is equivalent to minimizing the action in the Lagrangian formulation.

In conclusion, the Lagrangian formulation is a powerful tool in classical mechanics that allows us to understand the motion of a system by considering the energy associated with its position and motion. By minimizing the action, we can determine the path that the system takes that requires the least amount of energy.