Updated: Oct 24, 2019
This article uses advanced mathematical techniques. Deep calculus understanding is required as well as some knowledge in newtonian mechanics and in the concept of energy.
What is wrong with Newtonian mechanics ?
There is no doubt that Newtonian mechanics did succeed in describing the macroscopical dynamics of the world, but the more complex systems we encounter and the more variables that we introduce, the harder it becomes to reach the results needed. This pattern can be seen in all systems. Luckily, some ‘solving algorithms’ (or if want to call them theories) are more efficient than others depending on the area you’re working on.
To clarify, quantum mechanics is only used to describe systems that deal with quanta sized variables. Therefore, no one will use quantum mechanics to describe a falling apples motion. Every algorithm that tends to solve a certain problem has a limitation and Newtonian mechanics’s limitation is linear algebra.
Even Though Newton is famously known as the ‘father of calculus’, he still chose to use an excessive amount of linear algebra in his mechanics description. The problem with that is that linear algebra gets extremely complex when we deal with systems that require motion that’s depending on other motions. Systems like a pendulum that’s attached to the end of another pendulum to the end of another pendulum. If you were to use newtonian mechanics it’s extremely hard to solve two pendulum levels problems. If you were to solve for the kinematics of three pendulums that are attached to each other using newtonian mechanics. You probably won’t be able to do that. A hundred pendulum ? definitely not. Again, that’s because most newtonian mechanics variables (such as velocity, acceleration, force and displacement) are vectors.
A stronger replacement with less complexity
The realisation of the needingness to a mathematical model that’s more resistant to multivariable kinematic systems appeared after discovering the principle of conservation of energy. That’s because the new discovery was able to describe falling objects way more efficiently than newtonian mechanics and that’s because it deals only with scalar quantities (energies). Also, at the time, integration wasn’t well understood and widely used. It took a while till Riemann came up with his integration proximity model which helped in developing Lagrangian mechanics. Lagrange, who’s a very famous and respected mathematician and scientists, was the one who came up with a new mechanical model that can handle high level complexity. The biggest field in physics today (Quantum mechanics) is built on this model. Furthermore, the new model (the lagrangian model) is also able to bypass constraint forces in any coordinate system, which is revolutionary because we’re not bound to the cartesian plane anymore.
The principle of the lagrangian
Lagrange was looking for a way to describe all traveling bodies kinematics without the need of linear algebra. He focused on the concept of energy, which can be represented in different forms, but he figured out that there must be a relation between all these different mathematical representations of energies. He named this relation after his own name, The lagrangian. The lagrangian is the difference between the kinetic energy and potential energy. He realised that the kinetic energy has a universal form which is (1⁄2)mv^2 but potential energy is the evil variable here since it doesn’t have a constant form. Since every system has a potential and kinetic energy properties. The mathematical form of the lagrangian for any systems is :
Where KE and T represent kinetic energy meanwhile PE and V represents the potential energy of the system. The lagrangian is a value that’s used to exploit this benefit of energy by introducing the principle of stationary action.
The principle of stationary action or hamilton’s principle
Every object has a lagrangian value as long as it’s in motion, and that’s due to the mathematical property of the lagrangian. The only time this is not true is only when the kinetic energy is equal to the potential energy. If you were to plot an object's traveled distance as a function of time. You’ll see that every point along that function that the object chose to travel through has a lagrangian property, hence kinetic and gravitational potential energy are functions of time too. When lagrange looked at such graphs, he noticed that they always have a parabolic shape but he didn’t understand why they couldn’t be a bigger or a smaller parabolas. In other words he wanted to understand what’s special with the traces that objects take during their journey.
Figure 1 shows an object that started traveling at time t1 from the point q1 reaching the end of it’s journey at t2 q2. There is an infinite amount of traces that the object could have taken but it ‘chose’ to follow the path δq. Lagrange discovered that the special thing about the path δq is that it has the most minimized Action of all the possible traces. In other words, he noticed that the object took that path in an attempt to minimize the overall kinetic energy lost and instead maximize its potential energy and that’s why it’s correct to say that the universe is ‘lazy’. The action is a value that each trace has, it determines how much kinetic energy spent and how much potential energy is reserved. The mathematical form of Action is :
The principle of least action states that the path that the object will always choose to take is the path that has the least action. The action is the integral of the Lagrangian of the object with respect to the time traveled. Lagrange discovered all of this with no consideration of newtonian mechanics in mind. So to prove his mechanics, Lagrange derived Newtonian mechanics from the action principle.
Newtonian mechanics derivation
Since there is an infinite amount of possible paths, we have to represent them with one mathematical formula. They can be represented in the following way.
Where x(t) is every possible path, x1 is the traveled path and η(t) is the change in the distance between the original path and the fictional path. Substituting this in the lagrangian that’s in the action formula, we get:
Since the universal form of the kinetic energy is (mv^2 )/2, v can be written as the derivative of x with respect to time. So:
Since eita is an infinitesimal value, we can cancel it out when we square it and end up with a kinetic formula in this form :
Now, the potential energy doesn’t have only one mathematical form because different kinds of potential energy exists and therefore different formulas. In this case, all we can do is to approximate the potential energy. Lagrange chose to use taylor series approximation.
Luckily we can ignore the second order terms and up because we’re squaring an infinitesimal number which logically should result in 0. Therefore :
As you may have noticed we still have eta in the lagrangian formula which makes sense because we’re trying to describe all the possible path solutions. Plugging in the results of both kinetic and potential energy to the action formula we get :
The resultant terms can be rearranged to result in two integrals. One that represents the real path that the object will surely take and the other that represent every other path. We can do this because we have two terms that has eta in them. The first term (m/2)(dx/dt)2 is obviously the real kinetic of the traveling body. The second term is the fictional kinetic energy that the body should have in every other non real path. PE(x) is the potential energy of the object if it travels the least action path. Finally, the last term, which has eta in it, is the potential energy of the body if it doesn't take the least action path. Rearranging the equation we get :
Which can also be seen as :
That’s because the first term in the equation in figure 3, is an integral of the real lagrangian since it doesn’t have eta in it and the second term is the change in action because eta is present there. The principle of least action or the hamilton’s principle says that the object will take the path where delta A (or the change in action) is equal to 0. All we did in the last step was substituting the action terms with the integral values So that :
If we could factor out η in the delta action integral, we should get a conserved value that’s equal to 0. So let’s do that. By the way, don’t forget that η=0 in the beginning and the end of the traveling body journey.
Integrating on both sides with T_final and T_initial as boundaries we get:
Thereafter we can rewrite the change in action functional by factoring out eta
Therefore in order for this to be correct the factor component that’s multiplied by eta should equal to 0. Otherwise all of this will be wrong. Why can’t eta be equal to zero ? well that’s because, by definition, eta is the change in the distance between the original path and all the fictional paths.
And as you may already know the negative derivative of the potential energy is the force and the second derivative of position with respect to time is the acceleration. In other words we end up with
Which is the basic concept of newtonian mechanics. Lagrange did all we did but backward (because we started with the concept of stationary action or least action and worked our way to this result).
Deriving the Euler lagrange equation
Well, by now. You might be wondering, how could all this mess be useful as a sort of mechanics. Here comes the role of the Euler-Lagrange equation. This is one of the most useful equations in all physics because it converts the lagrangian of a body to it’s equations of motion. First you need to know that my way of derivation is probably not the most wonderful way to derive the Euler-Lagrange equation. If you want to see a beautiful derivation you’ll have to have some knowledge in calculus of variations. You can see this link for such derivation. So let’s start. Now that we know that :
Assuming the potential energy is a function of displacement, which is the case all the time. We can rewrite the components of the equation in terms of the lagrangian, such that :
Please notice that we’re working with partial derivatives here since the lagrangian is a functional) and not a function. Functionals are special kinds of functions that have the ability to take in a function or another functional as it’s argument. In the next step we want to rewrite the potential energy in terms of the lagrangian.
Finally we can substitute to get the euler lagrange equation which is :
Or the general form which is :
Where q is any possible coordinate system that the lagrangian is represented in terms of. So, to summarize the action principle in one equation:
To conclude, lagrangian mechanics may seem to involve a lot of advanced mathematical concepts and therefore it seems complex. Where in reality, it has the potential to describe the dynamics of any system. The action concept is super helpful not only in classical mechanics but also in quantum mechanics and advanced physics. You may be wondering about how to solve classical mechanics problems lagrangian mechanics. Therefore I'll do a problem solving article just for this purpose.