Lagrangian and Hamiltonian Mechanics : An Unseen Side of Classical Physics

Ever since, school and high-school till the early undergraduate years, students are rigorously made accustomed to the world of Newtonian Mechanics. In fact, those studying in other fields than the Physical Sciences may never see the world beyond this Newtonian Framework of Classical Physics. All of us have been taught and told to remember by heart the three most important laws of Mechanics - Newton's Laws of Motion. These laws provide an intuitive insight to how physical systems evolve and behave with time. They lay down the mathematical foundations to compute the trajectory of any object i.e. its motion through space as time passes, mainly - the Newton's Second Law of Motion which states that the magnitude of an external force experienced by any body is proportional to how much it accelerates. The proportionality constant here is the "inertial mass" of that body. Alternatively, one can say in the language of Calculus that the external force experienced by a body is equal to the product of its mass(m) and the derivative (rate of change) of its velocity with respect to time (dv/dt). But, velocity of that body is the rate of change of its displacement, or the derivative of displacement (x) with respect to time. By this definition, acceleration can be represented by the second derivative of position with respect to time, or - "The rate of change of  the rate of change of position". The first one sounds better.  This renders Newton's Second Law of Motion into a "Differential Equation" which can be solved by special methods. By solving this differential equation, the final solution is obtained in the form of a function x(t). Mapping this function against the time variable on a graph, shows visually how any physical system behaves with time and also predict its future states.

This formulation of Classical Physics works just fine in most scenarios and physical systems. However, there are some systems where applying Newton's formalism becomes too messy and rather inconvenient. Situations such as a double pendulum, a pendulum fixed to a moving support, an object sliding on an inclined plane which itself is moving, etc. are some examples where Newtonian Mechanics isn't a viable option. Such systems either comprise of too many coordinates and independent parameters or too many external forces or both, that it is extremely hard to keep track of individual forces and their effect on other forces. Life would get easier, if somehow it became possible to find our way around this problem by using some other physical quantity to predict the behavior of such dynamic systems. This other way around to the problem manifested itself in the form of Hamiltonian and Lagrangian Mechanics. The formalism was developed by Joseph-Louis Lagrange and Sir William Hamiltonian in the 18th and 19th centuries respectively. 

The core principle behind Lagrangian and Hamiltonian Mechanics is the mathematical concept of Calculus of Variations. The ordinary Calculus most of us are familiar with involves "functions" which is a one to one relation between two real number sets. They can be regarded as machines that take one real number as an input upon which certain mathematical operations act and spew out another or same number as an output. Calculus of Variations on the other hand deals with "Functionals" - they are machines that take an entire function as an input and yield a number as an output. The task in hand is then to minimize these "functionals" subject to certain constraints. A main goal of this article is not to hinge on the abstract mathematics but try to provide an intuition for the idea behind that mathematics. A central quantity of Lagrangian Mechanics is the "Lagrangian Function (L)" or simply the Lagrangian. This Lagrangian is equal to the difference between the Kinetic Energy (T) (yes, Kinetic Energy is represented by T) and the Potential Energy (V).

L = T - V   -- (1)

The integral of this Lagrangian in the limits of the starting time (t1) and ending time (t2) is equal to a quantity called as "Action". This quantity is the example of a "functional" which takes the Lagrangian function as input and gives a number. But where are we headed to by introducing such weird names for some mere mathematical operations on already existing quantities like Kinetic and Potential Energy from Newtonian Physics? Most standard introductory texts on the subject skip this intuition part and dive directly into the complicated derivations of mathematics. What we have here is the Lagrangian Function, which is the difference in Kinetic Energy and Potential Energy. The Kinetic Energy of a system is a direct representation of how much "motion" is happening in that system, while Potential Energy represents how much motion "could" happen in that system but isn't happening. A stretched rubber band or a ball placed on a height packs in more Potential Energy, but if you let go of the rubber band or allow the ball to fall from that height, this Potential Energy is converted to Kinetic Energy. From equation 1 we can infer that, if a system has greater Kinetic Energy its Lagrangian is greater and so the system is more dynamic, more "lively". Conversely, if its Potential Energy is greater then the system is less dynamic, less "lively". The integral (or summation) of this "liveliness"  of a system over some time period gives the "Action" of that system. 

Now, the main principle of Lagrangian Mechanics which is also known as the "Principle of Least Action" says that - the path followed by a system through space between any initial and final time is such that its Action is minimized or remains least. This implies that a system always goes through that path for which its Action is minimal or for which it's "liveliness" or "dynamicity" is least . 

Above diagram visualizes a few of all the possible paths between two points that a system can take while travelling through space and time. There is an infinite number of all such bizarre paths that one can draw. With each of such paths (represented by the different colours) a number is associated with them, which was introduced earlier as "Action". The system starts at point A at time t1 and reaches at point B at time t2.The "Action" of each individual paths between these points is then determined by the different Lagrangians of those paths. Some paths may have a greater Lagrangian value and thus the system going through such paths is more lively and in motion, while some paths have lesser overall Lagrangian and the system is less dynamic for such paths. Out of the seemingly many infinite paths, the Principle of Least Action tells us that this system will take that path for which the "Action' associated with it is least. It is as if there is an algorithm that already dictates what path a system shall choose.

Oddly enough what it means is that Nature is a bit lazy! Initially, this result was somehow attributed to the assertion that God chooses such a path for a system for which the action is minimum. The sense behind this statement is quite pragmatic. If I were to govern the motion of all objects in this Universe, I would most certainly prefer objects to not bounce around much without any reason. Either way, the principle did confirm with observed motions of dynamic bodies. Every spontaneous process in nature minimizes the Action of that process. The most important application of this principle was to explain the behavior of light. Fermat's principle which was a modified analog of Hamiltonian and Lagrangian Mechanics accounted for the path taken by any light ray, which is such that the time taken to travel between two points is minimized. By working through the Principle of Least Action, one arrives at the Euler - Lagrange Equation - An epitome of Lagrangian Mechanics. This equation represents Lagrangian Mechanics in the same way that F = ma represents Newtonian Mechanics. In fact, it is very much possible to extract the mathematical statement of Newton's Second Law from the Principle of Least Action and Euler - Lagrange Equation, even if the two appear distinct.

The above equation is the Euler - Lagrange Equation. It can be classified as a second order differential equation. The letter q in it represents the "generalized coordinate". Here generalized meaning that the coordinate can refer to anything. For example, one can use cartesian coordinates in case of linear motion, or polar coordinates for a pendulum or angular motion, etc. The fancy L like letter, is the symbolic representation of our Lagrangian Function given by :

Lagrangian is a function of the generalized coordinate q and the velocity of system, which is represented by the second q with a dot overhead. A convention in Physics is to denote the rate of change of a quantity(derivative) with respect to time by placing a dot over it. In the 19th century, Hamilton developed a similar modified version of Lagrangian Mechanics that became known as Hamiltonian Mechanics. Just like the Lagrangian, his version involves a "Hamiltonian" which is equal to the sum of Kinetic and Potential Energies of a system. 

H = T + V   --(4)

The Hamiltonian Equations of motion are :

Here q is the generalized coordinate and p is the momentum of system.

Lagrangian Mechanics along with Hamiltonian Mechanics proved crucial in supplementing the mathematics of modern Physics. The Hamiltonian became an operator which is extensively used in Schrodinger's Equation. The Principle of Least Action along with Lagrangian Mechanics was employed by Richard Feynman in his Path Integral Formulation of Quantum Physics. The Lagrangian is even found in Quantum Field Theory - one of the most precise theory ever discovered by mankind. Unfortunately, Lagrangian and Hamiltonian Mechanics wasn't able to gain the fame and recognition that Newton's Laws did in everyday lives. This could be explained perhaps by the fact that a physical intuition for these formulations is hard to explain. But, sometimes the physical intuition behind a theory isn't the most significant aspect as long as it works consistently, which is the case for Lagrange and Hamilton's theories.

- Thank You.