📂Classical Mechanics

Lagrangian Mechanics and Hamiltons Variational Principle

Overview

Hamilton’s principle, functionals, action, and variation are explained here in a way that is as simple as possible. If you have not found a satisfactory explanation elsewhere, it is recommended to read through to the end. This has been written so that even freshmen and sophomores in college can understand it.

Lagrangian Mechanics¹

When an object moves from time $t_{1}$ to $t_{2}$, the integral of the Lagrangian over the path of motion is called the action and is denoted as $J$.

$$ \begin{equation} J=\int_{t_{1}}^{t_{2}} L dt \end{equation} $$

Among all possible paths of motion, the action of the actual path of motion is the minimum. The Lagrangian is defined as the difference between the kinetic energy and potential energy and is commonly denoted as $L$.

$$ L = T-V $$

This is briefly referred to as Hamilton’s variational principle or the principle of least action. The reason it is called the principle of least action is because the integral in $(1)$ is called action. Although minimum and extremum are different concepts, let’s assume they mean the same thing here. Precisely, it would be correct to say extremum (either maximum or minimum). Based on the Marion textbook, this is probably the first content of Lagrangian mechanics you will learn in the first semester, and according to the Fowles textbook, in the second semester. However, it was very difficult to understand this content by sticking to the textbook alone. New concepts appear without kindly explaining what they are. For example, the Fowles textbook introduces the following equation:

$$ \begin{equation} \delta J =\delta \int_{t_{1}}^{t_{2}} L dt = 0 \end{equation} $$

And the explanation for the newly introduced symbol $\delta$ is as follows:

"$\delta$ is the variation of the total integral for the extremum."

How can one understand what $\delta$ means after reading this? They barely teach what variation is and then proceed with calculations recklessly. It’s too slow to read line by line without understanding why the equations hold, making it very difficult to grasp the content. Therefore, the author aims to explain Lagrangian mechanics as kindly as possible for students new to it. First, it is necessary to organize the terms used when describing Hamilton’s principle.

Functional

Many sources refer to the integral in $(2)$ as a functional, but it is normal for physics students to not know what a functional is. You might commonly know a function as something where if you input a real number, you get a real number (or a complex number) as the output.

$$ f(x)=x^2,\quad g(x)=e^{2x} $$

However, considering the mathematical definition of a function, there’s no need for the input to be a number and for the output to be a number. Since a function is something that gives a corresponding result when something is input, there’s no restriction on what can be input. If a function maps input functions to a number, that function is called a functional. For example, the function $F$ defined below is a functional.

$$ {\color{blue}F\big( {\color{orange}f(x)} \big)} := {\color{red}\int_{1}^{2} f(x) dx} $$

That is, the function $F$ takes any function and integrates it from $1$ to $2$ to get a function value. In reality, when you calculate

$$ {\color{blue}F( {\color{orange} e^{x} })} = \int_{1}^2 e^x dx = {\color{red}e^2-e},\quad {\color{blue}F({\color{orange}x^2})}=\int_{1}^2 x^{2} dx = {\color{red}\frac{7}{3} } $$

such a function that yields a real number (or a complex number) upon inserting a function is called a functional. The following content will discuss that action is precisely a functional because it produces some value when ’the Lagrangian for each path of motion’ is input, which is a function. There’s a post on the blog about the mathematical content of functionals, but it won’t be introduced here. It might be more confusing to read, so it’s recommended not to read it unless you’re curious. If you’re curious, search for functional in the top right search bar, read about it, and if you don’t understand, just forget it.

Action and Lagrangian

The difference between kinetic energy and potential energy is called the Lagrangian and is denoted as $L$.

$$ L=T-V $$

Since the Lagrangian depends on velocity, position, and time, if the position is denoted as $y$, it can also be denoted as follows:

$$ L=L(y^{\prime},\ y,\ t) $$

The name Lagrangian comes from the French mathematician Joseph Louis Lagrange. The integral of the Lagrangian over time is called action or act and is commonly denoted as $J$.

$$ J = \int_{t_{1}}^{t_{2}} L dt = \int_{t_{1}}^{t_{2}} L(y^{\prime},\ y,\ t) dt $$

Hamilton’s Principle

Devised by the British mathematician William Rowan Hamilton in 1834, the principle states that the path actually taken by a body will make the action minimal. This is not a provable fact but one of the basic principles existing in nature, as if $F=ma$. For example, let’s say we want to know the path a body takes when thrown from a high place to the ground. There could be countless paths we might imagine, but there’s something special about the actual path the body takes. That is, when integrating the Lagrangian over each path, the integral of the Lagrangian over the actual path the body takes is the smallest. That is, the path that minimizes the action is the actual path the body takes. Hence, Hamilton’s principle is also known as the principle of least action. Based on this principle, dealing with the motion of a body is Lagrangian mechanics. The amazing thing is that Lagrangian mechanics, though appearing entirely different, yields the same results as Newtonian mechanics. That is, only the method of expression is different, but the essence is the same. Newtonian mechanics deals with the motion of bodies based on vector calculations, whereas Lagrangian mechanics describes mechanics by calculating scalars (energy).

Variation

Simply put, the content explained above is organized mathematically. As an easy example, consider the problem of finding the minimum value of a quadratic function.

Let’s say we are given a quadratic function like the one in the picture above. The minimum value of the function is $1$, and the place where the function value is minimal is $x=3$. At the point where it has a minimum (extremum) value, the slope is $0$, so we know that when differentiated, $0$. Therefore,

$$ \dfrac{dy}{dx} \bigg|_{x=3}=0 $$

This content will be applied directly to the principle of least action.

In the picture above, let’s denote the actual path of motion of the body as $y(0, t)$. Let’s call any possible path the body can take as $y(\alpha, t)=y(0,t)+\alpha \eta (t)$, as shown in the picture above. Note that $\eta$ is the Greek letter eta. $\alpha \eta (t)$ can be thought of as the error when compared to the actual path. As can be seen from the picture and formula, when there is no error, that is, $\alpha=0$, the possible arbitrary path $y(\alpha, t)$ becomes the actual path. Furthermore, the principle of least action is the content that among all possible paths, the action for the actual path is the minimum value. Combining the two pieces of content and applying the example mentioned above, when differentiating the action and substituting $\alpha=0$, the result is that the value is $0$.

$$ \dfrac{\partial J}{\partial \alpha}=\dfrac{\partial }{\partial \alpha} \int_{t_{1}}^{t_{2}} L\big( y^{\prime}(\alpha,t),\ y(\alpha,t),\ t \big) dt =0 $$

This can be simply denoted as follows, and $\delta J$ is called the variation of $J$.

$$ \delta J = 0 $$

That is, it can be understood as $\delta=\dfrac{\partial }{\partial \alpha}$. Therefore, the following equation holds:

$$ \delta \dot{y}=\dfrac{\partial }{\partial \alpha}\frac{dy}{dt}=\dfrac{d}{dt}\frac{\partial y}{\partial \alpha}=\dfrac{d}{dt}\delta y $$

See Also

• Euler-Lagrange Equation