Newton is Still G.O.A.T.

In classical physics, you might have heard of Newton's laws of motion, where we solve problems related to the equations of motion by dealing with forces directly. But you know what? There is another way to deal with such problems, and they are called Lagrangian mechanics, which instead of looking at forces, deals with energies of the system. It uses just a single letter L, which we also call the Lagrangian, and it is defined as the difference between kinetic energy of the system we are studying, written using the letter T, and the potential energy of that system, written using the letter V. This method works in any coordinate system, and it gives the same results as Newton's laws, but often in a simpler and more flexible way, making it very powerful for solving complex problems in physics and engineering. Now let us understand it better with a simple example. Consider a point object of mass m at some height. y is in free fall, and it is falling with a velocity v, and acceleration is g in downward direction. So the kinetic energy T of this mass will be equal to half mv squared, right? And the potential energy V of this mass will be equal to mgy. So the Lagrangian L can now be written as T minus V, or this. Now how can we write the equation of motion of this mass? This is where the beauty of Lagrangian actually lies. We first compute the partial derivative of L with respect to Q, then we compute the partial derivative of L with respect to Q dot, and take its derivative with respect to time T in order to get this term. Now simply subtract both of them and equate it to zero. And that's it! Now you might be wondering, what is this Q? Okay, so here, Q is any generalized coordinate you choose to describe the system. For example, Q will be equal to x if the system is moving in a straight line along the x-axis, or it will be equal to y if the system is moving in a straight line along the y-axis, or it can be equal to theta if we are considering polar coordinate system, say for the angle of a pendulum. I think you got the point. And then this Q dot is simply the rate of change of Q with respect to time. Like if Q is position, then Q dot is the velocity, while Q double dot is the rate of change of Q dot with respect to time, like acceleration. So for our free fall example, Q is y, and Q dot is y dot, which is nothing but this v, or the velocity of the point mass. So this kinetic energy can be written as half m times y dot square, right? So L will be equal to this minus potential energy, or this. Also this equation of motion will become this, correct? We have simply replaced Q with y, and Q dot with y dot. Now the partial derivative of L with respect to y is zero, because there is no y term here, and the minus derivative of this will be mg. It is important to note that we took the partial derivative of this term with respect to y as zero, because that term has no y in it at all, and it only depends on the y dot. When we take the partial derivative with respect to y, we treat the y dot as independent from y, and so the derivative of that kinetic energy term with respect to y is simply zero. Similarly, when we take the partial derivative of L with respect to y dot, only the kinetic energy of one half m times y dot squared matters, and its derivative with respect to y dot gives m times y dot. Just treat the y dot as some other variable than y. Therefore this term will give us zero, because we have no y dot term here, right? Then, when we take the time derivative of this thing, we get m times time derivative of y dot, which is just the mass m times y double dot. Next, let us simply plug them into this equation to get m times y double dot. Minus of minus mg equals zero, or my double dot plus mg equals zero. Divide both sides by m to get y double dot plus g equals zero, or y double dot equals minus g. And folks, this is the magical moment. Without ever talking about forces, we arrived straight at the same law of free fall that Newton gave us. The acceleration y double dot comes out as a constant minus g, which means every object, no matter its mass, falls with the same acceleration. Let me quickly show you two more examples, so that you become well versed with Lagrangian mechanics. Suppose we have a mass m attached to a spring with stiffness k, and let x measure how far the mass is from the rest position, or the equilibrium position of the spring. So here, for the spring mass moving along a straight line, q will be equal to x. Then the q dot is simply the rate of change of q, with respect to time, which in this case is x dot, or the velocity of the mass. And q double dot is the acceleration, the rate of change of x dot, or x double dot. Now, the kinetic energy T can be written as one-half times m times x dot squared, since it depends on the mass moving with velocity x dot. The potential energy V stored in the spring is one-half times k times x squared. So the Lagrangian L will be T minus V, or this. Next we will apply this rule. The partial derivative of L with respect to x gives minus k times x, because only the potential energy part depends on x. The partial derivative of L with respect to x dot gives m times x dot, coming from the kinetic energy. Taking the time derivative of that gives m times x double dot. Plugging these into the equation gives us m times x double dot plus k times x equals zero, which is the famous spring mass equation. See, no forces were involved in this. Now, finally, let us find the equation of motion for a simple pendulum, where we have a massless string of length L and a point mass m attached at the end of the string, and it swings under gravity. For a simple pendulum, the motion is rotational. So q will be equal to theta, which is the angle that the pendulum makes with the vertical like this. Then the q dot is the rate of change of theta with respect to time, or theta dot, which is the angular velocity of the pendulum. Similarly, q double dot is the theta double dot, the angular acceleration, or the rate of change of angular velocity. Next, we will write the kinetic and potential energy. The kinetic energy, T, comes from the motion of the mass along the circular arc, and it is one-half times m times v squared. For circular motion, we know that the speed, v, is equal to L times theta dot, so the kinetic energy can be written in terms of the angular velocity, and it will be half times m times L squared times theta dot squared. The potential energy, v, comes from the height of the mass above the lowest point. If this is the lowest point, such that this is length L, and this is also L, so this length will be equal to L cos theta, right? Basic trigonometry. So this height can be written as L times one minus cos theta, and thus the potential energy can be written as m g times L times one minus cos theta. So the Lagrangian, L, is T minus v, or this minus this, which can be expanded like this. Next, we will apply this rule. The partial derivative of L with respect to theta is minus m g L times sine theta, because only this part of the potential energy depends on theta. Then the partial derivative of L with respect to q dot or theta dot is m L squared times theta dot, coming from the kinetic energy. Taking the time derivative of that gives m L squared times theta double dot. Plugging these into the equation gives m L squared times theta double dot plus m g L times sine theta equals zero. Dividing by m L squared gives theta double dot plus g divided by L times sine theta equals zero. For small angles, sine of theta is approximately theta, and we get the familiar small angle pendulum equation. Theta double dot plus g divided by L times theta equals zero. This shows that the pendulum oscillates back and forth with angular frequency of the square root of g divided by L. And that's the beauty of Lagrangian mechanics. By using energy instead of forces, it gives a simple, elegant method to find the equations of motion for any system. If you enjoyed this video, please don't forget to like, share, and subscribe to our channel. Also, you can support my channel by joining our community and becoming a member. So good! Thank you!