This is done through a quantity called the action. It is fundamentally a description of changes in energy. This is, in fact, the basis for Lagrangian mechanics. Take for example the trajectory of a projectile (such as a ball thrown in the air): In fact, the kinetic and potential energy of an object is all you need to know to fully predict where it will move next (not taking into account friction, for now). Now, the next question is, how can these two quantities be combined into a useful theory? The answer is actually simple energy. Once we know where an object is and which direction and how much velocity it has, we can fully predict where it will be at the next instant. If you really begin to think about it, there are only two quantities that we need in order to fully describe the motion of any object its position and velocity at any given point. When a ball rolls down a hill, its height (position) will decrease in some way, but its velocity, on the other hand, will increase as it rolls. Ultimately, dynamics and motion are optimization processes. If a net force is acting on you, your position will change and this is how we measure velocity and acceleration. Granted, they are intuitive and simple to understand, but after all, they are just one possible mathematical model to describe motion and dynamics.įundamentally though, motion is a description about changes. There is, however, nothing special about forces. Typically, we think of motion as being a result of different forces, which is practically what Newton’s laws are. To get started, let’s try to develop some intuition and reasoning behind what we’re going to be looking at in detail in this article.įor this, we’re going to rethink our notions of what motion really is in the most fundamental sense. These will both be explained in great detail in this article. This is known as the principle of stationary action, which is one of the most fundamental principles throughout all of physics. The important thing about the action is that it is required to be stationary in order to get the right equations of motion. The other important quantity is called action, which is used to define a path through space and time. The first one is called the Lagrangian, which is a sort of function that describes the state of motion for a particle through kinetic and potential energy. Lagrangian mechanics is practically based on two fundamental concepts, both of which extend to pretty much all areas of physics in some way. The equations of motion are then obtained by the Euler-Lagrange equation, which is the condition for the action being stationary. But what is Lagrangian mechanics, exactly?Īs a general introduction, Lagrangian mechanics is a formulation of classical mechanics that is based on the principle of stationary action and in which energies are used to describe motion. Second, since no transcendental number can be constructed with compass and straightedge, it is not possible to " square the circle".Often the most common approach to describing motion and dynamics is through Newton’s laws, however, there is a much more fundamental approach called Lagrangian mechanics. It is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although fractions such as 22 7 ). The number π appears in many formulae across mathematics and physics. The number π ( / p aɪ/ spelled out as " pi") is a mathematical constant that is the ratio of a circle's circumference to its diameter, approximately equal to 3.14159.
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