This is the start of a series I want to make to teach Lagrangian Mechanics. The target audience for this will be someone who has taken calculus, linear algebra, and introductory physics classes. I doubt I’ll work many problems unless they are particularly insightful, I intend for this to be a more “casual” read that can prompt you to research what interests you more.
What This Lesson Is
Part 1 is going to be a review of Newtonian Mechanics, if you’re comfortable with this topic feel free to skip it. For anyone who reading on, this post is not teaching anything. I’ll just be naming and summarizing everything you’ll need as a background, so if you are unfamiliar with a particular topic, you know what to get caught up on.
The Essentials
Classical mechanics is the study of the dynamics of macroscopic objects, and particularly, how these objects move. This is done through mathematical models that are based on what we observe in the world around, which allow us to predict things like position, velocity, and acceleration.
We describe position using the position vector \(r(t)\) that has three components. Generally, this is a vector that labels the position of an object and may change with time as the object moves around. We will use \(x(t)\) to denote functions that describe position. We may also want to know how this position changes over time, for that we use velocity denoted by \(v(t) = \frac{d r(t)}{dt}\). We can also find the acceleration, or the rate of change of velocity, denoted by \(a(t) = \frac{dv(t)}{dt}\).
Another important quantity is the mass of an object, labeled by \(m\). Mass is important as it describes the inertia of an object, which roughly speaking, the amount that an object resists acceleration. Related to mass is energy, which can broadly be related to various dynamical quantities. In introductory classes, I feel like it is generally taught as more of a “book-keeping” quantity that a system has to abide by. Traditionally, you are taught about kinetic energy (usually introduced as \(K\), but we will call \(T\)), and potential energy (which we will denote \(V\)). Generally speaking, kinetic energy relates to the amount of motion for an object, and potential energy describes the potential for motion. Together, these give the total energy of a system \(E = T + V\), which must be conserved for an isolated system.
Our next important topics are momentum (\(p = mv\)), or the tendency of an object to remain in motion. Similarly we have angular momentum (\(L = rp\)) which describes the tendency of an object to stay rotating or spinning. Both of these are vectors, and both of these are conserved quantities.
All of these come together under the Newtonian Framework which is based on the concept of forces. This framework is defined by the following laws:
- An object will remain at rest, or at a constant velocity in a straight line, unless acted upon by an outside force.
- The rate of change of an object’s momentum is proportional to the force acting on it.
- If two objects exert a force on each other, these two forces are equal in magnitude but opposite in direction.
In a lot of mechanics problems you solve in class, you would solve a problem with the following steps:
- Specify all the forces acting on a system
- Break down the forces into their individual vector components
- Simplify and solve the resulting equations of motion
Inertial and non-inertial frames of reference are fundamental concepts in Newtonian mechanics that determine how motion is perceived and described. An inertial frame is defined as one in which Newton’s laws of motion hold without modification: objects maintain their state of rest or uniform motion in the absence of external forces. Non-inertial frames accelerate or rotate relative to an inertial frame, necessitating the introduction of fictitious forces such as the Coriolis effect to accurately describe motion.
The last thing that we’ll talk about are conservation laws, which are fundamental principles that describe quantities that remain constant over time in a closed system. The three primary conservation laws are the conservation of energy, (linear) momentum, and angular momentum. These can all be derived from Newton’s Second Law, but also hint at deeper symmetries that we’ll cover later on.
If you recognize all of the bold words and can solve problems about them, you’re ready to move to. If not, between text books, website posts, and youtube videos, there is truly more than you would ever need to learn about Newtonian Mechanics. Please take some time to review those topics before moving on. Part 2 to this series if going to be pretty similar to this, but more centered around math.