## The Three Fundamental Principles of Mechanics

A major focus of introductory mechanics is understanding three fundamental principles of how objects interact (though they may have been called by different names): the momentum principle, the energy principle, and the angular momentum principle. These principles are central to the study of physics broadly, not simply mechanics; they are deep truths about our universe, which seem to hold in every observation we make and experiment we conduct.

As such, in Electricity & Magnetism (E&M for short), these fundamental principles do not change - we will still use these three principles. In addition, this semester we will add another deep principle (conservation of charge) and talk about two new types of interactions: the electric interaction and the magnetic interaction. As we talk about these new interactions and this new principle, we will be relying on the three fundamental principles to talk about how a system responds, so it is worth reviewing these principles and how they work.

### Lecture Video

### The Momentum Principle

The momentum principle describes how the momentum of a system will change as a result of external forces. Since momentum is a vector, this principle is really a set of three equations - one for momentum in the x-direction, one for momentum in the y-direction, and one for momentum in the z-direction. Therefore, it is also able to describe how an system will move in three dimensions. The momentum principle is the underlying theory behind Newton's Laws and the kinematic equations of motion.

$$\Delta \vec{p}_{sys} = \vec{F}_{ext} \Delta t$$

If you choose the system so that there are no external forces, or if the external forces can be neglected, the momentum of the system is conserved. This means that the initial momentum of the system has to equal the final momentum of the system.

$$\Delta \vec{p}_{sys} = 0 \longrightarrow \vec{p}_{sys,i} = \vec{p}_{sys,f}$$

### The Energy Principle

The energy principle describes how the energy of a system will change as a result of external work (W) and/or energy exchange due to a temperature difference (Q). It is a scalar principle as energy has no direction, only magnitude. This principle describes how energy is transferred in and out as well as around a system in different forms.

$$\Delta E_{sys} = W + Q$$

If you choose a system so that there are no exchanges of energy with the surroundings, or if these exchanges can be neglected, the energy of the system is conserved. This means that the initial energy of the system has to equal the final energy of the system.

$$\Delta E_{sys} = 0 \longrightarrow E_{sys,i} = E_{sys,f}$$

### The Angular Momentum Principle

The angular momentum principle describes how the angular momentum of a system will change as a result of external torques. It is a vector principle as it describes how a system will rotate or translate about an axis in 3 dimensions.

$$\Delta \vec{L}_{sys} = \vec{\tau}_{ext} \Delta t$$

If you choose a system so that there are no external torques, or if these external torques can be neglected, the angular momentum of the system is conserved. This means that the initial angular momentum of the system must equal the final angular momentum of the system.

$$\Delta \vec{L}_{sys} = 0 \longrightarrow \vec{L}_{sys,i} = \vec{L}_{sys,f}$$