Limitations of Newton’s Laws

A body remains at rest or in uniform motion unless acted on by an outside force.
A body acted on by a force moves in such a manner that the rate of change of motion equals the force. $\vec F = m\vec a$.
If two bodies exert forces on each other, these forces are equal in magnitude and opposite in direction.

Newton’s laws were powerful and useful in intro mechanics, but there are some serious limitations in their use!

Magnetic forces

Magnetic forces violate Newton’s third law. To understand exactly where the contradiction is, let’s first review Newton’s third law:

The mathematical statement of Newton’s third law is $\vec F_{12} = -\vec F{21}$. There is a lot going on in this deceptively simple formula. It defines a third law pair of forces that act on different objects – that is, $F_{12}$ acts on object 2 and $F_{21}$ acts on object 1. We require two objects acting on each other pairwise. The law clearly states that the magnitudes of the forces are equal, but also shows that they act in opposite directions along a common line.

Newton’s third law can be derived from our rules about isolated systems and Newton’s second law. Let’s consider two paired objects in an isolated system. In any isolated system, the net force is zero. Combining this with the momentum definition of Newton’s second law,

\[\begin{align} \vec F_{net} = \frac{d\vec p_1}{dt} + \frac{\vec p_2}{dt} &= 0 \\ \frac{d \left( \vec p_1 + \vec p_2 \right)}{dt} &= 0 \\ \frac{d\vec p_1}{dt} &= - \frac{d\vec p_2}{dt} \end{align}\]

Now we can redefine these momentum derivatives in terms of the force on each object. The momentum of object 1 will be the force of object 2 on object 1: $\frac{d\vec p_1}{dt} = \vec F_{21}$. Likewise, $\frac{d\vec p_2}{dt} = \vec F_{21}$. Then we get our expected formula for Newton’s third law, $\vec F_{12} = -\vec F_{21}$.

We cannot universally apply Newton’s third law when we are talking about forces generated from fields, such as electromagnetic fields.

\[\vec F = q \left( \vec E + \vec v \times \vec B \right)\]

The electric and magnetic fields are both generated by all other charged particles in the system – in other words, the force cannot be broken down into a sum of pairwise interactions between individual particles. There is also a velocity dependence to the electromagnetic force, which makes it impossible to assign pair-wise force interactions. This would have been an issue for our drag problems in the last class, too.

We just derived Newton’s third law as a consequence of conservation of momentum, and conservation of momentum must hold even if the third law does not. In electromagnetism, we would need to account for the momentum transfered to the field when we calculate the total momentum of the system. Any movement of the charges that produce the field will create a time-varying distortion and a momentum change:

\[\frac{d \left(\vec p_1 + \vec p_2\right)}{dt} + \frac{d \vec p_{field}}{dt} = 0\]

Inertial reference frames

The first law, and consequently the second law, both depend on having an inertial reference frame. This means the observer is not accelerating. If the observer is accelerating, we have to throw out all of our rules about how forces cause or describe acceleration. You might immediately think of problems in space travel with accelerating ships and observers, but this problem comes up much closer to home – we are all constantly accelerating with the earth’s rotation. This gives rise to the coriolis effect, which is responsible for hurricane rotation. It also means objects falling from high altitude will not follow the kinematic equations we would expect.

Contact forces

Introductory courses neglect to explain the origins of contact forces (normal force, tension, etc.). It can be confusing and unclear exactly how to apply these forces, and their direction may change in ways that are difficult to parameterize. In advanced mechanics, we will think of these “forces” as constraints of motion. As long as we can paramterize these constraints, we can use them consistently and without confusion.

(Non-)Linearity

In intro mechanics we studied linear systems, but mechanics is generally nonlinear. We can quickly move into territory where equations of motion (differential equations, typically second order in $x$) do not have linear solutions that can be added together to form new solutions. You can tell that your problem is going to be nonlinear if the equations of motion have higher powers of $x$ or $\dot{x}$. Some examples include complicated orbits of multiple bodies, a double hanging pendulum, and anharmonic oscillators. But really, nonlinear dynamics are much more common in nature than linear dynamics.

Nonlinear systems are often chaotic, meaning their time evolution has a sensitive dependence on the initial conditions. Compare this to the simple harmonic oscillator, which will do almost the same thing no matter how you start it off. Although these systems may be fully deterministic, their sensitivity makes their behavior difficult to predict. We look for general behaviors in their phase diagram and/or use numerical simulations.

I’m not sure we will have time to cover this topic, but both Taylor and Thornton have excellent chapters on nonlinear dynamics.

Energy review

The first “beyond Newtonian” topic we will cover is Lagrangian mechanics, which is a powerful and much more generalizable formulation of mechanics where we will derive the equations of motion from the energy state of the system. Today we will review energy concepts to prepare, and next week we will work on the mathematical foundations underlying the technique.

Conservation of Energy

Conservation of energy is a law based on universal symmetries, and is not linked specifically to Newtonian mechanics. We’re going to make great use of energy based pictures to reformulate mechanics in Lagrangian and Hamiltonian pictures. Let’s review.

Energy

We will define three types of energy:

Kinetic energy from the motion of particles. This one has two parts.
- linear kinetic energy $T = \frac{1}{2}mv^2$
- rotational kinetic energy $T_{rot} = \frac{1}{2}I\omega^2$
- we will mostly work with linear kinetic energy, unless rotation is specified.
Potential energy comes from fields and depends on the object’s position. Forces can be derived from potential energy as $F = -\bigtriangledown U$ Some examples include:
- Gravitation potential energy $U_g = mgh$ or $U = \frac{-GMm}{r}$
- Spring potential energy $U_{sp} = \frac{1}{2}kx^2$
Internal energy, $E_{int}$ which will describe energy losses or conversions such as sound, thermal, chemical, or light energy

Energy is conserved in any isolated system, meaning $\Delta E_{tot} = \Delta T + \Delta U + \Delta E_{int} = 0$ when comparing the total energy at any two points in time.

Forces, energy, and work

We define a quantity called work, with units Joules, that allows us to move between vector forces and scalar energy quantities. We know that energy depends on some combination of position (potential energy) and velocity (kinetic energy), so we want to include any changes in velocity (acceleration, forces) and changes in position (displacement). We also have an idea that any work a force does on an object only really “counts” if it results in some movement, or displacement. From these requirements, we define the work done by a single force on an object:

\[W_{\vec F} = \int_{\ell} \vec F \cdot d\vec\ell\]

If the force is a conservative force, meaning it is derived from a field, we can directly relate work and potential energy:

\[W_{\vec F} = -\Delta U\]

Any work done by nonconservative forces will show up as a change in internal energy:

\[W_{NC} = -\Delta E_{int}\]

Work and Kinetic Energy

We have related work to changes in potential energy and converted internal energy, but we need to also relate it to changes in kinetic energy. The total work done on an object is $W_{tot} = W_1 + W_2 + W_3 + \ldots$, where $W_i$ is the work done by an individual force over the path $\mathcal{S}$.

\[\begin{align} W_{tot} &= W_1 + W_2 + W_3 + \ldots \\ &= \int_{\mathcal{S}} \vec F_1 \cdot d\vec s + \int_{\mathcal{S}} \vec F_2 \cdot d\vec s + \int_{\mathcal{S}} \vec F_3 \cdot d\vec s + \ldots \\ &= \int_{\mathcal{S}} \left( \vec F_1 + \vec F_2 + \vec F_3 + \ldots \right) \cdot d\vec s \\ &= \int_{\mathcal{S}} \vec F_{net} \cdot d\vec s \end{align}\]

Since we can now relate the net work and the net force, we can use Newton’s second law to write

\[\begin{align} W_{tot} &= \int_{\mathcal{S}} \vec F_{net} \cdot d\vec s \\ &= \int_{\mathcal{S}} m\vec a \cdot d\vec s \\ &= \int_{\mathcal{S}} m \frac{d\vec v}{dt} \cdot \vec v dt \leftarrow d\vec s = \vec v dt \\ \end{align}\]

How do we treat this integral? We will “complete the square” on $\frac{dv}{dt} v$ using the product rule:

\[\begin{align} \frac{d}{dt} [v\cdot v] &= \frac{dv}{dt} v + v \frac{dv}{dt} = 2v \frac{dv}{dt} \\ \frac{dv}{dt} v &= \frac{1}{2}\frac{d}{dt}v^2 \end{align}\] \[\begin{align} W_{tot} &= \int_{\mathcal{S}} m \frac{d\vec v}{dt} \cdot \vec v dt\\ &= \int_{\mathcal{S}} m \frac{1}{2} \left[ \frac{d}{dt} v^2 \right] dt \\ &= \frac{1}{2}m\Delta v^2 = \Delta K \end{align}\]