Cars and Physics

When a child swings on a swing, have you ever thought how it is possible without touching the ground. There is nothing to push off against.. a naive application of conservation of momentum might indicate that it is not even possible to start swinging!

So how does it work? The catch is twofold. The first point is gravity, of course. Conservation of momentum applies in the absence of external forces. The second point is the person on the swing does not act as a point mass. Instead the simplest model one can use is a two point mass model, where the legs are one mass element and the torso is the other mass element.

When at the apex or top of the backward motion swing, the person would kick out their legs. This launches their torso and the swing backwards while projecting the leg forwards/downwards. Here is the crux.. the torso then does not decelerate by pulling back on the legs and hence conserving momentum. Instead, gravity decelerates the torso, storing up potential energy. This adds amplitude to the swing as desired. When the swing direction reverses then at the apex, the body follows the legs finally.

This bring us to a limitation of this technique. The person cannot detach his legs of course. So there is no benefit to making the kick too aggressive. The optimal velocity of the kick would be such that the knees reach full extension in the duration that gravity decelerates the torso to reverse swing direction. Any faster and the legs get pulled back by the body trying to not break apart and the additional force is for naught.

The final point is that all this is applicable only when starting with some small non-zero swing amplitude. Thus, it’s really difficult (not sure if it is impossible) to start swinging when perfectly stationary as an initial condition. At that stage, the person might just touch and use the ground, to get started.

This is a continuation to the previous post on Lagrange Formulation.

If the Lagrangian does not depend explicitly on time, then becuase the partial time derivative is zero, the change or total time derivative of the Lagrangian can be non-zero only because of motion (change in coordinate values; not to be confused with a coordinate transformation). We will use D_t to denote the total time derivative in order to distinquish it from the other partial derivatives.

D_t (L) = dL/dx (dx/dt) + dL/dv (dv/dt) = dL/dx (v) + dL/dv (dv/dt)

But, we are imposing and solving the equation of motion, dL/dx = d/dt (dL/dv)

Hence, D_tL = d/dt ( v*dL/dv)

Rearranging, D_t (L – v*dL/dv) = 0.

This quantity in the parenthesis is called the Hamiltonian and we showed that its change is zero.In other words, it is conserved over time when the Langrangian has no explicit time dependence. Also, the Hamilton itself cannot have explicit time dependence since it has to always conserved, even when the system is not in motion. This is essentially a weak version of the famous Noether’s theorem that proves any invariance of the Langrangian w.r.t a dependent variable results in a conserved quantity. We will now provide a proof outline for Noether’s theorem for coordinate transformations.

Since dL/dx = d/dt (dL/dv), when any linear combination of the LHS is zero, it follows that the associated linear combination of the parenthetical quanitity on the RHS would have zero time derivative (by the given equation). Hence, that quanitity would be invariant and be conserved over time.

Now we will address the teleological aspects of the least action principle using the Hamiltonian. Let us define a quantity, W = S(q) + Et, where S is the action time-integral of the Lagrangian and E is the constant energy. Note that the conserved Hamiltonian when the Lagrangian does not have explicit time dependence is the total energy of the system and S depends only on the coordinate as the velocities are integrated over time and becuase L is not directly dependent on t.

By the stationary action principle, S needs to be minimized or made stationary. From the above definition, it is equivalent to demand energy conservation and minimize W. Note that energy conservation is not a given if one simply minimizes W as both S and W are defined for physically impossible trajectories as well, as noted in the previous post. However, if S itself is directly minimized then energy conservation takes care of itself, from the symmetry of the Lagrangian seen above.

W is a potential function that is only state dependent (on the coordinate) and not on the path taken to arrive at the current point or state.

Phase space communicates both the motion and configuration (location with shape) of a system (x₁,x₂, x₁‘,x₂‘), wheras configuration space only mentions the values of the degrees of freedom of the system, therby decribing its location and configuration (x₁,x₂). Clearly the number of entries in the configuration space will be equal to the number of degrees of freedom of the system. The number of entries of phase space will be twice the number as it includes velocities as well.

Here is the key observation, shouldn’t we include accelerations as well or for that matter any number of higher order derivatives (x,x’,x”,x”’…) to make the description of motion complete? No! It is complete in phase space and any addition would be redunant. How?

It is because everything moves in Newtonian mechanics (or the equivalent Lagrangian formulation) according to second order differential equations. For example: x” = -bx’-kx. Hence, when we say the system is at a partciular point in phase space, the immediate future motion is determinied by the governing second order differential equation. In other words, there can be no intersection of trajectories in phase space. When the system arrives at a particular point, how the first derivative moves forward is dictated by the value of the second order derivative x”. Moreover, x” will evaluate to the same value at that point as long as the governing law does not change. And how x changes for the future trajectory is dictated by x’, which can be read off the current value of the point on the phase plot.

This is a good test if any imaginary system is truly second order. If the phase plot contains intersections with all the DOF accounted for, with their positions and velocities forming the axes then it points to a richer underlying (higher order) dynamics that is not captured adequately by the plot. In this case the plot is merely a lower dimension shadow or projection of the full dynamics in a higher dimension space.

The principle of stationary action from the previous post on Lagrangian mechanics can also be used outside pure mechanics. For example, for traveling light, including when it undergoes reflection or refraction. In this case, the action is not an energy but rather the time taken to traverse the distance under consideration. The principle states that the path taken will be the one that reduces the time taken to the least possible out of all possible paths, while respecting the physcial constrains of the system.

Example 1: Point light source

The time taken will be least if light travels in the shortest path between two possible points (if the velocity of light is constant throughout the path) so it implies that the travel path will be straight lines. If it takes any other path, it has not reached the end point (or any of the points enroute) in the shortest possible time. So a point light source emits straight rays in all directions outwards. However, the light remains straight only if the speed of light in the medium does not change! For example, in the Feynman lectures it is pointed out that when light traveling from a source to a person encounters cold air, it will bend to pass through another layer of warmer air faster and then finally curve back to the person’s eye level to maintain the fixed destination of the path.

Example 2: Reflection on a smooth surface

Even if the surface is curved, it will locally be flat at the point where a ray strikes (since it is smooth), so the curvature makes no difference to the law of reflection itself (only to the image when a bunch of rays strike different points on the surface). The angle at which the ray strikes the normal to the point of contact on the surface is called the incidence angle.

Firstly note, after reflection, the light has to continue in a straight line for reasons similar to the previous example. Let’s call the angle of this reflected ray the angle of reflection, t. Let c be the speed of light in the medium and d be the normal distance between the tangent to the surface and a point source. Then the time to be minimized is the sum of travel times before and after reflection and the constraint is that the distance covered has to be constant. Since the principle holds for any point in the reflected ray, we can easily pick the point where the vertical distance covered is zero and the horizontal distance covered (along the tangent to surface) is the constraint.

T= d/c (sec(i) +sec(r)) ……(1)

tan(i)+tan(r) = constant/d = constant …(2)

Differentiating (2), we get

dr/di= -sec²(i)/sec²(r) … (3)

To find the extremum of (1), we differentiate w.r.t to “i” and set to zero,

sec(i)tan(i)+sec(r)tan(r) dr/di= 0

Using (3) in above, we get that sin(i) = sin(r), or the law of reflection that the angles are equal!

Example 3: Refraction between non-dispersive media

Since the light has different velocities c1 and c2 in the two media, at the surface separating the media, the light refracts in order to maintain the shortest time principle. The governing equations would be as below and the minimization method would be similar to example 2. Note that again, without loss of generality, we have chosen a point at a far enough normal distance in the second medium that the refractive surface is equidistant to the chosen point and the point light source in the first medium.

T= t₁+t₂ = d/c₁*sec(i) + d/c₂*sec(r)

c₁*tan(i)+ c₂*tan(r) = constant (x/d)

Snell’s law: sin(i)/sin(r) = c₁/c₂ would yield the desired solution.

Pursuant to the preivous post on the pythagoras theorem, we continue asking ‘why’, in this series of posts. I attempt to share my feeble understanding of why electric forces and gravitational forces closely obey the inverse square law in classical physics.

I like the graivational wave interpretation. Simply put, this pins the cause on the world being three dimensional (disregarding sting theory). The power of any constant flux or flow outward from a central source (even a point source), will dissipate proportional to the area that it is spreading out over. For instance, a sound wave from a speaker suspended high up in the air would travel in all directions uniformly. At any given time, the wavefront would be a set distance away from the source. The distance itself would depend on the speed of sound. But the key point is that it would spread out in a sphere of radius equal to that distance. So the intensity of the sound energy drops proportional to the area of a sphere, whose radius is the distance from the source. Since the area of the sphere depends on the square of the radius, the wave is divided or distributed by a factor proportional to the square of that distance, leading to the inverse square law.

There is a theorem so powerful in geometry, that it lies at the heart of not only geometry, but also coordinate geometry, trigonometry, algebra, vector physics and more! I am talking about an ancient theorem known ages ago, called in modern day by a name none other than the Pythagorean theorem.

I am going to try to explain the beauty of this theorem and the breadth of its application in this post. This brings me to a key difference between mathematics and physics. In mathematics, one may never ask why, only if. You can ask if 11 is prime and how to determine if it is prime. But you may never meaningfully pose the question .. why is 11 prime. Yet I deeply believe, that all of human endeavor into the sciences relies on understanding the why. This is why I like physics so much, it allows us to ask why. My effort here is not really to prove such a well known theorem, but rather ask why the Pythagorean theorem is true. In the process, of course, it will be a non-rigorous proof.

So you may ask, am I claiming that prime numbers or other areas of relatively abstract math are unnatural? Not at all! In other words, what I am hinting at is the fact that we cannot answer why 11 is prime is just a sign of incomplete understanding. Even abstract math is often rooted in our observations of reality, if not specifically in nature. In fact, this is often the litmus test for something being the right way to define things or pick axioms.

So, let us start with a right triangle of side lengths a, b,c. Assume c is the hypotenuse, without loss of generality. Let us consider a light source at each vertex of the triangle, casting a vertical shadow. This is called a projection in physics. Specifically here, we are using an orthogonal projection. If it helps, one may imagine the hypotenuse as a ladder and the other two sides as a wall and the ground. So, with light sources present at each vertex, both sides cast shadows on the hypotenuse and the hypotenuse casts shadows onto each side from the light source at the opposite vertex. (In our example of the ladder, the ladder casts a shadow on the wall and on the ground. It is important to note that the shadows from the ladder fall vertically as the light sources are at the foot of the ladder and the top of the ladder)

Now we switch our attention to the shadows that fall from the sides onto the hyptoenuse. (Imagine a light source just behind the point where the ground meets the wall, the ground prevents part of the light from reaching the ladder and the other part of light is blocked by the wall, so no light reaches the ladder). Then we notice that the shadow of a and b fall onto c without any overlap!!! This is the key observation. So to get the side length of c , we need to simply add up the length of the shadows of a and b. This leads us to the question of determining the projection effect.

Note that when the shadow of c falls onto side a, it exactly covers the side a, so the shadow has length a. This leads to a length contraction. Now when the shadow of a falls back onto c, it is a second projection of the same contraction factor. A slight nuance is implicit: The projection factor depends only on the angle between two lines and not on which line is projected onto another. This is provable by symmetry, as if we extend the two lines to infinity, then both lines are indistinguishable from each other… we can reverse the construction to use the other line to cast the projection or shadow and the factor has to remain the same.

If c—> a in the first projection, then by the unitary method, 1–> a/c and consequentially, a —> a*a/c. But this is what happens in the second projection, when a casts a shadow on c! This gives us the length of the shadow cast by a. Repeating for side b, we get, c—> b ==> b—> b*b/c. Now adding up both shadows, we get c = a*a/c + b*b/c. Multiplying through by c, we get the usual form of the theorem!

So what have we learnt? We learnt that a good name for this theorem might be … the double projection theorem. The same factor by which the hypotenuse projects its shadow onto the sides is again the physical factor when the sides are projected back onto the hypotenuse. And the sides project back with no overlap, so they add up to the hypotenuse. This double projection or factor is the whole reason the exponent in the Pythagoras theorem is 2, and not some other power.

Now let’s study the generation of triples, which are basically sets of whole numbers which can be sides of a right triangle as they fit the equation, like (3,4,5). In general any set {(u-v)², 4uv, (u+v)²} will suffice, if 4uv is a perfect square. This condition is definitely possible to meet by starting with non-prime equal u,v and factorizing it. Obviously, with equal u,v the product is a square. Then, exchange some of the factors, to make them unequal and still keep the product the same .. i.e, a perfect square. For example, if we start with u=v=6, then exchange factors to set u=9 and v=4, then the triple (as per the above formula) would be 13, 5 and the square root of 144, which is 12. There is a proof that this is the only way to construct triples! To be precise, the claim is that this construction covers every possible set, upto a factor, so all the co-prime ones would be covered at the very least.

In terms of application, we can show that any directional quantity (known as vectors) for instance, velocity, adds up in this way. If an object is moving to the left at 3 units velocity and forward at 4 units velocity, how far will the object travel in total?

Let’s break it down in time: In unit time, the object moves 3 units to the left and 4 units forward. There is no impact due to the order of translation, as the distance covered is the same (regardless of whether the translation in both left and forward direction is simultaneous or one direction is covered first followed by the other). But following the sequential step of motion, essentially the sides of a right triangle are traced out, if we mark the object’s original position and trace the path to a new position (And if the motion is simulatenous, we know that the starting position and final position must be the same as the sequential motion). The final translation will be the shortest distance between the initial and final position, which is the straight line betwen the points, which in turn is the hypotenuse of the right triangle with aforemntioned sides. THe hypotenuse has length 5, covered in unit time. So, the effective velocity is 5 units, equivalent to adding the impact of the two velocities by the Pythagoras theorem.

Let me also take the opportunity to explain why orthogonal directions are considered independent in physics. (Independent things/directions are called dimensions). It is because we can meaningfully specify any vector value (like velocity) for one direction separately from the other. On the other hand, if we have two directions at 60 degrees with each other, then 5 units velocity along one line is not consistent with, for example, zero velocity in the other direction. An object moving along one of the lines is by default also moving along the other direction. (An object having zero velocity in a direction is defined as: if we draw a line parallel to the direction, the closest point on the line to the object does not change). This is however possible, if the lines are orthogonal. The closest point on the orthogonal line does not change and there is no motion in the direction that the line is pointing toward.

This is how the Pythagorean theorem derives it’s great importance and utility. It is a formula to combine actions along multiple dimensions. It is more than just a triangle! (… in some sense, but in another sense, it is all just a triangle)