integration

What will be the value of k to satisfy this integral equation

January 19, 2025 math and physics play No comments , ,

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Another problem from x/twitter [1]:

Find \( k \), where
\begin{equation}\label{eqn:trigProp:20}
\int_0^{2 \pi} \sin^4 x dx = k \int_0^{\pi/2} \cos^4 x dx.
\end{equation}
I initially misread the integration range in the second integral as \( 2 \pi \), not \( \pi/2 \), in which case the answer is just 1 by inspection. However, solving the stated problem, is not much more difficult.

Since sine and cosine are equal up to a shift by \( \pi/2 \)
\begin{equation}\label{eqn:trigProp:40}
\sin(u + \pi/2) = \frac{e^{i(u + \pi/2)} – e^{-i(u + \pi/2)}}{2i} = \frac{e^{i u} + e^{-i u}}{2} = \cos u,
\end{equation}
we can make an \( x = u + \pi/2 \) substitution in the sine integral.

Observe that \( \cos^4 x = \Abs{\cos x}^4 \), but the area under \( \Abs{\cos x} \) is the same for each \( \pi/2 \) interval. This is shown in fig. 1.

fig. 1. Plot of |cos x|

Of course, the area under \( \cos^4 x \), will also have the same periodicity, but those regions will be rounded out by the power operation, as shown in fig. 2.

fig. 2. Plot of cos^4 x.

Since the area under \( \cos^4 x \) is the same for each \( \pi/2 \) wide interval, we have
\begin{equation}\label{eqn:trigProp:60}
\boxed{
k = 4.
}
\end{equation}

References

[1] CalcInsights. What will be the value of k to satisfy this integral equation, 2025. URL https://x.com/CalcInsights_/status/1880932308108341443. [Online; accessed 19-Jan-2025].

New video: Elliptical motion from Newton’s law of gravitation.

September 14, 2023 math and physics play , , , , , , , , , ,

This blog post is a text version of the video below, available in a few forms:

 

We found previously that
\begin{equation}\label{eqn:solarellipse:20}
\mathbf{\hat{r}}’ = \inv{r} \mathbf{\hat{r}} \lr{ \mathbf{\hat{r}} \wedge \Bx’ }.
\end{equation}
Somewhat remarkably, we can use this identity to demonstrate that orbits governed gravitational force are elliptical (or parabolic, or hyperbolic.) This ends up being possible because the angular momentum of the system is a conserved quantity, and this immediately introduces angular momentum into the mix in a fundamental way. In particular,
\begin{equation}\label{eqn:solarellipse:40}
\mathbf{\hat{r}}’ = \inv{m r^2} \mathbf{\hat{r}} L,
\end{equation}
where we define the angular momentum bivector as
\begin{equation}\label{eqn:solarellipse:60}
L = \Bx \wedge \Bp.
\end{equation}
Our gravitational law is
\begin{equation}\label{eqn:solarellipse:80}
m \ddt{\Bv} = – G m M \frac{\mathbf{\hat{r}}}{r^2},
\end{equation}
or
\begin{equation}\label{eqn:solarellipse:100}
-\inv{G M} \ddt{\Bv} = \frac{\mathbf{\hat{r}}}{r^2}.
\end{equation}
Combining the gravitational law with our \( \mathbf{\hat{r}} \) derivative identity, we have
\begin{equation}\label{eqn:solarellipse:120}
\begin{aligned}
\ddt{ \mathbf{\hat{r}} }
&= \inv{m} \frac{\mathbf{\hat{r}}}{r^2} L \\
&= -\inv{G m M} \ddt{\Bv} L \\
&= -\inv{G m M} \lr{ \ddt{(\Bv L)} – \ddt{L} }.
\end{aligned}
\end{equation}
Since angular momentum is a constant of motion of the system, means that
\begin{equation}\label{eqn:solarellipse:140}
\ddt{L} = 0,
\end{equation}
our equation of motion is integratable
\begin{equation}\label{eqn:solarellipse:160}
\ddt{ \mathbf{\hat{r}} } = -\inv{G m M} \ddt{(\Bv L)}.
\end{equation}
Introducing a vector valued integration constant \( -\Be \), we have
\begin{equation}\label{eqn:solarellipse:180}
\mathbf{\hat{r}} = -\inv{G m M} \Bv L – \Be.
\end{equation}
We’ve transformed our second order differential equation to a first order equation, one that does not look easy to integrate one more time. Luckily, we do not have to integrate, and can partially solve this algebraically, enough to describe the orbit in a compact fashion.

Before trying that, it’s worth quickly demonstrating that this equation is not a multivector equation, but a vector equation, since the multivector \( \Bv L \) is, in fact, vector valued.
\begin{equation}\label{eqn:solarellipse:200}
\begin{aligned}
\Bv L
&= \Bv \lr{ \Bx \wedge (m \Bv) } \\
&\propto \mathbf{\hat{v}} \lr{ \mathbf{\hat{r}} \wedge \mathbf{\hat{v}} } \\
&= \mathbf{\hat{v}} \cdot \lr{ \mathbf{\hat{r}} \wedge \mathbf{\hat{v}} } + \mathbf{\hat{v}} \wedge \lr{ \mathbf{\hat{r}} \wedge \mathbf{\hat{v}} } \\
&= \mathbf{\hat{v}} \cdot \lr{ \mathbf{\hat{r}} \wedge \mathbf{\hat{v}} } \\
&= \lr{ \mathbf{\hat{v}} \cdot \mathbf{\hat{r}} } \mathbf{\hat{v}} – \mathbf{\hat{r}},
\end{aligned}
\end{equation}
which is a vector (i.e.: a vector that is directed along the portion of \( \Bx \) that is perpendicular to \( \Bv \).)

We can reduce \ref{eqn:solarellipse:180} to a scalar equation by dotting with \( \Bx = r \mathbf{\hat{r}} \), leaving
\begin{equation}\label{eqn:solarellipse:220}
\begin{aligned}
r
&= -\inv{G m M} \gpgradezero{ \Bx \Bv L } – \Bx \cdot \Be \\
&= -\inv{G m^2 M} \gpgradezero{ \Bx \Bp L } – \Bx \cdot \Be \\
&= -\inv{G m^2 M} \gpgradezero{ \lr{ \Bx \cdot \Bp + L } L } – \Bx \cdot \Be \\
&= -\inv{G m^2 M} L^2 – \Bx \cdot \Be,
\end{aligned}
\end{equation}
or
\begin{equation}\label{eqn:solarellipse:240}
r = -\frac{L^2}{G M m^2} – r e \cos\theta,
\end{equation}
or
\begin{equation}\label{eqn:solarellipse:260}
r \lr{ 1 + e \cos\theta } = -\frac{L^2}{G M m^2}.
\end{equation}
Observe that the RHS constant is a positive constant, since \( L^2 \le 0 \). This has the structure of a conic section, if we write
\begin{equation}\label{eqn:solarellipse:280}
-\frac{L^2}{G M m^2} = e d.
\end{equation}
This is an ellipse, for \( e \in [0,1) \), a parabola for \( e = 1 \), and hyperbola for \( e > 1 \) ([1] theorem 10.3.1).

fig. 1. Ellipse with e = 0.75

In fig. 1 is a plot with \( e = 0.75 \) (changing \( d \) doesn’t change the shape of the figure, just the size.)

References

[1] S.L. Salas and E. Hille. Calculus: one and several variables. Wiley New York, 1990.