Brachistochrone Solution
## The brachistochrone is the cycloid

### Johann Bernoulli’s solution

## References

A **brachistochrone curve** (Gr. βράχιστος, brachistos - *the shortest*, χρόνος, chronos - *time*) or curve of fastest descent, is the path that will carry a point-like body from one place to another in the least amount of time. The body is released at rest from the starting point and is constrained to move without friction along the curve to the end point, while under the action of constant gravity. The brachistochrone curve is the same as the tautochrone curve for a given starting point.

Given two points *A* and *B*, with *A* not lower than *B*, only one upside down cycloid passes through both points, has a vertical tangent line at *A*, and has no maximum points between *A* and *B*: the brachistochrone curve. The curve does not depend on the body’s mass or on the strength of the gravitational constant.

The problem can be solved with the tools from the calculus of variations and optimal control.^{}

If the body is given an initial velocity at *A*, or if friction is taken into account, then the curve that minimizes time will differ from the one described above.

According to Fermat’s principle: *The actual path between two points taken by a beam of light is the one which is traversed in the least time.* In 1697 Johann Bernoulli used this principle to derive the brachistochrone curve by considering the trajectory of a beam of light in a medium where the speed of light increases following a constant vertical acceleration (that of gravity *g*).^{[2]}

The Conservation of energy can be used to express the speed of a body in a constant gravitational field as:

,

where *y* represents the vertical distance the body has fallen. The speed of motion of the body along an arbitrary curve does not depend on the horizontal displacement.

Johann Bernoulli noted that the law of refraction gives a constant of the motion for a beam of light in a medium of variable density:

,

where *v _{m}* is the constant and represents the angle of the trajectory with respect to the vertical.

The equations above allow us to draw two conclusions:

- At the onset, the angle must be zero when the particle speed is zero. Hence, the brachistochrone curve is tangent to the vertical at the origin.
- The speed reaches a maximum value when the trajectory becomes horizontal and the angle θ = 90°.

Simplifyingly assuming that the particle (or the beam) with coordinates (x,y) departs from the point (0,0) and reaches maximum speed after a falling a vertical distance *D*:

.

Rearranging terms in the law of refraction and squaring gives:

which can be solved for *dx* in terms of *dy*:

.

Substituting from the expressions for *v* and *v _{m}* above gives:

which is the differential equation of an inverted cycloid generated by a circle of diameter *D*.

Johann’s brother Jakob showed how 2nd differentials can be used to obtain the condition for least time. A modernized version of the proof is as follows. If we make a negligible deviation from the path of least time, then, for the differential triangle formed by the displacement along the path and the horizontal and vertical displacements,

.

On differentiation with *dy* fixed we get,

.

And finally rearranging terms gives,

where the last part is the displacement for given change in time for 2nd differentials. Now consider the changes along the two neighboring paths in the figure below for which the horizontal separation between paths along the central line is *d ^{2}x* (the same for both the upper and lower differential triangles). Along the old and new paths, the parts that differ are,

For the path of least times these times are equal so for their difference we get,

And the condition for least time is,

** **

Ross, I. M. The Brachistochrone Paridgm, in *A Primer on Pontryagin’s Principle in Optimal Control*, Collegiate Publishers, 2009. ISBN 978-0-9843571-0-9.

Babb, Jeff; Currie, James (July 2008), "The Brachistochrone Problem: Mathematics for a Broad Audience via a Large Context Problem", *TMME* **5** (2&3): 169–184

Struik, J. D. (1969), *A Source Book in Mathematics, 1200-1800*, Harvard University Press, ISBN 0-691-02397-2

D.T.Whiteside, *Newton the mathematician*, in Bechler, *Contemporary Newtonian Research*, p. 122.

The Chemical Behind Catnip’s Effect on Cats
CATNIP: EGRESS TO OBLIVION?

☠**- 4aα,7α,7aα-Nepetalactone** is the specific isomer responsible for the catnip effect

☠ Firstly, nepetalactone will enter the cat’s nasal tissue, and there it will bind to certain receptors.

☠ These can then trigger particular sensory neurons to signal to other neurons, and eventually the brain; in particular, the ‘olfactory bulb’, a region at the front of the brain responsible for processing smells.

☠ This region then signals other regions of the brain, including the amygdala, responsible for emotional responses to stimuli, and the hypothalamus, responsible for behavioural responses to stimuli.

☠ This results in the observed response in cats – a response that is actually**similar to their response to natural sex pheromones**.

☠ The**effect of catnip lasts for around ten minutes**, and afterwards there will be a refractory period of around an hour where the cat will remain unaffected.

☠ Interestingly, not all cats are affected by catnip;**the response is genetic, and autosomal dominant**, which means if one parent passes on the gene, then the offspring will inherit the response.

**Article source:** http://goo.gl/lTHpY7

**Video by BBC Nature:** http://goo.gl/wLvZ5B

**Read more :** Inheritance of the catnip response in domestic cats , J. Hered (1962) 53 (2): 54-56 http://goo.gl/cCxU5a

☠

☠ Firstly, nepetalactone will enter the cat’s nasal tissue, and there it will bind to certain receptors.

☠ These can then trigger particular sensory neurons to signal to other neurons, and eventually the brain; in particular, the ‘olfactory bulb’, a region at the front of the brain responsible for processing smells.

☠ This region then signals other regions of the brain, including the amygdala, responsible for emotional responses to stimuli, and the hypothalamus, responsible for behavioural responses to stimuli.

☠ This results in the observed response in cats – a response that is actually

☠ The

☠ Interestingly, not all cats are affected by catnip;

**References & Further Reading**

- ‘How Does Catnip Work its Magic on Cats?’ – Scientific American
- ‘Catnip & The Catnip Response’ – A Tucker & S Tucker
- ‘Catnip – Its Uses & Effects, Past & Present’ – J Grognet
- Cat image: Sergiu Bacioiu, Flickr, Creative Commons licensed
- Catnip image: Liz West, Flickr, Creative Commons licensed

How to slice a bagel into two congruent, linked halves.