How do you prove that path of a satellite or a planet is a second degree curve? In other words, how do you prove Kepler's law which states that planets move in elliptical paths?
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How can analytically be  the ?   I found some extremely synthetic equations which from the  (in particular $\mathbf{F} = m \mathbf{a}$) tried to obtain the Kepler's laws, but even if it seemed to be a nice procedure they were too much incomplete.
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In classical geometry an ellipse is usually defined as the locus of points in the plane such that the distances from each point to the two foci have a given sum.  When we speak of an ellipse analytically, we usually describe it as a circle that has been squashed in one direction, i.e. something similar to the curve $x^2+(y/b)^2 = 1$.  "Everyone knows" that these two definitions yield the same family of shapes. But how can that be proved?
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So, my dilemma here is... I have an axis. This axis is given to me in the format of the slope of the axis in the x,y and z axes.  I need to come up with a parametric equation of a circle. This circle needs to have an axis of rotation at the given axis with a variable radius.  I've worked on this problem for days, and still haven't come up with a solution.  I'm using this circle to map the path of a satellite, programmed in C.  And help would be greatly appreciated.  Thanks!
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Show that if a curve $C ⊂ S$ is both a line of curvature and a geodesic, then $C$ is a plane curve.  Give an example of a line of curvature which is a plane curve and not a geodesic. (My thoughts: Take a sphere for $S$ and let the curve $C$ be any of the latitudes of $C$ that are not the equator or the poles.  Then $C$ will be a plane curve.  Additionally, since the curvature is constant on the whole of $S$, every direction is a principle direction (of sorts; this might be the flaw in my argument), so any curve is a line of curvature.  But, $C$ is not a segment of a great circle on $S$, so it is not a geodesic.)  Prove that a curve $C ⊂ S$ is both an asymptotic curve and a geodesic if and only if $C$ is a (segment of a) straight line.    Can I have some hints?  I am not very good at covariant derivatives yet, so hand-holding with detailed examples there would be nice.
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In classical General Relativity (meaning not modified) one can think of geodesics in two ways.    One way is to say that a geodesic is the curve which is the straightest (in analogy with the flat case) among all curves. More or less the story goes like this (correct me if I'm wrong): in the flat case geodesics are of the form $x^\mu(s)=st^\mu +b^\mu$ where $t$ and $b$ are constant vectors and $s$ is the curve parameter. The curve tangent vector is $\frac{dx^\mu}{ds}=t=const$ so $$\frac{dt^\mu}{ds}=t^\nu\partial_\nu t^\mu=0.$$ This is a tensor equation and by general covariance it  holds true in a general spacetime, mutatis mutandis: $$t^\nu\nabla_\nu t^\mu=0.$$ Another way is to obtain the equation by finding the minimum of the length functional $$L=\int \sqrt{g(t,t)} \,\, d\tau,$$ we get $$\frac{dt^\mu}{d\tau}+\Gamma^\mu_{\alpha\beta}t^\alpha t^\beta=0.$$ The above equation turns out to be just $t^\nu\nabla_\nu t^\mu=0$, with a certain parametrization choice.    The two formulations are thus equivalent. My question is when and why is it so? Is there a deep reason?  Looking up Wald's book I found the following, unclear to me, argument:     "On a manifold with a Riemannian metric, one can always find curves of arbitrarily long length connecting any two points. However, the length will be bounded from below, and the curve of shortest length connecting two points (assuming the lower bound in length is attained) is necessarily an extremum of length and thus a geodesic. Thus, the shortest path between two points is always a straightest possible path."    Later in the book he also says something about conjugate points.  I always thought it had to do with torsion: If we relax the condition $\Gamma^\mu_{\alpha\beta}-\Gamma^\mu_{\beta\alpha}=0$ then the connection is not Levi-Civita, hence $\delta_g L=0$ remains unchanged but $t^\nu\nabla_\nu t^\mu=0$ does and gives a different geodesic equation.. so Straightest $\neq$ Shortest anymore.  Can anyone clarify things to me?
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I'm trying to show that the derivative of $\sin\theta$ is equal to $\pi/180 \cos\theta$ if $\theta$ is measured in degrees. The main idea is that we need to convert $\theta$ to radians to be able to apply the identity $d/dx \sin x = \cos x $. So we need to express $ \sin \theta$  as  $$ \sin_{deg} \theta = \sin(\pi \theta /180), $$ where $\sin_{deg}$ is the $\sin$ function that takes degrees as input. Then applying the chain rule yields  $$ d/d\theta [ \sin(\pi\theta/180)] = \cos(\pi \theta/180) \pi/180 = \frac{\pi}{180}\cos_{deg}\theta. $$ Is this derivation formally correct?
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Is there a proof from the first principle that for the Lagrangian $L$,    $$L = T\text{(kinetic energy)} - V\text{(potential energy)}$$  in classical mechanics? Assume that Cartesian coordinates are used. Among the combinations, $L = T - nV$, only $n=1$ works. Is there a fundamental reason for it?  On the other hand, the variational principle used in deriving the equations of motion, Euler-Lagrange equation, is general enough (can be used to to find the optimum of any parametrized integral) and does not specify the form of Lagrangian. I appreciate for anyone who gives the  answer, and if possible, the primary source (who published the answer first in the literature).      Notes added on Sept 22: - Both answers are correct as far as I can find. Both answerers were not sure about what I meant by the term I used: 'first principle'. I like to elaborate what I was thinking, not meant to be condescending or anything near to that. Please have a little understanding if the words I use are not well-thought of. - We do science by collecting facts, forming empirical laws, building a theory which generalizes the laws, then we go back to the lab and find if the generalization part can stand up to the verification. Newton's laws are close to the end of empirical laws, meaning that they are easily verified in the lab. These laws are not limited to gravity, but are used mostly under the condition of gravity. When we generalize and express them in Lagrangian or Hamiltonian, they can be used where Newton's laws cannot, for example, on electromagnetism, or any other forces unknown to us. Lagrangian or Hamiltonian and the derived equations of motion are generalizations and more on the theory side, relatively speaking; at least those are a little more theoretical than Newton's laws. We still go to lab to verify these generalizations, but it's somewhat harder to do so, like we have to use Large Hadron Collider. - But here is a new problem, as @Jerry Schirmer pointed out in his comment and I agreed. Lagrangian is great tool if we know its expression. If we don't, then we are at lost. Lagrangian is almost as useless as Newton's laws for a new mysterious force. It's almost as useless but not quite, because we can try and error. We have much better luck to try and error on Lagrangian than on equations of motion. - Oh, variational principle is a 'first principle' in my mind and is used to derive Euler-Lagrange equation. But variational principle does not give a clue about the explicit expression of Lagrangian. This is the point I'm driving at. This is why I'm looking for help, say, in Physics SE. If someone knew the reason why n=1 in L=T-nV, then we could use this reasoning to find out about a mysterious force. It looks like that someone is in the future.
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Just to check that I'm on the right track:  Are all great-circles on the sphere and in equirectangular projection (i.e. latitude, longitude pairs) either:   meridians (i.e. going pole to pole) of the form tan latitude = sin360(longitude + rotation) * amplitude + offset   (with additional restrictions on the offset/amplitude combinations -- obviously, all great circle paths with an amplitude of 0 also have offset 0 - the equator).  Or are there great-circle paths that do not fit into this scheme (again, only in a longitude-latitude coordinate system, not on other map projections).  Note: I added the tan above after posting the question, in a reaction to whubers excellent reply. It turns out that offset then always is 0.
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I posted a similar  about geodesics on Math.SE. Many sources ( for instance) claim that the light could maximize the optical path length in some cases. But I don't think it's actually true since between two points I can always imagine a path with arbitrary length. Therefore, or the light minimizes the optical length or it's stationary (neither maximum nor minimum). My question is: is the trajectory in red in the figure below a local minimum or a stationary curve?   If I'm wrong, please provide some examples which the light propagates in trajectory of maximum length.
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My current idea is to use the dual graph, and prove that that is a bipartite graph, and so proving the statement. But I can't seem to figure out how even degree for all verticies means that it is bipartite.  Any help is appreciated. Thanks!
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Prove there's a simple path of length $k$ in a simple graph $G$ where all the vertices have degree of at least $k$.   Relevant definitions:  $G$ is a simple graph that consists of a vertex set $V(G) = \{v_1, v_2, ..., v_n\}$ and an edge set $E(G) = \{e_1, e_2, ..., e_m\}$ where each edge is an ordered pair of vertices. The edge $\{u,v\}$ is denoted $uv$.  A walk of length $k$ is a sequence $v_0,e_1,v_1,e_2,...,e_k,v_k$ of vertices and edges such that $e_i=v_{i-1}v_i$ for all $i$. A path is a walk with no repeated vertex.   My attempt:  Induction, for $k=1$ it's obvious.  Suppose for $k-1$ and we'll prove for $k$.  Let $G$ be a simple graph such that $\forall v \in V : d(v)\ge k$.  We can assume that there are at least $k+1$ vertices since there are $k$ neighbours to every vertex.  Let $v_0$ be some vertex, it has $k$ neighbours, we'll move to one of its neighbours say $v_1$ and remove $v_0$. So now there are $k$ vertices with a degree of at least $k-1$ and from the induction hypothesis we'll have a path of length $k$.  Is using only $k$ vertices from $n$ right? Does it keep the generality when using only all the neighbours of $v_0$?
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Last year, I took physics and over the summer I have started to wonder about why many phenomena work the way they do (such as why is $\mathrm{KE} =\frac{1}{2}mv^2$, etc.). I have found answers to all of my questions, except for this one:  Why are we allowed to use trigonometry to get vector components? I understand that if we draw a right triangle, then we can express the adjacent side with $\text{length of hypotenus} \times \cos(\theta)$. But why does this work with velocity and with forces? I have tried to reason through this many times, but I am not able to figure out why we are able to do this. Any ideas?   BTW: My attempts for reasoning through velocity were as follows: If we have a ball moving in two dimensions, and we are given that the ball has $\sqrt{3}$ the speed in the x-direction than the y-direction, then we can conclude that the ball will move in a 30-degree angle. I am not sure how to apply this to forces or acceleration though. For example, why is it that in the case of a ball spinning horizontally while attached to a string, that the y component of the strings tension gets smaller as the centripetal force increases? I can see how we can prove this by using trigonometry, but why does this trig work in the first place?
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I'm a maths teacher in Italian secondary school and I've been spending some time trying to construct "meaningful" problems about conic sections. I particularly like problems which focus on practical applications of the curves (in engineering, physics, finance, etc.), on their geometric properties, or both.  I know there's this really easy formula to find the latus rectum of a conic from its key parameters. It's quite nice and straightforward to derive it, but...  ... What's its use? Why should anyone be interested in knowing the latus rectum of a curve? Is there any physical phenomenon, any architectural trick, or anything else where its position and/or length plays a key role?   I know just the relation between angular momentum, standard gravitational parameter and latus rectum in an astronomical orbit ($l=\frac{h^2}{GM}$, where $l$ is half the latus rectum, and $h$ is the ratio between the angular momentum associated to the orbiting body and its mass). It's cool, but a bit tricky, and probably too far-fetched for secondary school students... Is there anything simpler? Or just anything else, at any difficulty level?  Many thanks!
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Might a planet perform figure-8 orbits around two stars?  I'm thinking that if the two stars were equal mass (and not orbiting each other) then a planet that were to go right between them would continue in a straight line, with no preference for either star. But since the two stars would in fact be orbiting each other, the system would be rotating and thus there would be a Coriolis preference for one of the stars. Might that preference be made to alternate stars?  Another possibility would be if each star were in turn orbited closely by another planet, which would perform three orbits for each orbit of our planet of interest. Then things could be timed where on one pass star A's inner planet were aligned right to pull the planet of interest into an orbit, and on another pass star B's inner planet would be aligned right to pull the planet of interest into an orbit. So we have a system of five bodies, two massive (stars) each orbited by a minor, and one minor performing figure-8s.  Is this at least plausible if contrived?
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I tried solving the above question but was unable to prove it. I used Descartes rule of sign, factorisation techniques, and many other things but could not figure out the solution.
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This is a problem occurs to me when I was trying to find the limit of a function with two variables(if only it exists). Please help.  When I have to show that the limit does not exist for some function. Then by showing along two paths have two different limits I can prove it since the functions with two variables have infinite number of directions of approach to the given point.   So selecting the paths, should I only take the linear ones or can I take non linear paths also?  As an example,consider the limit of $\frac{xy^3}{x^2+y^6}$ as $(x,y)$ goes to $(0,0)$ Then taking along $x$ axis I have the limit $0$. Can I take $y=x^{1/3}$ which gives me the limit $1/2$ ?  When I check the answer on some website it seems they haven't consider curved functions to find different limits.  Please explain. Where am I wrong?
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Hi my course specifically talks about :  Cartesian and Polar Coordinates in 3 Dim, second Degree eqns in 3 vars, reduction to canonical forms, straight lines, shortest distance between 2 skew lines, Plane, sphere, cone, cylinder, paraboloid, ellipsoid,hyperboloid of one and two sheets and their properties.  This is merely indicative, and I would really like if the book talks about these (atleast) and some more. Would appreciate if it contained solved examples and exercise problems as well  Help Much Appreciated, Soham