A bc dxb ds dxc ds 0 but the equivalent euler lagrange equations are d ds. The eulerlagrange equation is in general a second order di erential equation, but in some special cases, it can be reduced to a rst order di erential equation or where its solution can be obtained entirely by evaluating integrals. Generalized solutions for the euler equations in one and two. For example, if s is a circular cylinder, then there are three possible types of geodesic curves. Note that the extremal solution uis independent of the coordinate system you choose to represent it see arnold 3, page 59. Geodesic equation from euler lagrange stack exchange. For familiar surfaces, like the plane, sphere, cylinder, and cone, the results were also familiar because the integrals of the euler lagrange equation could be put in standard forms and worked out nicely. Eulerlagrange equations for geodesics physics forums.
Of course, it is just the derivation of the geodesic equations, i am not arguing that. Euler 17071783 was one of the most brilliant mathematicians of all time and he and lagrange 173618, a student of his and another great, invented variational calculus and the eulerlagrange equation. This was accomplished by first deriving the eulerlagrange equation. Time for both sides and do summation respect index of j for above equation. The smoothly varying inner product captures the idea of curved space. Solving the geodesic equation university of rochester. As the laws of physics can be written in any coordinate system, it is convenient to choose one that simplifies the geodesic equations. These are some brief notes on the calculus of variations aimed at undergraduate.
Lagrangian method christoffel symbols calculations. Browse other questions tagged geometry calculusofvariations integralequations geodesic eulerlagrangeequation or ask your own question. There are several ways to derive this result, and we will cover three of the most common approaches. Lecture notes on general relativity matthiasblau albert einstein center for fundamental physics institut fu. If the y variable is removed, we are back to a onedimensional rod. These equations together are the geodesic equation. We show that the eulerlagrange partial differential system on the energy function can be reduced. Calculus of variations and the geodesic equation springerlink. These are some brief notes on the calculus of variations aimed at undergraduate students in mathematics and physics. We get the geodesic equation in its most common form, which is more simple to use than its alternative form with the christoffel symbol as in this case the metric tensor g.
The eulerlagrange equation in this case is known as the geodesic equation. Derivation of the eulerlagrangeequation martin ueding. A bc dxb ds dxc ds 0 but the equivalent eulerlagrange equations are d ds. The euler lagrange pde and finsler metrizability arxiv. Deriving geodesic equation via action combine two parts of eulerlagrange equation together. Eulerlagranges equations in several variables so far we have studied one variable and its derivative let us now consider many variables and their derivatives i. In lecture, we derived the eulerlagrange equations the fundamental result of variational calculus. In this chapter, we will only have room to scratch the surface of this wide ranging and. Eulers equation are just formal derivatives as though y and y0 were unconnected and. The geodesic equation doesnt know which two points youre trying to find shortest distance between, so both orientations are possible. In this video, i introduce the calculus of variations and show a derivation of the eulerlagrange equation. A short history of calculus of variationsproblems from geometrynecessary condition. Carroll keeps using variational calculus and i think i understand it now.
And if so, why bother with euler lagrange to solve the problem. Calculus of variations 1 functional derivatives the fundamental equation of the calculus of variations is the eulerlagrange equation d dt. The main body of chapter 2 consists of well known results concerning necessary or su. Geodesic on a cylindrical surface checkpoints 3 a a cylindrical surface is centered on the zaxis and has radius b. We prove that uniqueness holds in dimension one whenever the pressure.
Euler lagrange s equations in several variables so far we have studied one variable and its derivative let us now consider many variables and their derivatives i. For familiar surfaces, like the plane, sphere, cylinder, and cone, the results were also familiar because the integrals of the eulerlagrange equation could be put in standard forms and worked out nicely. Lagrange multiplier from the incompressibility constraint. This gives us the method of lagranges undetermined multiplier. But the question is what is the maximal distance between geodesic and latitude lines. This is called the eulerlagrange equation for this variational problem, we see that in general it will be a secondorder ordinary di. So instead of doing the abstract formal derivation, lets look at an example, and see how it leads to the formation of the geodesic equation, and how one can read off the christoffel symbols from it. Yet both purport to give geodesic paths so both must ultimately. If you want to extremize a quantity s that is integrated over some path qi t of your system between. Putting all the terms together, our initial eulerlagrange equation becomes. Derivation of the eulerlagrange equation calculus of. There are several ways to derive the geodesic equation. So we could write the geodesic equation as before but where the dot now stands for derivative wrt the a. Thus apart from specifying a certain class of embedded curves, the geodesic equation also determines a preferred class of parameterizations on each of the curves.
Riemann discovered the essential features of metric geometry in arbitrary dimensions. It specifies the conditions on the functionalf to extremize the integral i. The euler lagrange equation is a necessary condition. Introductiontothecalculusofvariations university of minnesota. Lagrangian method christoffel symbols calculations mathstools. The mathematical techniques that have been developed to handle such optimization problems are fundamental in many areas of mathematics, physics, engineering, and other applications. Geodesic and euler lagrange equation stack exchange. One of which is the variational method which i seemed to understand it because it was written in great details. Feb 22, 2019 in this video, i introduce the calculus of variations and show a derivation of the euler lagrange equation. An introduction to lagrangian and hamiltonian mechanics. In this paper we investigate the following question. Einstein relatively easy geodesic equation from the. Then it was mentioned that the geodesic equation can be derived from the euler lagrange equations only.
Pdf the eulerlagrange pde and finsler metrizability. The eulerlagrange equation just guarantees that the integral is stationary. Thus, solving the geodesic equation here goes a long way toward motivating the basic techniques of riemannian geometry, which we will develop in the next chapter. Straightforward application of the euler lagrange equation d d. Euler equations and totally geodesic subgroups 81 the paper is organised as follows.
Phys 325 discussion 10 drilling eulerlagrange with. Eulerlagrange equation and geodesics classical physics. Herman march, 2008 we begin with the line element ds2 g. Find the euler lagrange differential equation satisfied by twice differentiable function yx which extremizes the functional 2 1, x x i y x f x y y dx. We would like to find a condition for the lagrange function, so that its integral, the action, becomes maximal or minimal for that, we change the coordinate by a little variation, although infinitesimal. Generalized solutions for the euler equations in one and. The euler lagrange equation in this case is known as the geodesic equation. Section 2 begins with a brief presentation of geodesic ow on groups and the eulerarnold equation.
Such a uis known as a stationary function of the functional j. A similar procedure to what we did in this section involving finding the geodesic of a cylinder can be generalized to find the geodesic along any surface. Then it was mentioned that the geodesic equation can be derived from the eulerlagrange equations only. We get the geodesic equation in its most common form, which is more simple to use than its alternative form with the christoffel symbol as in this case the metric tensor g get referenced directly. The only prerequisites are several variable calculus and the rudiments of linear algebra and di erential equations. The euler lagrange equation is in general a second order di erential equation, but in some special cases, it can be reduced to a rst order di erential equation or where its solution can be obtained entirely by evaluating integrals. Phys 325 discussion 10 drilling eulerlagrange with geodesics summary of variational calculus. Jul 16, 2017 in this video, i deriveprove the euler lagrange equation used to find the function yx which makes a functional stationary i. Deriving geodesic equation from lagrangian physics forums. Riemann discovered the essential features of metric geometry in ar. Ideas are the basis of the calculus of variations called principle of least action of eulerlagrange first of all, a.
Pdf the geodesic motion on a lie group equipped with a left or right invariant riemannian metric is governed by the eulerarnold equation. Eulerlagrange equationproblems from mechanicsmethod of lagrange multipliera problem from springmass systemsa problem from elasticitya problem from uid mechanicsa problem from image science compressed sensing geodesic curve on the plane. Calculus of variations university of california, san diego. Find the curve between two given points in the plane that yields a surface of. A symmetry implies that there is a conserved quantity along a geodesic. This can be seen by looking at the eulerlagrange equation, from which the geodesic equation is derived. Suppose the given function f is twice continuously di erentiable with respect to all of its arguments. Worked example geodesics on the surface of a sphere.
Euler introduced a condition on the path in the form of differential equations, which we later introduce as euler s equation. Sep 09, 2019 euler 17071783 was one of the most brilliant mathematicians of all time and he and lagrange 173618, a student of his and another great, invented variational calculus and the euler lagrange equation. Let p x y1 1, and q x y2 2, be two fixed points in xy plane. The key idea is that the distance between nearby points ds2 g.
After thinking about this a bit i realize the question was due to a rather silly misconception,the converse was not mentioned in the usual references because its trivial from the proof of the usual direction. Variational calculus and the eulerlagrange equation. Brief notes on the calculus of variations jose figueroaofarrill abstract. Passing the above lagrangian through the eulerlagrange equations. Mathematically, this means a coordinate chart is chosen in which the geodesic equations have a particularly tractable form. We will see in this section, the lagrangian method allows us to obtain the geodesic equations and hence obtain the chistoffel symbols. Just saying that it may be confusing to the op to start talking about proper time for a classical mechanics problem g is specified to be positive definite in the op and it. The integral is the parametric equation of the geodesic. This is the eulerlagrange equation, the basic equation of this theory. Generalized solutions for the euler equations in one and two dimensions marc bernot. It is a differential equation which determines y as a function of x. Consequently, in this case we must have from equation 5 that 2 1 1 2.
Ideas are the basis of the calculus of variations called principle of least action of euler lagrange first of all, a. Using variational methods as seen in classical dynamics, we obtain the eulerlagrange equations in the form. The euler lagrange equation can be used to find the geodesic on any curved surface. The calculus of variations and the eulerlagrange equation. Phys 325 discussion 10 drilling eulerlagrange with geodesics. Solving geodesic problems with eulerlagrange equation. Equation is invariant under affine reparameterizations. He began a systematic study of extreme value problems and was aware of developments by joseph louis lagrange. Euler introduced a condition on the path in the form of differential equations, which we later introduce as eulers equation. Euler lagrange equationproblems from mechanicsmethod of lagrange multipliera problem from springmass systemsa problem from elasticitya problem from uid mechanicsa problem from image science compressed sensing geodesic curve on the plane.
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