Calculus of Variations: New Applications
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en-us2019 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemTue, 16 Jul 2019 08:41:59 GMTTue, 16 Jul 2019 08:41:59 GMTNew applications in the Calculus of Variations categoryhttps://www.maplesoft.com/images/Application_center_hp.jpgCalculus of Variations: New Applications
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Parameterizing Motion along a Curve
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<p>We use the Euler-Lagrange equation to parameterize the motion of a bead on a parabola, a helix, and a piecewise defined combination of the two.</p><img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Parameterizing Motion along a Curve" style="max-width: 25%;" align="left"/><p>We use the Euler-Lagrange equation to parameterize the motion of a bead on a parabola, a helix, and a piecewise defined combination of the two.</p>https://www.maplesoft.com/applications/view.aspx?SID=130465&ref=FeedWed, 08 Feb 2012 05:00:00 ZShawn HedmanShawn HedmanClassroom Tips and Techniques: Nonlinear Fit, Optimization, and the DirectSearch Package
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In this month's article, I revisit a nonlinear curve-fitting problem that appears in my Advanced Engineering Mathematics ebook, examine the role of Maple's Optimization package in that problem, and then explore the DirectSearch package from Dr. Sergey N. Moiseev.<img src="https://www.maplesoft.com/view.aspx?si=122760/thumb.jpg" alt="Classroom Tips and Techniques: Nonlinear Fit, Optimization, and the DirectSearch Package" style="max-width: 25%;" align="left"/>In this month's article, I revisit a nonlinear curve-fitting problem that appears in my Advanced Engineering Mathematics ebook, examine the role of Maple's Optimization package in that problem, and then explore the DirectSearch package from Dr. Sergey N. Moiseev.https://www.maplesoft.com/applications/view.aspx?SID=122760&ref=FeedWed, 15 Jun 2011 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Geodesics on a Surface
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<p>Several months ago we provided the article Tensor Calculus with the Differential Geometry Package in which we found geodesics in the plane when the plane was referred to polar coordinates. In this month's article we find geodesics on a surface embedded in R<sup>3</sup>. We illustrate three approaches: numeric approximation, the calculus of variations, and differential geometry.</p><img src="https://www.maplesoft.com/view.aspx?si=34940/thumb.jpg" alt="Classroom Tips and Techniques: Geodesics on a Surface" style="max-width: 25%;" align="left"/><p>Several months ago we provided the article Tensor Calculus with the Differential Geometry Package in which we found geodesics in the plane when the plane was referred to polar coordinates. In this month's article we find geodesics on a surface embedded in R<sup>3</sup>. We illustrate three approaches: numeric approximation, the calculus of variations, and differential geometry.</p>https://www.maplesoft.com/applications/view.aspx?SID=34940&ref=FeedTue, 08 Dec 2009 05:00:00 ZDr. Robert LopezDr. Robert LopezKarush Khun Tucker (KKT)
https://www.maplesoft.com/applications/view.aspx?SID=33095&ref=Feed
<p>Resolución de problemas de optimización con el método de KKT , dicho método es una generalización del método de los multiplicadores de Lagrange , que asume ciertas condiciones para las variables.</p><img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Karush Khun Tucker (KKT)" style="max-width: 25%;" align="left"/><p>Resolución de problemas de optimización con el método de KKT , dicho método es una generalización del método de los multiplicadores de Lagrange , que asume ciertas condiciones para las variables.</p>https://www.maplesoft.com/applications/view.aspx?SID=33095&ref=FeedMon, 08 Jun 2009 04:00:00 ZNicolas PalmaNicolas PalmaComputing ODE symmetries as abnormal variational symmetries
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We give a new computational method to obtain symmetries of ordinary differential equations. The proposed approach appears as an extension of a recent algorithm to compute variational symmetries of optimal control problems [P.D.F. Gouveia, D.F.M. Torres, Automatic computation of conservation laws in the calculus of variations and optimal control, Comput. Methods Appl. Math. 5 (4) (2005) 387-409], and is based on the resolution of a first order linear PDE that arises as a necessary and sufficient condition of invariance for abnormal optimal control problems. A computer algebra procedure is developed, which permits one to obtain ODE symmetries by the proposed method. Examples are given, and results compared with those obtained by previous available methods.<img src="https://www.maplesoft.com/view.aspx?si=6881/1.jpg" alt="Computing ODE symmetries as abnormal variational symmetries" style="max-width: 25%;" align="left"/>We give a new computational method to obtain symmetries of ordinary differential equations. The proposed approach appears as an extension of a recent algorithm to compute variational symmetries of optimal control problems [P.D.F. Gouveia, D.F.M. Torres, Automatic computation of conservation laws in the calculus of variations and optimal control, Comput. Methods Appl. Math. 5 (4) (2005) 387-409], and is based on the resolution of a first order linear PDE that arises as a necessary and sufficient condition of invariance for abnormal optimal control problems. A computer algebra procedure is developed, which permits one to obtain ODE symmetries by the proposed method. Examples are given, and results compared with those obtained by previous available methods.https://www.maplesoft.com/applications/view.aspx?SID=6881&ref=FeedTue, 11 Nov 2008 00:00:00 ZProf. Delfim TorresProf. Delfim TorresAutomatic Computation of Conservation Laws in the Calculus of Variations and Optimal Control
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We present analytic computational tools that permit us to identify, in an automatic way, conservation laws in optimal control. The central result we use is the famous Noetherâ€™s theorem, a classical theory developed by Emmy Noether in 1918, in the context of the calculus of variations and mathematical physics, and which was extended recently to the more general context of optimal control. We show how a Computer Algebra System can be very helpful in finding the symmetries and corresponding conservation laws in optimal control theory, thus making useful in practice the theoretical results recently obtained in the literature. A Maple implementation is provided and several illustrative examples given.<img src="https://www.maplesoft.com/view.aspx?si=4805/gouveia-torres-CLsOptCont_5.gif" alt="Automatic Computation of Conservation Laws in the Calculus of Variations and Optimal Control" style="max-width: 25%;" align="left"/>We present analytic computational tools that permit us to identify, in an automatic way, conservation laws in optimal control. The central result we use is the famous Noetherâ€™s theorem, a classical theory developed by Emmy Noether in 1918, in the context of the calculus of variations and mathematical physics, and which was extended recently to the more general context of optimal control. We show how a Computer Algebra System can be very helpful in finding the symmetries and corresponding conservation laws in optimal control theory, thus making useful in practice the theoretical results recently obtained in the literature. A Maple implementation is provided and several illustrative examples given.https://www.maplesoft.com/applications/view.aspx?SID=4805&ref=FeedWed, 26 Jul 2006 00:00:00 ZProf. Delfim TorresProf. Delfim TorresInverted Pendulum on an Oscillating Table
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We model the motion of an inverted pendulum supported on a table that oscillates vertically with motion y=A cos(wt). We show that for high enough frequency, the pendulum does not fall.<img src="https://www.maplesoft.com/view.aspx?si=4490/1294.jpg" alt="Inverted Pendulum on an Oscillating Table" style="max-width: 25%;" align="left"/>We model the motion of an inverted pendulum supported on a table that oscillates vertically with motion y=A cos(wt). We show that for high enough frequency, the pendulum does not fall.https://www.maplesoft.com/applications/view.aspx?SID=4490&ref=FeedThu, 25 Mar 2004 13:46:08 ZFrank WangFrank WangThe VariationalCalculus Package
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The new VariationalCalculus package in Maple 8 provides routines for solving problems in the calculus of variations, which studies nature's most "efficient" curves and surfaces. Examples include: find the shortest path between two points on a 3-D surface, shape a ramp between two heights such that a ball rolling down it reaches the bottom in minimum time, and find the shape of a soap film having minimum surface area spanning a given wire frame.
Such problems can often be solved with the Euler-Lagrange equation, which generalizes the Lagrange Multiplier Theorem. The Euler-Lagrange equation is easy to write down in general but notoriously difficult to write down and solve for most practical problems. The VariationalCalculus package automates the construction and analysis of the Euler-Lagrange equation.<img src="https://www.maplesoft.com/view.aspx?si=4269//applications/images/app_image_blank_lg.jpg" alt="The VariationalCalculus Package" style="max-width: 25%;" align="left"/>The new VariationalCalculus package in Maple 8 provides routines for solving problems in the calculus of variations, which studies nature's most "efficient" curves and surfaces. Examples include: find the shortest path between two points on a 3-D surface, shape a ramp between two heights such that a ball rolling down it reaches the bottom in minimum time, and find the shape of a soap film having minimum surface area spanning a given wire frame.
Such problems can often be solved with the Euler-Lagrange equation, which generalizes the Lagrange Multiplier Theorem. The Euler-Lagrange equation is easy to write down in general but notoriously difficult to write down and solve for most practical problems. The VariationalCalculus package automates the construction and analysis of the Euler-Lagrange equation.https://www.maplesoft.com/applications/view.aspx?SID=4269&ref=FeedThu, 25 Apr 2002 14:28:54 ZMaplesoftMaplesoftTrajectory Near a Black Hole: an application of Lagrangian mechanics
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The lagrangian formulation of mechanics has great advantages in practical use. Calculations of this type require finding the derivative of a function with respect to another function. Although our method is a pedagogic approach that might involve longer steps, it is a straightforward attack on this problem, and practically all problems in classical mechanics can be solved once the lagrangian is found. In most real physics problems, there are no analytic solutions to differential equations. We particularly emphasize forming plots numerically. We introduce an example in general relativity, to find the trajectory of a particle near a black hole, which corresponds to the shortest path between two points in a curved space.<img src="https://www.maplesoft.com/view.aspx?si=4240//applications/images/app_image_blank_lg.jpg" alt="Trajectory Near a Black Hole: an application of Lagrangian mechanics " style="max-width: 25%;" align="left"/>The lagrangian formulation of mechanics has great advantages in practical use. Calculations of this type require finding the derivative of a function with respect to another function. Although our method is a pedagogic approach that might involve longer steps, it is a straightforward attack on this problem, and practically all problems in classical mechanics can be solved once the lagrangian is found. In most real physics problems, there are no analytic solutions to differential equations. We particularly emphasize forming plots numerically. We introduce an example in general relativity, to find the trajectory of a particle near a black hole, which corresponds to the shortest path between two points in a curved space.https://www.maplesoft.com/applications/view.aspx?SID=4240&ref=FeedTue, 12 Mar 2002 11:24:18 ZProf. J. OgilvieProf. J. OgilvieMotion of a Heavy Symmetric Top
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Calculus of variations involves finding a derivative of a function with respect to a function, which Maple does not directly support. With simple substitution, we can, however, solve easily a problem of this type without invoking an external library. We use a problem in classical mechanics to illustrate a procedure to solve the Euler-Lagrange equation, which is the most important application of calculus of variations.<img src="https://www.maplesoft.com/view.aspx?si=4161//applications/images/app_image_blank_lg.jpg" alt="Motion of a Heavy Symmetric Top" style="max-width: 25%;" align="left"/>Calculus of variations involves finding a derivative of a function with respect to a function, which Maple does not directly support. With simple substitution, we can, however, solve easily a problem of this type without invoking an external library. We use a problem in classical mechanics to illustrate a procedure to solve the Euler-Lagrange equation, which is the most important application of calculus of variations.https://www.maplesoft.com/applications/view.aspx?SID=4161&ref=FeedMon, 05 Nov 2001 10:42:57 ZProf. J. OgilvieProf. J. OgilvieGeodesic on a Cone
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This worksheet shows how to draw geodesics on the cone as well as how to calculate the arclength of a geodesic. <img src="https://www.maplesoft.com/view.aspx?si=3575//applications/images/app_image_blank_lg.jpg" alt="Geodesic on a Cone" style="max-width: 25%;" align="left"/>This worksheet shows how to draw geodesics on the cone as well as how to calculate the arclength of a geodesic. https://www.maplesoft.com/applications/view.aspx?SID=3575&ref=FeedMon, 18 Jun 2001 00:00:00 ZJohn OpreaJohn Oprea