[{"data":1,"prerenderedAt":32},["ShallowReactive",2],{"topic-en-theoretical-mechanics\u002Fdynamics\u002Fanalytical-mechanics-and-oscillations\u002Flagranges-equations-of-the-second-kind":3},{"topic":4,"trail":13,"children":27,"tasks":28,"alternates":29},{"id":5,"name":6,"locale":7,"path":8,"seo_title":6,"seo_description":9,"seo_text":10,"content_html":11,"content_chunks":12},268,"Lagrange's Equations of the Second Kind","en","theoretical-mechanics\u002Fdynamics\u002Fanalytical-mechanics-and-oscillations\u002Flagranges-equations-of-the-second-kind","Generalized coordinates, generalized forces, and Lagrange's equations for mechanical systems.","This topic introduces generalized coordinates and forces and develops Lagrange's equations of the second kind as a systematic method for deriving equations of motion.","\u003Cp>\u003Cstrong>Lagrange's equations of the second kind\u003C\u002Fstrong> provide a systematic way to derive equations of motion using independent generalized coordinates without explicitly introducing reactions of ideal constraints.\u003C\u002Fp>\u003Ch2>Generalized coordinates\u003C\u002Fh2>\u003Cp>If a system has $s$ degrees of freedom, its configuration can be described by independent coordinates $q_1,\\ldots,q_s$. These may be linear displacements, angles, or other parameters that uniquely determine configuration.\u003C\u002Fp>\u003Ch2>Generalized velocities\u003C\u002Fh2>\u003Cp>The derivatives $\\dot q_j$ are generalized velocities. The kinetic energy is written as a function $T(q_j,\\dot q_j,t)$.\u003C\u002Fp>\u003Ch2>Generalized forces\u003C\u002Fh2>\u003Cp>The virtual work of active forces is written:\u003C\u002Fp>\u003Cp>$$\\delta A=\\sum_{j=1}^{s}Q_j\\delta q_j,$$\u003C\u002Fp>\u003Cp>where $Q_j$ is the generalized force corresponding to coordinate $q_j$.\u003C\u002Fp>\u003Ch2>Lagrange's equations\u003C\u002Fh2>\u003Cp>For a system with ideal constraints:\u003C\u002Fp>\u003Cp>$$\\frac{d}{dt}\\left(\\frac{\\partial T}{\\partial\\dot q_j}\\right)-\\frac{\\partial T}{\\partial q_j}=Q_j,\\qquad j=1,\\ldots,s.$$\u003C\u002Fp>\u003Cp>The number of independent equations equals the number of degrees of freedom.\u003C\u002Fp>\u003Ch2>Conservative forces\u003C\u002Fh2>\u003Cp>If forces have potential energy $\\Pi(q,t)$, their conservative generalized-force contribution is $Q_j=-\\partial\\Pi\u002F\\partial q_j$. With the Lagrangian $L=T-\\Pi$, the equations may be written $d(\\partial L\u002F\\partial\\dot q_j)\u002Fdt-\\partial L\u002F\\partial q_j=Q_j^{nc}$, where $Q_j^{nc}$ are nonconservative generalized forces.\u003C\u002Fp>\u003Ch2>Solution procedure\u003C\u002Fh2>\u003Col>\u003Cli>determine the number of degrees of freedom;\u003C\u002Fli>\u003Cli>choose independent $q_j$;\u003C\u002Fli>\u003Cli>express positions and velocities through $q_j,\\dot q_j$;\u003C\u002Fli>\u003Cli>calculate $T$;\u003C\u002Fli>\u003Cli>determine $Q_j$ or $\\Pi$;\u003C\u002Fli>\u003Cli>write one Lagrange equation for each coordinate.\u003C\u002Fli>\u003C\u002Fol>\u003Ch2>Example\u003C\u002Fh2>\u003Cp>For a mass $m$ on a horizontal spring with coordinate $x$, $T=m\\dot x^2\u002F2$ and $\\Pi=kx^2\u002F2$. Lagrange's equation gives $m\\ddot x+kx=0$.\u003C\u002Fp>\u003Ch2>Common mistakes\u003C\u002Fh2>\u003Cul>\u003Cli>choosing dependent coordinates as independent;\u003C\u002Fli>\u003Cli>assuming every generalized force has units of newtons — for an angular coordinate it has units of moment;\u003C\u002Fli>\u003Cli>omitting coordinate dependence of kinetic energy;\u003C\u002Fli>\u003Cli>counting the same conservative force both through $\\Pi$ and through $Q_j$.\u003C\u002Fli>\u003C\u002Ful>",[],[14,18,22,26],{"id":15,"name":16,"path":17},81,"Theoretical Mechanics","theoretical-mechanics",{"id":19,"name":20,"path":21},138,"Dynamics","theoretical-mechanics\u002Fdynamics",{"id":23,"name":24,"path":25},296,"Analytical Mechanics and Oscillations","theoretical-mechanics\u002Fdynamics\u002Fanalytical-mechanics-and-oscillations",{"id":5,"name":6,"path":8},[],[],{"en":30,"uk":31},"https:\u002F\u002Fmechclassroom.com\u002Fen\u002Ftopics\u002Ftheoretical-mechanics\u002Fdynamics\u002Fanalytical-mechanics-and-oscillations\u002Flagranges-equations-of-the-second-kind","https:\u002F\u002Fmechclassroom.com\u002Ftopics\u002Fteoretychna-mekhanika\u002Fdynamika\u002Fanalitychna-mekhanika-ta-kolyvannia\u002Frivniannia-lahranzha-druhoho-rodu",1787778343545]