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$\newcommand{\pp}[2][ ]{\frac{\partial #1}{\partial #2}}\newcommand{\dd}[2][ ]{\frac{d #1}{d #2}}$
1 Why Lagrangian Mechanics
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1.1 Limitations of Newtonian Mechanics Some limitations of Newtonian mechanics are pointed out. |
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1.2 Constraints, Degrees of Freedom, Generalized Coordinates The equations of motion in Newtonian mechanics are always written in Cartesian coordinates. If there are constraints among the coordinates, these have to be taken into account separately. A special type of constraints, called holonomic constraints and the number of degrees of freedom is defined. |
2. Lagrangian Form of Dynamics
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2.1 From Newton's EOM to Euler Lagrange EOM Euler Lagrange equations are obtained using Newton's laws and D' Alembert's principle. |
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2,2. Lagrangian for Conservative Forces The Lagrangian for conservative systems is defined as \(L=T-V\) and the Euler Lagrange equations take the form \[\begin{equation*} \frac{d}{dt}\frac{\partial{L}}{\partial\dot{q_k}}-\frac{\partial{L}}{\partial{ q_k}}=0. \end{equation*}\] |
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2.3. Lagrangian For Velocity Dependent Forces For systems for which the generalized forces can be derived from a generalized potential \(U\), the Lagrangian can be defined as \(L=T-U\) and the Euler Lagrange equations take the form \[\begin{equation*} \frac{d}{dt}\frac{\partial{L}}{\partial\dot{q_k}}-\frac{\partial{L}}{\partial{ q_k}}=0. \end{equation*}\] |
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2.4. Generalized Force and Lagrange Equations -- An Example Whenever the generalized force \(Q_k\) \[\begin{equation}\label{AB01} Q_k=\sum_\alpha F^\text{(e)}_\alpha \pp[\vec{x}_\alpha]{q_k}. \end{equation}\] can be written in the form \[\begin{equation}\label{AB02} Q_k = \dd{t}\Big(\pp[U]{\dot{q}_k}\Big)-\pp[U]{q_k} . \end{equation}\] the equations of motion take the Euler Lagrange form with \(L=T-U\). It is demonstrated that the Lorentz force,the force on a charged particle in e.m. fields, has the form as in \(\eqref{AB02}\). An expression for the generalized potential \(U\) is derived. |
3. Constants of Motion and Integration of Equations of Motion
| 3.1 Conservation of Energy
If the Lagrangian does not depend on time explicitly, the Hamiltonian \(H=\sum_{k=1}\frac {\partial{L}}{\partial{\dot q_k}}\dot q_k-L\) is a constant of motion. For conservative systems of many particles, the Hamiltonian coincides with the total energy. is conserved. |
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3.2. Cyclic Coordinates A generalized coordinate,\(q_k\), is called cyclic if the Lagrangian is independent of the coordinate \(q_k\). It is shown that the corresponding canonical momentum is a constant of motion. A simple example of cyclic coordinate is given. |
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3.3. Integration of EOM by Quadratures We discuss an example of particle in two dimensions in a potential independent of \(\theta\). By working in plane polar coordinates, we show how solution of equations of motion can be reduced to quadratures. |
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3.4. Eliminating Cyclic Coordinates Eliminating cyclic coordinates using Routh's procedure is presented. |
