https://www.physik.uni-leipzig.de/jvwikis/mechanics/ 2026-05-28T17:44:23+00:00 https://www.physik.uni-leipzig.de/jvwikis/mechanics/ https://www.physik.uni-leipzig.de/jvwikis/mechanics/_media/wiki/favicon.ico text/html 2022-04-15T15:41:11+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.physik.uni-leipzig.de/jvwikis/mechanics/book/chap6/6.1_motivation_and_outline?rev=1650030071&do=diff 6.1 Motivation and outline: How to deal with constraint motion? Figure 6.2: Forces acting for the motion of a swing, or its equivalent idealization of of a mathematical pendulum. Almost all interesting problems in mechanics involve constraints due to rails or tracks, and due to mechanical joints of particles. The most elementary example is a swing ($M$$L$$M \, \mathbf g$$\mathbf F_r$$N$$D$$D$$M < D \, N$$\mathbf x \in \mathbb{R}^{D\,N}$$\mathbf x( \mathbf q(t) )$$\mathbf q \in \mathbb{R}^{M… text/html 2022-02-14T14:06:41+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.physik.uni-leipzig.de/jvwikis/mechanics/book/chap6/6.2_lagrange_formalism?rev=1644844001&do=diff 6.2 Lagrange formalism The Lagrange formalism provides an effective approach to derive the EOM for generalized coordinates. We first provide a derivation in a Cartesian coordinate frame. Then we discuss how the EOM for generalized coordinates are determined.$\mathbf x$$\delta\mathbf x \, \cdot \nabla \Phi(\mathbf x)$$\Phi(\mathbf x)$$\delta\mathbf x$$f : \mathbb D \times [t_I, t_E] \to \mathbb R$$\mathbb D \subset R^D$$\mathbf x \in \mathbb D$$\delta\mathbf x$$\mathbf x$$\mathbf x + \delta\mat… text/html 2022-04-12T13:53:06+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.physik.uni-leipzig.de/jvwikis/mechanics/book/chap6/6.3_dynamics_with_one_degree_of_freedom?rev=1649764386&do=diff 6.3 Dynamics with one degree of freedom We will now illustrate the application of the Lagrange formalism for three examples with a single degree of freedom of the motion: * The mathematical pendulum, Example 6.1, will give a first idea of how to find EOMs with the Lagrange formalism. This \begin{align*} T = \frac{M}{2} \, \dot{\mathbf x}^2 = \frac{M}{2} \, L^2 \, \dot\theta^2 \: \hat{\boldsymbol\theta}(\theta(t))^2 = \frac{M}{2} \, L^2 \, \dot\theta^2 \end{align*}\begin{align*} … text/html 2022-04-12T14:50:05+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.physik.uni-leipzig.de/jvwikis/mechanics/book/chap6/6.4_several_dof_and_conservation_laws?rev=1649767805&do=diff 6.4 Several degrees of freedom and conservation laws In Section 6.3 we discussed the EOM of systems with one degree of freedom. In the present section this analysis is extended to systems with two and more degrees of freedom. Again the discussion will be based on examples: \begin{align*} T &= \frac{m}{2} \: \dot{\mathbf x}^2 = \frac{m}{2} \; \sum_i \dot x_i^2\\ V &= 0 \end{align*}$\mathcal L = T-V$$\dot{\mathbf q}$$\mathbf q$$x_i$\begin{align*} m \: \ddot x_i = \frac{\mathrm{d}}{\ma… text/html 2022-06-22T10:27:21+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.physik.uni-leipzig.de/jvwikis/mechanics/book/chap6/6.5_dynamics_of_2-particle_systems?rev=1655886441&do=diff 6.5 Dynamics of 2-particle systems In Section 6.3 and Section 6.4 we discussed the motion of single particles whose motion is constraint by tracks, arms and joints. Now we revisit the treatment of the Kepler problem, Section 4.6, in order to explore settings with two interacting particles. The central idea in this endeavor is a representation of the particle position as a sum of the position of the center of mass $\mathbf Q$$\mathbf R$$m_1$$m_2$$\mathbf x_1$$\mathbf x_2$\begin{align*} T &= … text/html 2022-06-22T10:36:19+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.physik.uni-leipzig.de/jvwikis/mechanics/book/chap6/6.6_worked_problems?rev=1655886979&do=diff 6.6 Worked problems: Foucault pendulum A Foucault pendulum is a mathematical pendulum set up on the Earth surface where we consider only small amplitude oscillations $\theta \ll 1$ with vanishing angular momentum, $C=0$. According to Equation 6.4.11 one expects that it follows the \begin{align} \ddot \theta(t) &= -\frac{g}{\ell} \; \sin\theta(t) \simeq -\frac{g}{\ell} \: \theta(t) \, . \tag{6.6.1} \end{align}$\omega = \sqrt{g/\ell}$$\theta(t) = \theta_{\text{max}} \: \sin\bigl( \alpha + \o… text/html 2022-06-22T16:34:23+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.physik.uni-leipzig.de/jvwikis/mechanics/book/chap6/6.7_problems?rev=1655908463&do=diff 6.7 Problems 6.7.1 Rehearsing Concepts Problem 6.16: Two masses hanging at a rubber band Two weights of the same mass $m$ are attached on opposite ends of a rubber band that is hanging over a roll. The weights are at height $h_1$ and $h_2$. They move only vertically, either one up and one down at a fixed length of the band, or stretching the band, or releasing tension on the band. We assume that friction and the mass of the band are negligible. $H = h_1+h_2$$D = h_1 - h_2$\[ \mathcal L ( … text/html 2022-02-14T14:05:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.physik.uni-leipzig.de/jvwikis/mechanics/book/chap6/lagrange?rev=1644843956&do=diff 6. Integrable Dynamics Marguerite Martyn, 1914 Figure 6.1: The point-particle idealization of a girl on a swing is the mathematical pendulum of Figures 1.2 and 1.3. In Chapter 5 we considered objects that consist of a mass points with fixed relative positions, like a flying and spinning ping-pong ball. Rather than providing a description of each individual mass element, we established equations of motion for their center of mass and the orientation of the body in space. From the perspective…