https://www.physik.uni-leipzig.de/jvwikis/mechanics/ 2026-05-01T05:49:22+00:00 https://www.physik.uni-leipzig.de/jvwikis/mechanics/ https://www.physik.uni-leipzig.de/jvwikis/mechanics/_media/wiki/favicon.ico text/html 2021-12-02T16:07:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.physik.uni-leipzig.de/jvwikis/mechanics/book/chap4/4.1_motivation_and_outline?rev=1638457677&do=diff 4.1 Motivation and outline: EOM are ODEs From the mathematical point of view the equation of motion is an ordinary differential equation (ODE). Definition 4.1 Ordinary Differential Equation (ODE) An ordinary differential equation (ODE) of $n^{\text{th}}$ order for a function $f(t)$$n^{\text{th}}$\begin{align*} f^{(n)}(t) = \frac{\mathrm{d}^n}{\mathrm{d} t^n} f(t) \end{align*}$f^{(n-1)}(t)$$\dots$$f^{(1)}(t) = \frac{\mathrm{d}}{\mathrm{d} t} f(t)$$f^{(0)}(t) = f(t)$\begin{align*} f^{… text/html 2022-12-07T23:01:07+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.physik.uni-leipzig.de/jvwikis/mechanics/book/chap4/4.2_integrating_odes?rev=1670450467&do=diff 4.2 Integrating ODEs — Free flight We first discuss the motion of a single particle moving in a gravitational field that gives rise to the constant gravitational acceleration$\mathbf g$. Hence, the particle position $\mathbf q(t)$ obeys the EOM \begin{align} \ddot{\mathbf q} &= \mathbf g \tag{4.2.1} \end{align} The right hand side of this equation is constant. It neither depends on $\dot{\mathbf q}$$\mathbf q$$t$$q_\alpha$$\mathbf q$\begin{align*} \dot q_\alpha = g_\alpha \end{align*}$D… text/html 2024-01-31T23:30:36+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.physik.uni-leipzig.de/jvwikis/mechanics/book/chap4/4.3_separation_of_variables?rev=1706740236&do=diff 4.3 Separation of variables — Settling with Stokes drag The settling of a ball in a viscous medium can be described by the equations of motion \begin{align} m \, \ddot h (t) = -m \, g - \mu \, \dot h (t) \, . \tag{4.3.1a} \end{align} Here $h(t)$ is the vertical position of the ball (height), $g$ is the acceleration due to gravity, and the contribution $-\mu \, \dot h (t)$$\mu$$\eta$$[\eta]$$ \, \text{Pa} = \, \text{kg/m}\,\text{s}$$\eta_{\text{air}} \simeq 2 \times 10^{-5}\, \, \text{kg/… text/html 2021-12-06T16:45:30+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.physik.uni-leipzig.de/jvwikis/mechanics/book/chap4/4.4_worked_example_free_flight?rev=1638805530&do=diff 4.4 Worked example: Free flight with turbulent friction In Example 4.6 we reached the puzzling conclusion that --- for all physically relevant parameters --- Stokes friction plays no role for the motion of a steel ball in air and water. On the other hand, we know from experience that friction arises to the very least for large velocities, like for gun shots. This apparent contradiction is resolved by observing that the drag is not due to Stokes drag. Rather for most settings in our daily liv… text/html 2024-02-01T00:50:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.physik.uni-leipzig.de/jvwikis/mechanics/book/chap4/4.5_linear_odes?rev=1706745001&do=diff 4.5 Linear ODEs — Particle suspended from a spring There are two forces acting on a particle is suspended from a spring: the gravitational force $-m\, g$ and the spring force $-k\, z(t)$ where $z(t)$ measures the displacement of the spring from its rest position. Hence, the \begin{align} m \, \ddot z(t) = -m \, g - k \, z(t) \tag{4.5.1} \end{align}$z(t)$$z(t)$$\dot z(t)$$z(t)$$N^{\text{th}}$$z(t)$\begin{align*} I(t) = z^{(N)}(t) + c_{N-1}(t) \: z^{(N-1)}(t) + \cdots + c_{0}(t) \… text/html 2022-02-10T21:10:07+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.physik.uni-leipzig.de/jvwikis/mechanics/book/chap4/4.6_the_center_of_mass_cm_inertial_frame?rev=1644523807&do=diff 4.6 Employing constants of motion - the center of mass (CM) inertial frame One of the most important objectives of physics is the description of the motion of interacting particles. As a first step in this direction we discuss how to employ constants of motion to determine the motion of two point particles that interact with a conservative force depending only on the scalar distance between the particles, the interaction most commonly encountered in physical systems. The impact of spatial exte… text/html 2021-12-18T12:48:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.physik.uni-leipzig.de/jvwikis/mechanics/book/chap4/4.7_worked_example_the_kepler_problem?rev=1639828138&do=diff 4.7 Worked example: the Kepler problem The first problem tackled in theoretical mechanics was the motion of two point particles with gravitational interaction. It is formulated in terms of three laws. The second law holds for all central forces, the 3rd law is a consequence of mechanical similarity, and the 1st law is based on a solution of the \begin{align*} \mathbf Q = \frac{m_1}{m_1+m_2} \: (\mathbf Q + \mathbf r_1) + \frac{m_2}{m_1+m_2} \: (\mathbf Q + \mathbf r_2) = \mathbf Q +… text/html 2024-01-08T17:30:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.physik.uni-leipzig.de/jvwikis/mechanics/book/chap4/4.8_mechanical_similarity_kepler_s_3rd_law?rev=1704731459&do=diff 4.8 Mechanical similarity -- Kepler's 3rd Law Two solutions of a differential equations are called similar when they can be transformed into one another by a rescaling of the time-, length-, and mass-scales. We indicate the rescaled quantities by a prime, and denote the scale factors as $\tau$$\lambda$$\alpha$\begin{align*} t' = \tau t \, , \qquad\qquad \mathbf q_i ' = \lambda \mathbf q_i \, , \qquad\qquad m_i ' = \alpha m_i \end{align*}$\mathbf F$\begin{align*} \Phi( |\mathbf R| … text/html 2021-12-26T03:03:09+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.physik.uni-leipzig.de/jvwikis/mechanics/book/chap4/4.9_solving_odes_by_coordinate_transformations?rev=1640484189&do=diff 4.9 Solving ODEs by coordinate transformations: Kepler's 1st law In polar coordinates $\mathbf R = (R,\theta)$ the kinetic energy takes the form $\mu \dot{\mathbf R}^2 / 2 = \mu \, \bigl( \dot R^2 + (R \dot\theta)^2 \bigr)/2$ while the conservation of angular momentum implies $R \dot\theta = L / (\mu R)$ with $L = |\mathbf L|$. Consequently, \begin{align} E = \frac{\mu}{2} \: \dot R^2(t) + \frac{ L^2 }{2\mu \, R^2(t)} - \frac{ m_1 m_2 G }{R(t)} \tag{4.9.1} \end{align} Figure 4.15: Effec… text/html 2022-01-04T03:25:33+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.physik.uni-leipzig.de/jvwikis/mechanics/book/chap4/4.10_problems?rev=1641263133&do=diff 4.10 Problems 4.10.1 Rehearsing Concepts ---------- Problem 4.23: Maximum distance of flight There is a well-known rule that one should through a ball at an angle of roughly $\theta = \pi/4$ to achieve a maximum width. * Solve the equation of motion of the ball thrown in $x$$z$$(x,z)$$\pi/4$$H$$L$$H/L$$\theta$$\Phi(x) = 1-1/\cosh x$$x \in \mathbb{R}$$\ddot x = -\partial_x\Phi(x)$$E = \frac{1}{2} \, \dot x^2 + \Phi(x)$$(x, \dot x)$\begin{align*} \ddot x = a \, x \qquad \text{… text/html 2022-01-02T15:02:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.physik.uni-leipzig.de/jvwikis/mechanics/book/chap4/4.11_further_reading?rev=1641132179&do=diff 4.11 Further reading An introductory mathematical treatise of the theory of ODEs that is well-accessible for physicists is given by * J. David Logan: A First Course in Differential Equations (Springer, 2015) PDF from author Beware that there were complaints about typos in the Amazon Reviews. text/html 2021-12-18T12:48:36+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.physik.uni-leipzig.de/jvwikis/mechanics/book/chap4/eom-ode?rev=1639828116&do=diff 4. Motion of Point Particles In Chapter 3 we learned how to set up a physical model based on finding the forces acting on a body, and thus determining the acceleration of its motion. For a particle of mass~$m$ and position~$\vec q$ Newton's second law relates its acceleration $\ddot{\vec q}$$\vec F( \vec q, \dot{\vec q}, t )$$\vec q$$\dot{\vec q}$$t$