Class Information

Moving coordinate systems, three-body problems, partial differential equations, wave propagation (strings, membranes, fluids), boundary value problems, normal modes, fluid equations of motion, viscosity; virtual work, Lagrange’s equations, Hamilton’s equations; angular momentum of a rigid body, inertia tensor, Euler’s equations, Euler angles, tops and gyroscopes, small vibrations.

Textbook: Classical Mechanics by John Taylor

(This description is a bit outdated. The inertia tensor and other rigid body stuff like Euler angles, tops, etc. was covered in Classical Mechanics).

Mechanics in Non-Inertial Frames

For a point on a rigid body
$$\overrightarrow{v}=\overrightarrow{\omega }\times \overrightarrow{r}$$

Proof

$$v=\omega (\rho \sin \theta )=\overrightarrow{\omega }\times \overrightarrow{r}$$

Example Problems

Example 1: Double Pendulum – Lagrangian Mechanics

For the first mass:
$$\overrightarrow{v_1}=\dot{{\varphi }_1}{l}_1\widehat{{\varphi }_1}$$
$$⟹T=(\dot{{\varphi }_1}{l}_1{)}^2\frac{m_1}{2}$$
For the second mass:
$$\overrightarrow{v_2}=\dot{{\varphi }_1}{l}_1\widehat{{\varphi }_1}+\dot{{\varphi }_2}{l}_2\widehat{{\varphi }_2}$$
$$v_2^2=a^2+b^2+2\overrightarrow{a}\cdot \overrightarrow{b}$$
$$v_2^2=l_1^2{\dot{{\varphi }_1}}^2+l_2^2{\dot{{\varphi }_2}}^2+2l_1{l}_2{\varphi }_1{\varphi }_2(\widehat{{\varphi }_1}\cdot \widehat{{\varphi }_2})$$
$$v_2^2=l_1^2{\dot{{\varphi }_1}}^2+l_2^2{\dot{{\varphi }_2}}^2+2l_1{l}_2{\varphi }_1{\varphi }_2\cos ({\varphi }_1-{\varphi }_2)$$

Example 2: 7.29 From Taylor – Lagrangian Mechanics

A particle with mass $m$ moving with fixed $\omega$

Just use cartesian coordinates, since there is no symmetry we can take advantage of
$$\overrightarrow{r}=x\hat{x}+y\hat{y}$$
$$x=l\sin \varphi +R\cos \theta =l\sin \varphi +R\cos \omega t$$
$$y=-l\cos \varphi +R\sin \theta =-l\cos \varphi +R\sin \omega t$$
$$v_x=\hat{x}$$
$$v_y=\hat{y}$$

$$v^2=v_x^2+v_y^2$$

In the end, we have $T(\varphi ,t)$ with the other variables being constants

Example 3: Mass on Cylindrical Surface – Hamiltonian Mechanics

A bead of mass $m$, is constrained to move on the surface of a cone
$$\rho =cz$$
where $c$ is a constant

See the Derivation of Velocity for Polar Coordinates
$$\overrightarrow{v}=\dot{\rho }\hat{\rho }+\rho \dot{\varphi }\hat{\varphi }+\dot{z}\hat{z}$$

Then we get
$$v^2={\dot{z}}^2+(\rho \dot{\varphi }{)}^2+{\dot{\rho }}^2$$

Note that the particle is not part of a rigid body, since it has freedom to move more freely.

There are no cross terms since $\hat{z}$, $\hat{\varphi }$,$\hat{\rho }$ are all orthogonal, so

$$\hat{\rho }\cdot \hat{\varphi }=0$$