Thomas Antony

1.6.2 Systems with Rigid Constraints So far we have found $L = T-V$ to be a suitable Lagrangian for systems of point particles subject to forces derived from a potential.

Exercise 2.1: Rotational kinetic energy Show that the rotational kinetic energy can also be written as: $$ T = \frac{1}{2} I\omega^2\\ $$ where $I$ is the moment of inertia about the line through the center of mass with direction $\hat{\omega}$ and $\omeg…

2.3 Moments of Inertia The rotational kinetic energy consists of the sum of the kinetic energies of all the constituent particles.

1.8.4 The Restricted Three-Body Problem Consider two bodies of masses $M_0$ and $M_1$ in circular orbits about their barycenter.

1.6 How to Find Lagrangians It is possible to work back from Newton’s second law and obtain the Lagrangian $L = T - V$.

1.6.1 Coordinate Transformations Assume we have a mechanical system whose motion is described by a Lagrangian that depends on time, position and velocities.

2.2 Kinematics of Rotation The motion of a rigid body about a center of rotation (a body-fixed reference point) is characterized at each point in time by an axis of rotation and a rate of rotation According to Euler’s theorem of rotations, we can go from…

Exercise 1.12: Lagrange’s Equations Compute Lagrange’s equations for the Lagrangians in exercise 1.9 using the Lagrange-equations procedure. a.

Exercise 1.14: Coordinate-independence of Lagrange equations Check that the Lagrange equations for central force motion in polar coordinates and in rectangular coordinates are equivalent.

Exercise 1.13: Higher-derivative Lagrangians a.

Exercise 1.11: Kepler’s third law A Lagrangian suitable for studying the relative motion of two particles, of masses $m_1$ and $m_2$, with potential energy $V$, is: $$ L = \frac{1}{2} m \left(\dot{r}^2 + (r\dot{\phi})^2\right) + V(r) $$ The argument $m$ is

Exercise 1.15: Equivalence Here we use the p->r conversion function to convert the Lagrangian in rectangular coordinates to polar coordinates (and compare to results from notes and exercise 1.14 where it was computed explicitly). ;; The Coordinate convers…

Exercise 1.16: Central force motion Find Lagrangians for central force motion in three dimensions in rectangular coordinates and in spherical coordinates.

Exercise 1.17: Bead on a helical wire A bead of mass $m$ is constrained to move on a frictionless helical wire.

Exercise 1.19: Two-bar linkage The two-bar linkage shown in the figure below is constrained to move in the plane.

Exercise 1.20: Sliding pendulum Consider a pendulum of length $l$ attached to a support that is free to move horizontally, as shown in the figure.

Exercise 1.18: Bead on a triaxial surface A bead of mass $m$ moves without friction on a triaxial ellipsoidal surface.

Exercise 1.40: Bead on a triaxial surface Consider again the motion of a bead constrained to move on a triaxial surface (exercise 1.18).

2 Rigid Bodies The polhode rolls without slipping on the herpolhode lying in the invariable plane.

Exercise 1.39: Combining Lagrangians a.

Summary This chapter was primarily introducing the Lagrangian method of analyzing mechanical systems.

1.10.3 Non-holonomic Systems Systems with constraints that are no integrable are called non-holonomic systems.

1.10.2 Derivative Constraints In this section we look at a specific type of velocity-dependent constraints, which are “total-time derivatives” of a velocity independent function.

1.10.1 Coordinate Constraints In the previous section, we showed that for motion of a system with the constraint $\varphi(t, q(t), Dq(t)) = 0$, for stationary action, the path variations, $\eta(t)$, must be tangent to the constraint surface.

1.10 Constrained Motion When using the Lagrangian method, we may choose the coordinate system that is the most convenient - the number of coordinates may match exactly with the number of degrees of freedom of the system, or we may pick a system that has r…

Exercise 1.38: Properties of $\mathscr{E}$, the Euler-Lagrange Operator Refer to Section 1.9 for more details Let $F$ and $G$ be two Lagrangian-like functions of a local tuple, $C$ be a local-tuple transformation function, and $c$ a constant.

Exercise 1.38: Velocity transformation Use the procedure Gamma-bar to construct a procedure that transforms velocities given a coordinate transformation.

1.9 Abstraction of Path Functions Note: This section may seem rather dry from its title.

Exercise 1.36: Noether integral Consider motion on an ellipsoidal surface.

1.8.5 Noether’s Theorem If a dynamical system has a symmetry, a coordinate system can be chosen so that the Lagrangian does not depend on a coordinate associated with the symmetry.

Exercise 1.21: A dumbbell In this exercise we will recapitulate the derivation of the Lagrangian for constrained systems for a particular simple system.

A coordinate function $\chi$ maps a point in a coordinate patch of a manifold to a coordinate tuple of real numbers: $$ x = \chi(m) $$ where $x$ represents a convenient tuple structure.

Note: This is just my current understanding while studying Functional Differential Geometry.

A manifold is a generalization of the idea of a smooth surface embedded in Euclidean space.

This is an easier/less computationally expensive method compared to the one described in Points in a plane using null space.

For two vectors $\vec{a}$ and $\vec{b}$, the cross product $\vec{a} \times \vec{b}$ can be computed using matrix multiplcation1 as: $$ \vec{a} \times \vec{b} = A_\times \vec{b} $$ where $A_\times$ is defined as: $$ A_\times = \begin{bmatrix} 0 & -a_3 & a_…

A “bdot” (or $\mathbf{\dot{B}}$) attitude controller is used to detumble a spacecraft using just a magnetometer and torque rods.

The “Hack” computer built during this course has its own instruction set consisting of two instructions: the A-command and the C-command.

All of the chips described in the previous course notes used what is called “combinational logic”, where the output is available instantenously and there is no time-component to their operation.

The Airthmetic and Logic Unit is a core part of a computer based on the Von Neumann architecture.

Two’s complement is a format for representing negative numbers in binary.

Elementary Logic Gates NOT AND OR XOR => output 1 if inputs different MUX: => a, b, sel; if sel == 0: out = a, else: out = b DMUX: => in, sel (outputs a and b): if sel == 0: a = in else b = in 16-bit variants NOT16 : 16 bits in and out AND16 : Two 16-bit…

Nand2tetris is a course where we build a full general purpose computer up from basic logic gates - in fact, just NAND gates.

A plane can be defined using its normal vector n.

Allocating torques to RWAs are accomplished by first defining a “distribution matrix” which contains the spin axes of the individual wheels, and then using this matrix to allocate torques.Each column of the distribution matrix consists of the unit vector…

TLA+ is a language for creating “specifications” for computer programs/algorithms and then verify all the possible execution paths and mathematically prove assertions.

Reactions wheels are used for controlling spacecraft attitude to a very high precision.

The TRIAD or Tri-Axial Attitude Determination algorithm is used for attitude determination given two body-frame vectors ($b_1$ and $b_2$) and two corresponding inertial or “reference” frame vectors ($r_1$ and $r_2$).

Used a walkthrough for tips here.

Run the VMs with “Host-Only Networking”.