MTH2007 Lagrangian Mechanics Cheat Sheet DRAFT TWO

MTH2007 Lagrangian Mechanics - Cheat Sheet

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Made by William Fayers :)

Make sure to read this before the exam - I recommend completing a practice test with it so you learn where everything is and can ask if you don’t understand something. I might’ve made mistakes! There’s a sudoku at the end in case you finish early, and the cheat sheet is generated based on analysis of past exams and given material.

Small warning: the following makes the most sense if you read it all first!

Section 0: Crash Course in Prerequisites

Things you should already know, but might forget!

1. Calculus & Linear Algebra…
   • Integration by Parts:$\int udv=uv-\int vdu$.
   • Product Rule: $\frac{d}{dx}[u(x)\cdot v(x)]=u^{\prime }(x)\cdot v(x)+u(x)\cdot v^{\prime }(x)$.
   • Quotient Rule: $\frac{d}{dx}\left[\frac{u(x)}{v(x)}\right]=\frac{u^{\prime }(x)\cdot v(x)-u(x)\cdot v^{\prime }(x)}{[v(x){]}^2}$.
   • Partial Derivatives:$\frac{\partial }{\partial x}f(x,y)=\text{Derivative of }f\text{ w.r.t. }x,\text{ treat }y\text{ as constant}$.
     • Mixed Derivatives: Partially differentiate with respect to the first variable, then the second.
   • Matrix Diagonalization:
     1. Find eigenvalues by solving $\det (M-\lambda I)=0$.
     2. Eigenvectors solve $(M-\lambda I)\mathbf{v}=0$.
   • Polar Coordinates:
     • Convert from Cartesian to Polar: $x=r\cos \theta ,y=r\sin \theta$.
     • Area: $A=\frac{1}{2}{\int }_{{\theta }_1}^{{\theta }^2}{r}^{2}d\theta$.
     • Velocity…
       • tangential: $v=r\dot{\theta }$.
       • radial: $v=\dot{r}$.
       • Note that these are often summed for the total velocity, depending on the system.
2. Physics Basics…
   • Momentum: $p=mv$.
   • Newton’s Law: $F=ma$.
   • Kinetic Energy: $T=\frac{1}{2}m{v}^2=\frac{1}{2}m({\dot{x}}^2+{\dot{y}}^2+{\dot{z}}^2)$ for Cartesian, $T=\frac{1}{2}\mathbf{I}{\omega }^2$ for rotational.
   • Potential Energy: $U=mgh$ (gravity) or $U=\frac{1}{2}k{x}^2$ (spring).
   • Angular Velocity: $\omega =\frac{d\theta }{dt}$, measured in rad/s, note that the same variable is used for Eigenfrequencies.
   • Density: $\sigma =\frac{M}{A}$.
3. Key Symbols…
   • Reduced Mass ($\mu$): $\mu =\frac{m_1{m}_2}{m_1+m_2}$. Used in oscillatory frequency $\omega =\sqrt{\frac{k}{u}}$.
   • Total Mass ($M$): Sum of all masses in the system.
   • Mass Matrix ($M$): The masses along a diagonal matrix, e.g. $\left[\begin{array}{cc}{m}_1& 0\\ 0& m_2\end{array}\right]$.
   • Perpendicular Distance ($r$): The distance measured at $90^{\circ }$ from a line/point.
   • Complex Number ($i$): Defined as $i^2=-1$, double check context though!

Section 1: Degrees of Freedom & Coordinates

Often used when setting up a problem!

1. Degrees of Freedom (d.f.): $\text{d.f.}=(\text{Dimensions}\times \text{Number of Particles})-\text{Holonomic Constraints}$.
   • Example: A pendulum on a 2D plane has 1 d.f. (angle $\theta$).
   • Holonomic Constraints: Equations that depend only on coordinates (not velocities) and time.
     • Example: Fixed distances (rigid rods), surfaces (particles sliding on a curve).
2. Generalised Coordinates: Minimal independent variables (e.g., angles, lengths).
   • Check Validity:
     1. No constraints relate them.
     2. They span all possible motions.

Section 2: Lagrangians & Euler-Lagrange

Used to describe motion without forces.

1. Lagrangian ($L$): $L=T-U$, using polar coordinates for central forces.
   • Example: 1D oscillator, $L=\frac{1}{2}m{\dot{x}}^2-\frac{1}{2}k{x}^2$.
2. Euler-Lagrange Equations: $\frac{d}{dt}\left(\frac{\partial L}{\partial {\dot{q}}_i}\right)-\frac{\partial L}{\partial q_i}=0$.
   • Example: For $L=\frac{1}{2}m{\dot{x}}^2-U(x)$…
     1. $\frac{\partial L}{\partial \dot{x}}=m\dot{x}$,
     2. $\frac{d}{dt}(m\dot{x})=m\ddot{x}$,
     3. $-\frac{\partial L}{\partial x}=-\frac{dU}{dx}$,
     4. $m\ddot{x}=-\frac{dU}{dx}$.

Section 3: Hamiltonians & Phase Space

Converts Lagrangians to use generalised momenta instead of coordinates.

1. Hamiltonian ($H$): $H={\sum }_i{p}_i{\dot{q}}_i-L,p_i=\frac{\partial L}{\partial {\dot{q}}_i}$.
   • Example derivation from $L$ (e.g., 1D oscillator):
     1. $p=m\dot{x}$,
     2. $H=p\dot{x}-L=\frac{p^2}{2m}+\frac{1}{2}k{x}^2$.
2. Hamilton’s Equations: ${\dot{q}}_i=\frac{\partial H}{\partial p_i},{\dot{p}}_i=-\frac{\partial H}{\partial q_i}$.
   • Derivation: From $dH={\sum }_i({\dot{q}}_{i}d{p}_i-{\dot{p}}_{i}d{q}_i)$, match coefficients.

Section 4: Center of Mass & Inertia

Simplifies calculations of multiple particles.

1. Center of Mass (Discrete/Continuous): ${\vec{R}}_{\text{cm}}=\frac{1}{M}{\sum }_i{m}_i{\vec{r}}_i\text{or}\frac{1}{M}\int \vec{r}dm$, where $m$ is the center of mass of a particle and $dm=\sigma dr$.
   • Example (2 particles): $X_{\text{cm}}=\frac{m_1{x}_1+m_2{x}_2}{m_1+m_2}$.
2. Moment of Inertia (Discrete/Continuous): $I={\sum }_i{m}_i{r}_i^2\text{or}\int r^{2}dm$.
   • Parallel Axis Theorem: $I=I_{\text{cm}}+M{d}^2$, useful for simplifying calculations.
   • Example: Rod about center, $I=\frac{1}{12}M{L}^2$.
   • Example: Two masses on rod, $I=m_1{r}_1^2+m_2{r}_2^2$.

Section 5: Small Oscillations

Analysing systems near stable equilibrium.

1. Stiffness Matrix ($K$): $K_{ij}={\frac{{\partial }^{2}U}{\partial q_i\partial q_j}|}_{\text{eq}}$ (evaluated at equilibrium, i.e. ${\frac{\partial U}{\partial q_i}|}_{eq}=0$).
   • Meaning:
   • Example: For $U=\frac{1}{2}k(x_1^2+x_2^2)$, $K=\left[\begin{array}{cc}k& 0\\ 0& k\end{array}\right]$.
2. Harmonic Solutions: For small oscillations, you can assume $q_n(t)=A_n{e}^{i\omega t}$ - useful to substitute into EL equations and then divide through by $e^{i\omega t}$. This system, written as a matrix equation (for $A_n$), is equivalent $(K-{\omega }^{2}M)\mathbf{A}=0$.
3. Eigenfrequencies: Solve $\det (K-{\omega }^{2}M)=0$.
   • Meaning: The eigenvalues ${\omega }^2$ correspond to the squared frequencies of the system’s normal modes (directions where the system oscillates independently).
   • Example: For two masses coupled by a spring, ${\omega }_1=0$ (translational mode) and $\omega =\sqrt{\frac{k}{\mu }}$ (oscillatory mode).
   • Periodic Motion: If $\frac{{\omega }_1}{{\omega }_2}$ is rational, period $T=\frac{2\pi }{\text{gcd}({\omega }_1,{\omega }_{2)}}$.

Section 6: Central Potentials & Orbits

Motion under radial forces (e.g., gravity, electromagnetism).

1. Effective Potential: $U_{\text{eff}}(r)=U(r)+\frac{L^2}{2\mu r^2}$ (sum of potential energy and the centrifugal term).
   • Equilibrium: Solve $\frac{d{U}_{\text{eff}}}{dr}=0$.
2. Poisson Brackets: $\{A,B\}={\sum }_i\left(\frac{\partial A}{\partial q_i}\frac{\partial B}{\partial p_i}-\frac{\partial A}{\partial p_i}\frac{\partial B}{\partial q_i}\right)$.
   • Use: Determines how observables evolve over time.
   • If $\{A,B\}=0$… $A,B$ are independent. If $B=H$, $A$ is conserved.
   • If $\{A,B\}<0$… $A,B$ are negatively correlated over time.
   • If $\{A,B\}>0$… $A,B$ are positively correlated over time.

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