Euler-Lagrange Equation
Integrate by parts;
A differentiable functional is stationary at a local maximum or minimum.
Lagrange Multipliers
Lagrange Multipliers allow optimization without explicit parameterization. Look for the stationary points of a function
![$f(x)$](img8.svg)
subject to the constraint
![$\displaystyle \mathcal{L}(x,\lambda) = f(x) + \lambda g(x)$](img10.svg) |
(1) |
This can also be formed by modifying Lagrange's equations, rather than the lagrangian itself.
![$\displaystyle \frac{d}{dt} \frac{\partial\mathcal{L}}{\partial\dot{x}} - \frac{\partial\mathcal{L}}{\partial x} = \lambda \frac{\partial f}{\partial x}$](img11.svg) |
(2) |
Example: Atwood's Machine
![$\displaystyle \mathcal{L} = \frac{1}{2}m_1\dot{x}_1^2 + \frac{1}{2}m_2\dot{x}_2^2 + m_1 g x_1 + m_2 g x_2$](img12.svg) |
(3) |
const. |
(4) |
Turn the E-L crank. |
|
![$\displaystyle m_1 \ddot{x_1} - m_1 g = \lambda$](img14.svg) |
|
![$\displaystyle m_2 \ddot{x_2} - m_2 g = \lambda$](img15.svg) |
|
![$\displaystyle \ddot{x}_1 = - \ddot{x}_2;$](img16.svg) |
(5) |
![$\displaystyle m_1 \ddot{x}_1 - m_1 g = -m_2 \ddot{x}_1 - m_2 g$](img17.svg) |
|
![$\displaystyle \rightarrow \boxed{\ddot{x}_1 = \frac{(m_1 - m_2) g}{m_1 + m_2}}$](img18.svg) |
(6) |
This is also of extreme use in statistical mechanics, where lagrange multipliers often appear in the form of the chemical potential
![$\mu$](img19.svg)
.
Hamiltonian Mechanics
One can obtain the Hamiltonian via a Legendre transform of the Lagrangian:
![$\displaystyle \mathcal{H}(q,p) = p\dot{q} - \mathcal{L}$](img20.svg) |
(7) |
The equations of motion are
The Hamiltonian is the cotangent bundle over t.
Poisson Bracket
![$\displaystyle \{A,B\}_{q,p} = \frac{\partial A}{\partial q}\frac{\partial B}{\partial p} - \frac{\partial A}{\partial p}\frac{\partial B}{\partial q}$](img24.svg) |
(8) |
![$\displaystyle \frac{df}{dt} = \{f,\mathcal{H} \} + \frac{\partial f}{\partial t}$](img25.svg) |
(9) |
There is a strong relation between the poisson bracket of classical mechanics and the commutator of quantum mechanics.
Moment of Inertia
![$\displaystyle I = \int r^2 dm = \int r^2 \rho(x) dV$](img26.svg) |
(10) |
![$\displaystyle I_{ij} = \int \rho(x) [\delta^i_j \left\vert\mathbf{x}\right\vert^2 - x_ix_j] d^3x$](img27.svg) |
(11) |
Parallel Axis Theorem: ![$\displaystyle I_{parallel} = I_{cm} + MR^2$](img28.svg) |
(12) |
- point mass:
- rod through center:
- rod through end:
- thin hoop:
in x,y
- thin hoop (symmetric):
- sphere:
Euler Equations
For rigid body rotations:
This is especially useful when motion is constrained to be torque free; the
![$\Gamma$](img40.svg)
dependence drops out and the equations are much simpler.
Euler Angles
TODO: PICTURE
- Rotate about
by
;
after rotation.
- Rotate about
by
- Rotate around
by
The dashed line is the line of nodes.
Note: The angle projections always depend on the previous angle's rate of change, never a later one; i.e.,
![$\dot{\alpha} \cos{\beta}$](img51.svg)
will occur, but NOT
Differential Scattering Cross Section
Gauss' Law for Newtonian Gravity
![$\displaystyle \mathcal{L}_{g} = -\rho(\mathbf{x},t)\phi(\mathbf{x},t) - \frac{1}{8\pi G}(\nabla\mathbf{\phi}(\mathbf{x},t))^2$](img57.svg) |
(19) |
![$\displaystyle \rightarrow -\rho = \frac{2}{8\pi G} \nabla^2{\phi}$](img58.svg) |
|
![$\displaystyle \boxed{\nabla^2\phi = 4\pi G \rho}$](img59.svg) |
(20) |
![$\displaystyle \boxed{\mathbf{g} = -\nabla\mathbf{\phi}}$](img60.svg) |
(21) |
Gauss' Law differential form: ![$\displaystyle \nabla\cdot\mathbf{\mathbf{g}} = -4\pi G \rho$](img61.svg) |
(22) |
Gauss' Law integral form: ![$\displaystyle \oint \nabla\cdot\mathbf{\mathbf{g}} dV$](img62.svg) |
(23) |
![$\displaystyle = \oint \mathbf{g} \cdot d\mathbf{\sigma} = -4\pi G M$](img63.svg) |
|
Relativistic Doppler Shift
![$\displaystyle p^{\mu} = \{ E, p^x, p^y, 0 \}$](img64.svg) |
(24) |
![$\displaystyle v_{obs}^{\mu} = \{ \frac{dt}{d\tau}, \frac{dx}{d\tau} \} = \{ \frac{dt}{d\tau}, \gamma v_x \}$](img65.svg) |
(25) |
![$\displaystyle E = p^{\mu}V_{\mu} = \omega \frac{dt}{d\tau} - \omega \cos{\theta}v_x\gamma$](img66.svg) |
(26) |
![$\displaystyle \omega' = \omega(\gamma - \cos{\theta}\gamma v_x)$](img67.svg) |
(27) |
transverse: ![$\displaystyle \boxed{ \frac{\omega'}{\omega} = \frac{1 - v\cos{\theta}}{\sqrt{1-v^2}} }$](img68.svg) |
(28) |
parallel: ![$\displaystyle \boxed{ \frac{\omega'}{\omega} = \sqrt{ \frac{1 - v}{1 + v}}}$](img69.svg) |
(29) |
Schwarzschild Metric
![$\displaystyle ds^2 = -\bigg( 1 - \frac{2GM}{r} \bigg) dt^2 + \bigg(1 + \frac{2GM}{r} \bigg)^{-1} dr^2 + r^2 d\Omega^2$](img70.svg) |
(30) |
Gravitational field outside a spherical mass; no charge/ angular momentum. Universal cosmological constant = 0.
FLRW Metric
Friedmann-LemaƮtre-Robertson-Walker metric for the curvature of the universe:
![$\displaystyle ds^2 = -dt^2 + \frac{dr^2}{1-kr^2} + r^2 d\Omega^2$](img71.svg) |
(31) |
- k = -1
Hyperbolic curvature (negative). Substitute
.
- k = 0
Flat space. (No need to sub to integrate.)
- k = 1
Elliptical curvature (positive). Sub
.