Euler-Lagrange Equation

perturb slightly; | |

Integrate by parts;

if stationary. | |

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 subject to the constraint

Lagrange Multipliers allow optimization without explicit parameterization. Look for the stationary points of a function subject to the constraint

(1) |

This can also be formed by modifying Lagrange's equations, rather than the lagrangian itself.

(2) |

Example: Atwood's Machine

(3) | |

const. | (4) |

Turn the E-L crank. | |

(5) | |

(6) |

This is also of extreme use in statistical mechanics, where lagrange multipliers often appear in the form of the chemical potential .

Hamiltonian Mechanics

One can obtain the Hamiltonian via a Legendre transform of the Lagrangian:

One can obtain the Hamiltonian via a Legendre transform of the Lagrangian:

(7) |

The equations of motion are

The Hamiltonian is the cotangent bundle over t.

Poisson Bracket

(8) | |

(9) |

There is a strong relation between the poisson bracket of classical mechanics and the commutator of quantum mechanics.

Moment of Inertia

(10) | |

(11) | |

Parallel Axis Theorem: | (12) |

- point mass:
- rod through center:
- rod through end:
- thin hoop: in x,y
- thin hoop (symmetric):
- sphere:

Euler Equations

For rigid body rotations:

For rigid body rotations:

(13) | |

cyclic; | (14) |

This is especially useful when motion is constrained to be torque free; the 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.

(15) | |

(16) | |

(17) |

Note: The angle projections always depend on the previous angle's rate of change, never a later one; i.e., will occur, but NOT

Differential Scattering Cross Section

(18) | |

Gauss' Law for Newtonian Gravity

(19) | |

(20) | |

(21) | |

Gauss' Law differential form: | (22) |

Gauss' Law integral form: | (23) |

Relativistic Doppler Shift

(24) | |

(25) | |

(26) | |

(27) | |

transverse: | (28) |

parallel: | (29) |

Schwarzschild Metric

(30) |

Gravitational field outside a spherical mass; no charge/ angular momentum. Universal cosmological constant = 0.

FLRW Metric

Friedmann-LemaĆ®tre-Robertson-Walker metric for the curvature of the universe:

Friedmann-LemaĆ®tre-Robertson-Walker metric for the curvature of the universe:

(31) |

- k = -1 Hyperbolic curvature (negative). Substitute .
- k = 0 Flat space. (No need to sub to integrate.)
- k = 1 Elliptical curvature (positive). Sub .