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
subject to the constraint
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(1) |
This can also be formed by modifying Lagrange's equations, rather than the lagrangian itself.
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(2) |
Example: Atwood's Machine
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(3) |
const. |
(4) |
Turn the E-L crank. |
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(5) |
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(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:
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(7) |
The equations of motion are
The Hamiltonian is the cotangent bundle over t.
Poisson Bracket
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(8) |
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(9) |
There is a strong relation between the poisson bracket of classical mechanics and the commutator of quantum mechanics.
Moment of Inertia
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(10) |
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(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:
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.
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
Gauss' Law for Newtonian Gravity
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(19) |
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(20) |
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(21) |
Gauss' Law differential form: |
(22) |
Gauss' Law integral form: |
(23) |
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Relativistic Doppler Shift
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(24) |
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(25) |
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(26) |
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(27) |
transverse: |
(28) |
parallel: |
(29) |
Schwarzschild Metric
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(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:
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(31) |
- k = -1
Hyperbolic curvature (negative). Substitute
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- k = 0
Flat space. (No need to sub to integrate.)
- k = 1
Elliptical curvature (positive). Sub
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