1. Hamiltonian Mechanics1 1
Hamiltonian Mechanics
Jeffrey Eldred
Classical Mechanics and Electromagnetism
June 2018 USPAS at MSU 1 1 1 1 1 1

2. 2
Einstein Sum Notation
The Goldstein textbook makes use of Einstein Sum Notation and I will try to follow the same notation.
Three different meanings of indices: 2
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3. 3 3
Lagrangian Mechanics 3
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4. Deriving the Lagrangian from Newton’s Equation4
Deriving the Lagrangian from Newton’s Equation
Newton’s equation is given by:
We can write this instead as a differential operation on a single variable known as the Lagrangian: 4
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5. Least Action Principle5
Least Action Principle 5
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6. Least Action Principle6
Least Action Principle 6
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7. Least Action Principle Example7
Least Action Principle Example
Example: Harmonic Oscillator with a periodic deviation 7
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8. Lagrangian under coordinate transformations8
Lagrangian under coordinate transformations
Example: Polar Coordinates 8
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9. 9
Kinetic Energy
Why is the kinetic energy term of the Lagrangian take this form?
Lagrangian invariant under velocity rotations:
It means orthogonal system of coordinates. 9
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10. Orthogonal Coordinates10
Orthogonal Coordinates
Polar coordinates are also orthogonal:
Consider some nonorthogonal coordinates: 10
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11. Relativistic Lagrangian11
Relativistic Lagrangian
The form of the kinetic term comes from the Newtonian equations of motion that we started with.
But now consider relativistic momentum:
Still invariant under velocity rotation, but no longer uses simple velocity addition (we will instead need a Lorentz transform). 11
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12. Lagrangian for a Continuum System 12 12 12 12 12 12 12 12 12
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13. Lagrangian in the Continuum Limit13
Lagrangian in the Continuum Limit
The equations of a motion from a Lagrangian are:
But what if the Lagrangian has a dependence? Then the equations of motion become:
Example: Wave Equation for a String
N springs with constant k,
separating each of N-1
masses m by distance a,
given by N-1 coordinates η 13
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14. Wave Equation for a String14
Wave Equation for a String
The kinetic energy of
each mass is given by:
The potential energy of
each spring is given by:
The Lagrangian and equations of motion are: 14
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15. Wave Equation for a String (cont.)15
Wave Equation for a String (cont.)
Now we take the continuum limit:
Notice the Lagrangian density becomes the quantity of interest:
And where we had qi, now we have η(x). 15
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16. Hamiltonian Mechanics16 16
Hamiltonian Mechanics 16
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17. Lagrangian to Hamiltonian17
Lagrangian to Hamiltonian
The Lagrangian is a function of and our equations of motion yield N second-order differential equations.
For each degree of freedom, we can define a new variable the canonical momentum.
This will permit us to write 2N first-order differential equations. 17
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18. Deriving the Hamiltonian18
Deriving the Hamiltonian
Look for an expression H, which depends on but has the same differential behavior as L: 18
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19. Obtaining the Hamiltonian19
Obtaining the Hamiltonian
Now we have a way to obtain H from L:
The equations of motion from H have the same underlying dynamics, but are expressed in terms of 2N variables. 19
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20. Obtaining the Hamiltonian20
Obtaining the Hamiltonian
Another way to obtain the Hamiltonian is by directly integrating the equations of motion.
Example: Synchrotron focusing
This works for any equation of the form: 20
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21. The Hamiltonian and the Energy21
The Hamiltonian and the Energy
If L = L0 – V, does that mean H = L0 + V?
That is, when L depends on quadratically.
Example: Polar Coordinates 21
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22. The Hamiltonian and the Energy22
The Hamiltonian and the Energy
If L = L0 – V, does that mean H = L0 + V?
That is, when L depends on quadratically.
Another-Example: Relativistic Hamiltonian 22
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23. The Hamiltonian and the Energy23
The Hamiltonian and the Energy
Is the Hamiltonian a constant of motion?
The Hamiltonian is an invariant of motion if and only if it is not explicitly time-dependent. 23
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24. Example: Conservation of Angular Momentum24
Example: Conservation of Angular Momentum
Hamiltonians, as 2N first-order differential equations rather than N second-order differential equations, sometimes make conserved momentum more explicit.
Example: Central Force 24
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