Lagrangian and Hamiltonian Dynamics. Chapter 7 Claude Pruneau Physics and Astronomy. Minimal Principles in Physics. Hero of Alexandria 2nd century BC. Law governing light reflection minimizes the path length. Fermat’s Principle
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Physics and Astronomy
Of all possible paths along which a dynamical system may move from one point to another within a specified time interval (consistent with any constraints), the actual path followed is that which minimizes the time integral of the difference between the kinetic and potential energy.
Problem: Obtain the Lagrange Equation of motion for the one-dimensional harmonic oscillator.
“m” constraint equations
Force derivable from one/many potential
Constraint Eqs connect coordinates, may be fct(t)
2 degrees of freedom only!
2 generalized coordinates.
L is independent of q.
is the angular momentum relative to the axis of the cone.
Non holonomic constraints unless eqs can be integrated to yield constrains among the coordinates.
Generalized force defined through virtual work dW
Whenever the potential energy is velocity independent:
Result extended to define the Generalized Momenta:
Given Euler-Lagrange Eqs:
One also finds:
The Hamiltonian may then be considered a function of the generalized coordinates, qj, and momenta pj:
To “convert” from the Lagrange formulation to the Hamiltonian formulation, we consider:
One can also write:
We then conclude:
H is a constant of motion
If, additionally, H=U+T=E, then E is a conserved quantity.:
One thus get the simple (trivial) solution:
The solution for a cyclic variable is thus reduced to a simple integral as above.
The simplest solution to a system would occur if one could choose the generalized coordinates in a way they are ALL cyclic. One would then have “s” equations of the form :
Such a choice is possible by applying appropriate transformations – this is known as Hamilton-Jacobi Theory.
Some remarks on the calculus of variation
The above integral becomes after integration by parts:
Which gives rise to Euler-Lagrange equations:
Which evaluates to:
Integrate by parts:
The variation may then be written: