Finding inveriance size, equilibrium and fluctuations

In summary: The equations of motion relate the time derivatives of the coordinates and momenta to the Hamiltonian and can be used to study the dynamics of the system.
  • #1
Gbox
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Homework Statement
Let there be a pendulum which is free to move in space, the pendulum is connected to a wire with a constant length, which is connected to a fixed point. will use ##\theta, \phi## to point its location.
1. Write ##\vec{r}(\theta,\phi)##
2. Find the lagrangian
3. write equations of motion
4. Which coordinate is cyclic? and which size is invariant
5. plugin the invariant size and find a. equilibrium b. small fluctuations
Relevant Equations
##E_k=\frac{m\dot{r}^2}{2}##
##E_p=mgh##
##L=E_K-e_P##
##\frac{\partial L}{\partial q_i }-\frac{d}{dt}\frac{\partial L}{\partial \dot{q_1}}##
##H(p,q)=p\dot{q}(p.q)-L##
So I answered 1 and 2, got:

1. ##\vec(r)(\theta,\phi)=l(sin \theta cos \phi, sin \theta sin \phi, -cos \theta)##
2. ##L=\frac{ml^2 (\dot{\theta}^2+\dot{\phi}^2 sin^2 \theta)}{2}+mglcos \theta##
3. a ##mlsin \theta -mgsin \theta =l^2 \ddot{\theta}## , b. ##ml^2 \ddot{\phi}=0##
4. I know that ##\phi## is a cyclic coordinate, because only its derivative is in the lagrangian. which mean that the momentum in the ##\phi## axis is constant.

Now it is seems the "invariant size" (sorry it is a direct translation) is related to the Hamiltonian (which I do not know why).
The Hamiltonian is defined as ##H(p,q)=\sum_{i=1}^{n}(p_i\dot{q_i})-L## where n is the number of coordinates.

So we first need to find ##p_\theta=\frac{\partial L}{\partial \dot{\theta}}=ml^2\dot{\theta}## and ##p_\theta=\frac{\partial L}{\partial \dot{\phi}}=ml^2\dot{\phi}##

No the Hamiltonian should not have ##\dot{\theta}## or ##\dot{\phi}## so ##\frac{p_\theta}{ml^2}=\dot{\theta}## and ##\frac{p_\phi}{ml^2}=\dot{\phi}##

So the Hamiltonian ##H=p_\theta*\frac{p_\theta}{ml^2}+p_\phi*\frac{p_\phi}{ml^2}+\frac{ml^2 (\dot{\theta}^2+\dot{\phi}^2 sin^2 \theta)}{2}+mglcos \theta## ?

I know that there are ##\dot{q_i}=\frac{\partial H}{\partial p_i}## and ##\dot{p_i}=-\frac{\partial H}{\partial q_i}## but how are they related to the "invariant size", equilibrium and "small fluctuations" (again direct translation)

Thanks
 
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  • #2
for your help!The "invariant size" is related to the Hamiltonian because it is a measure of the total energy of the system, which can be calculated by summing up the kinetic and potential energies. The equilibrium refers to the state in which the system is at rest, i.e. when the velocities of all particles in the system are 0. The small fluctuations refer to small oscillations around the equilibrium state, which can be described using the equations of motion derived from the Hamiltonian.
 

Related to Finding inveriance size, equilibrium and fluctuations

1. What is meant by "invariance size" in scientific research?

Invariance size refers to the scale at which a particular phenomenon or system remains consistent or unchanged. In other words, it is the size or range of a variable that does not affect the outcome or behavior of the system being studied.

2. How is equilibrium defined in scientific terms?

Equilibrium refers to a state of balance or stability in a system where the forces or factors acting on it are equal and opposing, resulting in a constant or unchanging state. In scientific research, equilibrium is often used to describe the balance between different variables or components of a system.

3. What is the significance of studying fluctuations in scientific research?

Fluctuations are variations or changes in a system or phenomenon over time. Studying fluctuations can provide important insights into the underlying mechanisms and dynamics of a system, as well as its potential for change or adaptation. This can be especially relevant in fields such as ecology, economics, and climate science.

4. How do scientists measure invariance size, equilibrium, and fluctuations?

The methods and techniques for measuring invariance size, equilibrium, and fluctuations vary depending on the specific research question and field of study. In some cases, mathematical models and simulations may be used, while in others, data collection and analysis techniques such as statistical analysis or experimental design may be employed.

5. Can invariance size, equilibrium, and fluctuations be applied to all scientific disciplines?

Yes, the concepts of invariance size, equilibrium, and fluctuations can be applied to a wide range of scientific disciplines, including physics, biology, chemistry, and social sciences. However, the specific definitions and methods of measurement may differ depending on the field of study and research goals.

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