What other coordinate is that?Orodruin said:First of all, the Hamiltonian is a function of the generalised coordinates and the corresponding canonical momenta (or more generally, of coordinates on phase space). You should not write the Hamiltonian as a function of derivatives of the momenta. Second, it is unclear to me what "it" that is supposed to mean that the system has no cyclic coordinates is. What property of the expression are you referring to? Clearly, none of the coordinates that you have are cyclic, but that does not mean that there are necessarily other coordinates you could use where one of them would be cyclic (as in fact there is in this case).
What symmetries does the Hamiltonian have? In what coordinates would the Hamiltonian therefore not depend on one of the coordinates?digogalvao said:What other coordinate is that?
Yes, the frequency is ##\frac{1}{\sqrt{m}}##. Should I put everything in polar coordinates?Orodruin said:What symmetries does the Hamiltonian have? In what coordinates would the Hamiltonian therefore not depend on one of the coordinates?
Hint: What you have is quite clearly a 2-dimensional harmonic oscillator.
Cyclic variables in Hamiltonian systems are quantities that do not affect the dynamics of the system. They have a constant value and do not contribute to the equations of motion. These variables are also known as ignorable coordinates or constants of motion.
Cyclic variables are important in Hamiltonian systems because they simplify the equations of motion and allow for easier analysis and understanding of the system. They can also provide insight into the symmetries and conservation laws of the system.
Cyclic variables are closely related to integrals of motion, which are quantities that remain constant throughout the motion of the system. In fact, cyclic variables are a type of integral of motion, where the corresponding constant is known as a first integral.
To identify cyclic variables in a Hamiltonian system, you can use the Poisson bracket to determine if a variable has a zero bracket with the Hamiltonian function. If it does, then it is a cyclic variable. Additionally, cyclic variables often have symmetries or invariances in the system.
Yes, cyclic variables can exist in non-conservative Hamiltonian systems. In these systems, the Hamiltonian function is not a constant and therefore does not represent a conserved quantity. However, there may still be variables that do not contribute to the equations of motion and have a constant value, making them cyclic variables.