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JD_PM
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- Homework Statement
- Consider a Hamiltonian system of ##N## degrees of freedom ##q_i##, where ##i = 1,......,N##. The momentum is ##p_i##. Suppose the Hamiltonian has the following specific form:
$$H = H[f_1(q_1, p_1), f_2(q_2, p_2), ..., f_N(q_N, p_N)]$$
Where ##f_i## is some function of ##q^i## and ##p_i## alone.
Show that ##f_i(q_i, p_i)## is a constant of motion.
- Relevant Equations
- $$f(q_1, q_2, ...., q_N, p_1, p_2,...., p_N) = \text{constant}$$
Alright my idea is that, in order to show that ##f_i(q_i, p_i)## is a constant of motion, it would suffice to show that the Hamiltonian is equal to a constant.
Well, the Hamiltonian will be equal to a constant iff:
$$f(q_1, q_2, ..., q_N, p_1, p_2,..., p_N) = \text{constant}$$
Which is what we have to prove... So I am in a closed loop.
Could you give me a hint?
I am actually checking out Goldstein, section 2.6. He says:
'In many problems a number of first integrals of the equations of motion can be obtained immediately; by this we mean relations of the type (note that his ##f## is in function of ##q_i## and its derivatives wrt time alone, while mine is in function of ##q_i## and ##p_i## alone):
$$f(q_1, q_2, ..., \dot q_1, \dot q_2, ..., t) = \text{constant}$$
...These first integrals are of interest because they tell us something physically about the system.'
OK so we should be able to show ##f(q_1, q_2, ..., p_1, p_2) = \text{constant}## by Hamilton's principle then?
Thanks.
Well, the Hamiltonian will be equal to a constant iff:
$$f(q_1, q_2, ..., q_N, p_1, p_2,..., p_N) = \text{constant}$$
Which is what we have to prove... So I am in a closed loop.
Could you give me a hint?
I am actually checking out Goldstein, section 2.6. He says:
'In many problems a number of first integrals of the equations of motion can be obtained immediately; by this we mean relations of the type (note that his ##f## is in function of ##q_i## and its derivatives wrt time alone, while mine is in function of ##q_i## and ##p_i## alone):
$$f(q_1, q_2, ..., \dot q_1, \dot q_2, ..., t) = \text{constant}$$
...These first integrals are of interest because they tell us something physically about the system.'
OK so we should be able to show ##f(q_1, q_2, ..., p_1, p_2) = \text{constant}## by Hamilton's principle then?
Thanks.
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