First Integrals in Classical Mechanics: Definition & Procedure

[KFC]

In my text of classical mechanics, it reads: for all relations

$$f(q_1, q_2, \cdots; \dot{q}_1, \dot{q}_2, \cdots, t) = const.$$

are called first integrals. This definition is very vague. I wonder if it means

1) Any quantities which are not changing with time are called "first integrals" ?

2) If a generalized coordinate is cyclic in Lagrangian equations, can I say this coordinate is one of the first integrals?

3) If the Hamiltonian is not time-dependent explicitly, can I said the energy is one of the first integrals?

What is the general procedure to find all first integrals?

BTW, if there is so called *first* integrals, so is there *second*-integrals?

 

[Dr.D]

Motion problems involve second order differential equations. First integrals are the result of integrating one time, to reduce the second order equations to first order differential equations. Thus if you look carefully at the form you gave,
f(q1,q2,q2,...q1dot,q2dot,q3dot...,t) = const
you will see that it does not involve any second derivatives; it is a "first integral" by definition. If you are able to find a "second integral" you will have solved the equation of motion.

 

[IPart]

1), 2) and 3) are all correct. Starting from Lagrange's equations of motion, a second order DE, if you have found a 'first integral' of motion, then one of the integrations has already been performed and your equation of motion effectively becomes a first order equation. I think that's the origin of the term.