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(1) We often say (for a conservative system like this one) that the
mechanical energy is the first integral of the differential equation.
In light of your comments, would you say this statement is misleading
or at least incomplete?
"The strength sounding name first integral is a relic of the times
when the mathematicians tried to solve all differential equations
by integration. In those days the name integral (or a partial
integral) was given to what we now called a solution." (See
"Ordinary Differential Equations" by Vladimir I. Arnol'd)
If you integrate the given differential equation for the first time
you got the first integral . If you again integrate the first
integral you got an equation called the second integral and so on.
Ultimately by this you will get your desire solution of the
there are plenty of problems that are more easily solved by directly
proceeding to mechanical energy than by trying to solve Newton’s
second law. Students will ask me sometimes how to tell which kinds of
problems those are. I admit that even I am not always sure.
A good example is objects rolling without slipping down an inclined plane.
You hinted at, but didn’t elaborate on, Lagrangian methods as a third alternative