Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

If a particle of mass $m$ moves in the $x$-$y$ plane, then its equations of motion are

$$m\frac{d^2x}{dt^2}=f(t,x,y)\space \space \text{and} \space \space m\frac{d^2y}{dt^2}=g(t,x,y).$$

Here $f$ and $g$ represent the $x$ and $y$ components, respectively, of the force acting on the particle. Replace this system of two second-order equations by an equivalent system of four first order equations of the form:




I understand how replace a differential equation by an equivalent system of first order equations when the differential looks something like


An equivalent system is:

$$y_0'=y_1$$ $$y_1'=x^2y_0+xy_1$$

Therefore, I need someone to point me in the right direction for my question stated at the top.

share|cite|improve this question

It's the same thing, you just state $x'(t) = r(t)$, and $y'(t) = s(t)$ and you'll have the system \begin{align} x'(t) &= r(t)\\ y'(t) &= s(t)\\ r'(t) &= \frac{1}{m}f(t,x,y)\\ s'(t) &= \frac{1}{m}g(t,x,y) \end{align}

Note that if $f=f(t,x,y,x',y')$ then the substitution would lead to $r' = \frac{1}{m}f(t,x,y,r,s)$; the same thing for $g$.

On a side note, if you take $\dot{x_i}(t) = \frac{1}{m_i} \dot{p_i}(t)$, $1 \le i \le n$, you'll have the problem formulated clearly in the language of classical mechanics. In that case, for a $n$ particle system \begin{align} \dot{x_1}(t) &= \frac{1}{m_1}p_1(t)\\ \\ \dot{x_2}(t) &= \frac{1}{m_2}p_2(t)\\ &\vdots \\ \dot{x_n}(t) &= \frac{1}{m_n}p_n(t)\\ \\ \\ \dot{p_1}(t) &= f_1(t,x_1,\ldots,x_n,p_1,\ldots,p_n)\\ \\ \dot{p_2}(t) &= f_2(t,x_1,\ldots,x_n,p_1,\ldots,p_n)\\ &\vdots \\ \dot{p_n}(t) &= f_n(t,x_1,\ldots,x_n,p_1,\ldots,p_n)\\ \end{align} This are Newton's equations rewritten in terms of momenta. The space $\{x_1, \ldots, x_n, p_1,\ldots, p_n\}$ is called the Configuration Space, and it's very important for the study of the behavior of the system.

share|cite|improve this answer

Given the physical flavour of the problem, I'll explain as follws: If you know the Lagrangian of the system, say $L\equiv L(q,\dot{q})=T-V$, the equations of motion are (2nd order)

$$ \frac{d}{dt}\frac{\partial L}{\partial{\dot{q}}}=\frac{\partial L}{\partial q} $$

Now, state the problem in Hamiltonian Mechanics. Define the momenta as

$$ p=\frac{\partial L}{\partial \dot{q}} $$

and solve for $\dot{q}=\dot{q}(p)$. Write the Hamiltonian in that new variables:

$$ H\equiv H(p,q)=\displaystyle\sum_{i}p_i\dot{q}_i -L$$

Then the equations of motion are (1st order)

$$ \dot{p}=-\frac{\partial H}{\partial q}, \dot{q}=\frac{\partial H}{\partial p} $$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.