The following question originates from a Physics problem, so I apologize if I am not using a suitable mathematical jargon.

The original system involves $N$ massless electric charges at position $\boldsymbol{r}_1$, $\boldsymbol{r}_2$, ..., $\boldsymbol{r}_N$ which can move on a plane pierced by a uniform and constant transverse magnetic field $\boldsymbol{B}$. The motion equation governing the dynamics of the $j$-th charge is: \begin{equation} \label{eq:Motion_eq} \boldsymbol{0}=q_j\,\dot{\boldsymbol{r}}_j\times\boldsymbol{B}\, +\, q_j\boldsymbol{E}(\boldsymbol{r}_j) \end{equation} where $q_j$ is the charge of the $j$-th electric charge and $\boldsymbol{E}(\boldsymbol{r}_j)$ is the electric field generated at position $\boldsymbol{r}_j$ by all the remaining charges. Moreover, at any time, the electric field at position $\boldsymbol{E}(\boldsymbol{r}_j)$ can be written as $$ \boldsymbol{E}(\boldsymbol{r}_j) = \sum_{i=1\\i\neq j}^N q_i\frac{\boldsymbol{r}_j-\boldsymbol{r}_i}{|\boldsymbol{r}_j-\boldsymbol{r}_i|^3}. $$ The motion equation above is manifestly of first order in time and the only initial conditions one needs are the initial positions of the massless charges, i.e. $\boldsymbol{r}_1(t=0)$, $\boldsymbol{r}_2(t=0)$, ..., $\boldsymbol{r}_N(t=0)$.

Now let us switch from massless to massive charges. This means that the aforementioned motion equation transforms as: \begin{equation} M_j \ddot{\boldsymbol{r}}_j=q_j\,\dot{\boldsymbol{r}}_j\times\boldsymbol{B}\, +\, q_j\boldsymbol{E}(\boldsymbol{r}_j) \end{equation} We assume that all the masses $M_j$ are very small. As far as I understand, the introduction of a non-zero mass constitutes a singular perturbation as it manifestly alters the order of the differential equations. Accordingly, the number of initial conditions which one should set doubles.

My question is: if one knows the solution $\{\boldsymbol{r}_1(t),\,\boldsymbol{r}_2(t),\,\dots,\,\boldsymbol{r}_N(t)\}$ of the massless problem (i.e. of the system of first-order differential equations), what can be said about the solution of the massive problem (i.e. of the system of second-order differential equations)?

For simplicity we can assume $M_j=M\,\forall j$ and we can also make additional assumptions, if needed (e.g. that the energy $H_\mathrm{massive}=T+V=\sum_{j=1}^N\frac{1}{2}M_j\dot{\boldsymbol{r}}_j^2 + V$ of the massive system is limited and does not significantly depart from that of the massless system $H_\mathrm{massless}=V=\frac{1}{2}\sum_{j=1}^N\sum_{i\neq j}^N q_iq_j/|\boldsymbol{r}_i-\boldsymbol{r}_j|$). Another possible and reasonable assumption would be that the initial velocities of the 2nd-order problem deviate only little from the fixed initial velocities of the 1st order problem.

I guess that a term $\mathcal{O}(M^{-1})$ will show up in the solutions of the perturbed system. Is this correct? Apart from this term, is it possible to write a sort of Taylor expansion involving powers of $M$? Is there a general framework to approach this singular-perturbation problem?

The following question originates from a Physics problem, so I apologize if I am not using a suitable mathematical jargon.

The original system involves $N$ massless electric charges at position $\boldsymbol{r}_1$, $\boldsymbol{r}_2$, ..., $\boldsymbol{r}_N$ which can move on a plane pierced by a uniform and constant transverse magnetic field $\boldsymbol{B}$. The motion equation governing the dynamics of the $j$-th charge is: \begin{equation} \label{eq:Motion_eq} \boldsymbol{0}=q_j\,\dot{\boldsymbol{r}}_j\times\boldsymbol{B}\, +\, q_j\boldsymbol{E}(\boldsymbol{r}_j) \end{equation} where $q_j$ is the charge of the $j$-th electric charge and $\boldsymbol{E}(\boldsymbol{r}_j)$ is the electric field generated at position $\boldsymbol{r}_j$ by all the remaining charges. Moreover, at any time, the electric field at position $\boldsymbol{E}(\boldsymbol{r}_j)$ can be written as $$ \boldsymbol{E}(\boldsymbol{r}_j) = \sum_{i=1\\i\neq j}^N q_i\frac{\boldsymbol{r}_j-\boldsymbol{r}_i}{|\boldsymbol{r}_j-\boldsymbol{r}_i|^3}. $$ The motion equation above is manifestly of first order in time and the only initial conditions one needs are the initial positions of the massless charges, i.e. $\boldsymbol{r}_1(t=0)$, $\boldsymbol{r}_2(t=0)$, ..., $\boldsymbol{r}_N(t=0)$.

Now let us switch from massless to massive charges. This means that the aforementioned motion equation transforms as: \begin{equation} M_j \ddot{\boldsymbol{r}}_j=q_j\,\dot{\boldsymbol{r}}_j\times\boldsymbol{B}\, +\, q_j\boldsymbol{E}(\boldsymbol{r}_j) \end{equation} We assume that all the masses $M_j$ are very small. As far as I understand, the introduction of a non-zero mass constitutes a singular perturbation as it manifestly alters the order of the differential equations. Accordingly, the number of initial conditions which one should set doubles.

My question is: if one knows the solution $\{\boldsymbol{r}_1(t),\,\boldsymbol{r}_2(t),\,\dots,\,\boldsymbol{r}_N(t)\}$ of the massless problem (i.e. of the system of first-order differential equations), what can be said about the solution of the massive problem (i.e. of the system of second-order differential equations)?

For simplicity we can assume $M_j=M\,\forall j$ and we can also make additional assumptions, if needed. E.g. that the energy $$H_\mathrm{massive}=T+V=\sum_{j=1}^N\frac{1}{2}M_j\dot{\boldsymbol{r}}_j^2 + V$$ of the massive system is limited and does not significantly depart from that of the massless system $$H_\mathrm{massless}=V=\frac{1}{2}\sum_{j=1}^N\sum_{i\neq j}^N \frac{q_iq_j}{|\boldsymbol{r}_i-\boldsymbol{r}_j|}\,.$$ Another possible and reasonable assumption would be that the initial velocities of the 2nd-order problem deviate only little from the fixed initial velocities of the 1st order problem.

I guess that a term $\mathcal{O}(M^{-1})$ will show up in the solutions of the perturbed system. Is this correct? Apart from this term, is it possible to write a sort of Taylor expansion involving powers of $M$? Is there a general framework to approach this singular-perturbation problem?

The following question originates from a Physics problem, so I apologize if I am not using a suitable mathematical jargon.

The original system involves $N$ massless electric charges at position $\boldsymbol{r}_1$, $\boldsymbol{r}_2$, ..., $\boldsymbol{r}_N$ which can move on a plane pierced by a uniform and constant transverse magnetic field $\boldsymbol{B}$. The motion equation governing the dynamics of the $j$-th charge is: \begin{equation} \label{eq:Motion_eq} \boldsymbol{0}=q_j\,\dot{\boldsymbol{r}}_j\times\boldsymbol{B}\, +\, q_j\boldsymbol{E}(\boldsymbol{r}_j) \end{equation} where $q_j$ is the charge of the $j$-th electric charge and $\boldsymbol{E}(\boldsymbol{r}_j)$ is the electric field generated at position $\boldsymbol{r}_j$ by all the remaining charges. Moreover, at any time, the electric field at position $\boldsymbol{E}(\boldsymbol{r}_j)$ can be written as $$ \boldsymbol{E}(\boldsymbol{r}_j) = \sum_{i=1\\i\neq j}^N q_i\frac{\boldsymbol{r}_j-\boldsymbol{r}_i}{|\boldsymbol{r}_j-\boldsymbol{r}_i|^3}. $$ The motion equation above is manifestly of first order in time and the only initial conditions one needs are the initial positions of the massless charges, i.e. $\boldsymbol{r}_1(t=0)$, $\boldsymbol{r}_2(t=0)$, ..., $\boldsymbol{r}_N(t=0)$.

Now let us switch from massless to massive charges. This means that the aforementioned motion equation transforms as: \begin{equation} \end{equation} \begin{equation} M_j \ddot{\boldsymbol{r}}_j=q_j\,\dot{\boldsymbol{r}}_j\times\boldsymbol{B}\, +\, q_j\boldsymbol{E}(\boldsymbol{r}_j) \end{equation} We assume that all the masses $M_j$ are very small. As far as I understand, the introduction of a non-zero mass constitutes a singular perturbation as it manifestly alters the order of the differential equations. Accordingly, the number of initial conditions which one should set doubles.

My question is: if one knows the solution $\{\boldsymbol{r}_1(t),\,\boldsymbol{r}_2(t),\,\dots,\,\boldsymbol{r}_N(t)\}$ of the massless problem (i.e. of the system of first-order differential equations), what can be said about the solution of the massive problem (i.e. of the system of second-order differential equations)? For simplicity we can assume $M_j=M\,\forall j$ and we can also make additional assumptions, if needed (e.g. that the energy $H_\mathrm{massive}=T+V=\sum_{j=1}^N\frac{1}{2}M_j\dot{\boldsymbol{r}}_j^2 + V$ of the massive system is limited and does not significantly depart from that of the massless system $H_\mathrm{massless}=V=\frac{1}{2}\sum_{j=1}^N\sum_{i\neq j}^N q_iq_j/|\boldsymbol{r}_i-\boldsymbol{r}_j|$). I guess that a term $\mathcal{O}(M^{-1})$ will show up in the solutions of the perturbed system. Is this correct? Apart from this term, is it possible to write a sort of Taylor expansion involving powers of $M$? Is there a general framework to approach this singular-perturbation problem?

The following question originates from a Physics problem, so I apologize if I am not using a suitable mathematical jargon.

The original system involves $N$ massless electric charges at position $\boldsymbol{r}_1$, $\boldsymbol{r}_2$, ..., $\boldsymbol{r}_N$ which can move on a plane pierced by a uniform and constant transverse magnetic field $\boldsymbol{B}$. The motion equation governing the dynamics of the $j$-th charge is: \begin{equation} \label{eq:Motion_eq} \boldsymbol{0}=q_j\,\dot{\boldsymbol{r}}_j\times\boldsymbol{B}\, +\, q_j\boldsymbol{E}(\boldsymbol{r}_j) \end{equation} where $q_j$ is the charge of the $j$-th electric charge and $\boldsymbol{E}(\boldsymbol{r}_j)$ is the electric field generated at position $\boldsymbol{r}_j$ by all the remaining charges. Moreover, at any time, the electric field at position $\boldsymbol{E}(\boldsymbol{r}_j)$ can be written as $$ \boldsymbol{E}(\boldsymbol{r}_j) = \sum_{i=1\\i\neq j}^N q_i\frac{\boldsymbol{r}_j-\boldsymbol{r}_i}{|\boldsymbol{r}_j-\boldsymbol{r}_i|^3}. $$ The motion equation above is manifestly of first order in time and the only initial conditions one needs are the initial positions of the massless charges, i.e. $\boldsymbol{r}_1(t=0)$, $\boldsymbol{r}_2(t=0)$, ..., $\boldsymbol{r}_N(t=0)$.

Now let us switch from massless to massive charges. This means that the aforementioned motion equation transforms as: \begin{equation} M_j \ddot{\boldsymbol{r}}_j=q_j\,\dot{\boldsymbol{r}}_j\times\boldsymbol{B}\, +\, q_j\boldsymbol{E}(\boldsymbol{r}_j) \end{equation} We assume that all the masses $M_j$ are very small. As far as I understand, the introduction of a non-zero mass constitutes a singular perturbation as it manifestly alters the order of the differential equations. Accordingly, the number of initial conditions which one should set doubles.

My question is: if one knows the solution $\{\boldsymbol{r}_1(t),\,\boldsymbol{r}_2(t),\,\dots,\,\boldsymbol{r}_N(t)\}$ of the massless problem (i.e. of the system of first-order differential equations), what can be said about the solution of the massive problem (i.e. of the system of second-order differential equations)? 

For simplicity we can assume $M_j=M\,\forall j$ and we can also make additional assumptions, if needed (e.g. that the energy $H_\mathrm{massive}=T+V=\sum_{j=1}^N\frac{1}{2}M_j\dot{\boldsymbol{r}}_j^2 + V$ of the massive system is limited and does not significantly depart from that of the massless system $H_\mathrm{massless}=V=\frac{1}{2}\sum_{j=1}^N\sum_{i\neq j}^N q_iq_j/|\boldsymbol{r}_i-\boldsymbol{r}_j|$). Another possible and reasonable assumption would be that the initial velocities of the 2nd-order problem deviate only little from the fixed initial velocities of the 1st order problem.

I guess that a term $\mathcal{O}(M^{-1})$ will show up in the solutions of the perturbed system. Is this correct? Apart from this term, is it possible to write a sort of Taylor expansion involving powers of $M$? Is there a general framework to approach this singular-perturbation problem?