Siméon Denis Poisson
Early Life and Education
Siméon Denis Poisson was born on June 21, 1781, in Pithiviers, France. His father, a former soldier, encouraged him to pursue a career in medicine, but Poisson's interests lay in mathematics. At the age of 17, he enrolled at the École Polytechnique in Paris, where he studied under prominent mathematicians such as Lagrange and Laplace. His exceptional talent was recognized early on, and he quickly became a protégé of these influential figures.
Academic Career
Early Contributions
Poisson's academic career began with his appointment as a répétiteur at the École Polytechnique in 1802. His early work focused on the calculus of variations, where he made significant contributions that laid the groundwork for future developments in the field. In 1808, he was appointed to the chair of pure mathematics at the Faculty of Sciences of Paris, a position he held for the rest of his life.
Poisson's Equation
One of Poisson's most notable contributions to mathematics is the formulation of Poisson's equation, a partial differential equation of broad applicability in electrostatics, mechanical engineering, and theoretical physics. The equation is expressed as:
\[ \nabla^2 \phi = f \]
where \(\nabla^2\) is the Laplace operator, \(\phi\) is the potential function, and \(f\) is a source term. This equation extends the concept of Laplace's equation by incorporating a non-zero source term, making it applicable to a wider range of physical phenomena.
Contributions to Physics
Poisson's Ratio
In the field of elasticity, Poisson introduced the concept of Poisson's ratio, a measure of the deformation of a material in directions perpendicular to the direction of applied force. This dimensionless quantity is crucial in understanding the mechanical properties of materials and is defined as:
\[ \nu = -\frac{\varepsilon_{\text{transverse}}}{\varepsilon_{\text{axial}}} \]
where \(\varepsilon_{\text{transverse}}\) and \(\varepsilon_{\text{axial}}\) are the transverse and axial strains, respectively.
Poisson Distribution
Poisson also made significant contributions to probability theory, particularly with the Poisson distribution. This discrete probability distribution expresses the probability of a given number of events occurring in a fixed interval of time or space, given that these events happen with a known constant mean rate and independently of the time since the last event. The probability mass function is given by:
\[ P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!} \]
where \(\lambda\) is the average number of events in the interval, \(k\) is the number of occurrences, and \(e\) is Euler's number.
Influence on Mathematics and Science
The Poisson Bracket
In classical mechanics, Poisson introduced the concept of the Poisson bracket, a fundamental construct in Hamiltonian mechanics. The Poisson bracket is used to describe the time evolution of a dynamical system and is defined for two functions \(f\) and \(g\) in phase space as:
\[ \{f, g\} = \sum_{i} \left( \frac{\partial f}{\partial q_i} \frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i} \frac{\partial g}{\partial q_i} \right) \]
where \(q_i\) and \(p_i\) are the generalized coordinates and momenta, respectively.
Poisson's Integral Formula
Poisson's work in potential theory led to the development of Poisson's integral formula, which provides a solution to the Dirichlet problem for the unit disk in the complex plane. This formula is instrumental in solving boundary value problems and is expressed as:
\[ u(r, \theta) = \frac{1}{2\pi} \int_{0}^{2\pi} \frac{1 - r^2}{1 - 2r\cos(\theta - \phi) + r^2} u(1, \phi) \, d\phi \]
where \(u(r, \theta)\) is the potential function in polar coordinates.
Legacy and Honors
Poisson's work had a profound impact on both mathematics and physics, influencing a wide range of fields from fluid dynamics to quantum mechanics. His legacy is preserved in the numerous mathematical concepts and physical laws that bear his name. Poisson was elected to the Académie des Sciences in 1812 and received numerous honors throughout his career, including the Legion of Honor.