The conservation laws in physics form the cornerstone of our understanding of the universe. Among these, the conservation of energy and momentum are two of the most profound and universally applicable principles. These laws arise from the symmetries of nature—energy conservation from time invariance and momentum conservation from space homogeneity—as articulated by Noether’s theorem. These principles are valid in both classical and modern physics, from the predictable motion of planets to the probabilistic realm of subatomic particles.
Table of Contents
Conservation of Energy
Mathematical Formulation
For a closed system the total energy \(E\) can be written as:
\[ E = K + U + E_{\mathrm{other}} = \text{constant}, \]
where \(K\) denotes kinetic energy and \(U\) potential energy. In more advanced contexts, other terms may include internal energy, rest mass energy \(m c^2\), and radiation energy.
Applications
- Classical Mechanics: In elastic collisions total kinetic energy is conserved. In inelastic collisions some kinetic energy converts to heat or deformation, but the total energy of the closed system remains constant.
- Thermodynamics: The First Law can be written as \(\Delta U = Q – W\), where \(Q\) is heat added and \(W\) is work done by the system.
- Quantum Mechanics: Energy conservation controls processes like photon emission/absorption, pair production and nuclear decay.
- Astrophysics: Stellar energy release (e.g., in the Sun) converts mass to energy via \(E = m c^2\).
Conservation of Momentum
The conservation of momentum states that the total momentum of an isolated system remains constant when no external forces act. Momentum is a vector quantity — it has magnitude and direction — defined for a particle as the product of its mass and velocity.
Mathematical Formulation
For a discrete system of particles:
\[ \mathbf{P} = \sum_i m_i \mathbf{v}_i = \text{constant} \]
For a continuous medium the time rate of change of momentum equals external force:
\[ \frac{d\mathbf{P}}{dt} = \mathbf{F}_{\mathrm{external}}. \]
Applications
- Collisions: Total momentum is conserved for elastic and inelastic collisions — useful in crash analysis and ballistic problems.
- Rocket Propulsion: Thrust arises from momentum conservation (Newton’s Third Law): expelled propellant carries momentum opposite to the rocket’s motion.
- Particle Physics: Momentum conservation helps reconstruct events and detect missing particles.
- Astronomy: Orbital dynamics and interactions of celestial bodies follow momentum conservation in the absence of external torques and forces.
Connection Between Energy and Momentum
In relativistic mechanics, energy and momentum combine into a four-vector. The invariant relation is:
\[ E^2 = (pc)^2 + (m_0 c^2)^2, \]
where \(p\) is momentum, \(c\) the speed of light and \(m_0\) the rest mass. For massless particles (e.g. photons) this reduces to \(E = pc\).
This relation is central in high-energy physics and cosmology because it links the energy available in collisions to momenta of products and can be used to infer particle masses and kinematics.