Principle of Least Action

see also:

• Principle of Stationary Action (should probably merge)

Action-based physics is a framework within theoretical physics that uses the principle of least action, or more broadly, the action principle, as its foundation for describing and predicting the dynamics of physical systems. This principle is a cornerstone in modern physics and underlies much of classical mechanics, quantum mechanics, and field theory, including the general theory of relativity. Let’s delve into the concept of action and how it shapes our understanding of the physical universe.

What is Action?

In physics, action is a quantity that is used to describe the dynamics of a system. It is defined through an integral that combines the configuration of a system and its kinetic and potential energies over time. The action, $S$, for a system is given by the integral of the Lagrangian $L$ (which is the difference between the kinetic and potential energies, $T-V$) over time:

$S={\int }_{t_1}^{t_2}Ldt$

Principle of Least Action

The principle of least action, or the action principle, states that the path taken by a system between two points in its configuration space is the one for which the action is minimized (or more generally, is stationary, meaning it does not vary to first order for small variations of the path). This principle can be mathematically expressed using the calculus of variations, leading to the Euler-Lagrange equation, which provides the equations of motion for the system:

$\frac{d}{dt}\left(\frac{\partial L}{\partial {\dot{q}}_i}\right)-\frac{\partial L}{\partial q_i}=0$

where $q_i$ represents the generalized coordinates of the system, and ${\dot{q}}_i$ their time derivatives.

Applications and Significance

• Classical Mechanics: In classical mechanics, the action principle provides a powerful method for deriving the equations of motion of a system. It is particularly useful in systems with constraints and in generalized coordinates, offering a more straightforward approach than Newton’s laws in many cases.

• Quantum Mechanics: In quantum mechanics, the action principle underlies the path integral formulation introduced by Richard Feynman. Here, the probability amplitude of a particle’s path is related to the action, with the classical path of least action being of central importance.

• Field Theory: In field theory, including electromagnetism and quantum field theory, the action principle is used to derive the field equations. The Lagrangian is formulated in terms of fields rather than particles, and the action integral extends over all spacetime.

• General Relativity: In general relativity, the action principle is embodied in the Einstein-Hilbert action, where the Lagrangian is a function of the metric tensor that describes the geometry of spacetime. Variation of the action with respect to the metric tensor yields Einstein’s field equations.

Conclusion

The action-based approach to physics highlights the elegance and unifying power of the action principle across different domains of physics. By focusing on the stationary action rather than forces or energy directly, it provides a versatile framework for formulating the laws of physics in a wide variety of contexts. This approach has been fundamental in the development of modern physics and continues to be a vital tool in theoretical research, including efforts to unify gravity with quantum mechanics and in the exploration of new theories of fundamental forces and particles.

how is the action principle connected to noether’s theorem

The connection between the action principle and Noether’s theorem is one of the most profound and illuminating relationships in theoretical physics. Noether’s Theorem links symmetries of physical systems to conserved quantities, revealing deep insights into the fundamental nature of our universe. This relationship is made explicit through the framework of action-based physics, particularly through the use of the principle of least action.

Understanding Noether’s Theorem

Noether’s Theorem, formulated by Emmy Noether in 1915, states that every differentiable symmetry of the action of a physical system corresponds to a conservation law. In simpler terms, for every symmetry in the laws of physics, there is an associated quantity that is conserved over time. Symmetries and conservation laws are thus two sides of the same coin.

Types of Symmetries and Conserved Quantities

1. Temporal Symmetry: If the laws of physics do not change over time, then energy is conserved. This symmetry under time translations means that the physics of a system remains the same regardless of when you observe it.

2. Spatial Symmetry: If the laws of physics are the same everywhere in space, then momentum is conserved. This spatial translation symmetry implies that moving a system from one location to another without rotation or distortion does not change the physical predictions.

3. Rotational Symmetry: If the laws of physics are invariant under rotation, then angular momentum is conserved. This means that the orientation of a physical system does not affect its behavior.

The Role of the Action Principle

The action principle, or the principle of least action, provides a unified framework within which these symmetries and conservation laws can be identified and understood. When the action of a system is invariant under some transformation (e.g., shifting time, space, or rotating the system), Noether’s Theorem tells us that there is a corresponding conserved quantity.

Mathematically, the invariance of the action under a continuous symmetry leads to a specific form of the Euler-Lagrange equations, which encapsulate the equations of motion of the system. Applying Noether’s Theorem then directly yields expressions for conserved quantities. This method is especially powerful because it applies broadly across physics, from classical mechanics to quantum field theory and general relativity.

Implications

The implications of the connection between the action principle and Noether’s Theorem are far-reaching:

• Unification of Physics: It highlights the deep connection between the symmetries of the universe and conservation laws, unifying different areas of physics under a common principle.

• Predictive Power: It provides a powerful tool for predicting conserved quantities in complex systems, simply by identifying the system’s symmetries.

• Theoretical Development: This relationship has guided the development of modern theoretical physics, including the standard model of particle physics and attempts to formulate a theory of quantum gravity.

Conclusion

The connection between the action principle and Noether’s Theorem is foundational to modern physics, illustrating how symmetries underpin the fundamental laws of the universe and lead to conservation laws. This relationship enriches our understanding of the physical world, offering a lens through which we can explore and unify diverse physical phenomena.

that was a fairly hand-wavy response. i have graduate degrees in philosophy and math, please respond with deep technical formalizations and nuanced details

Given your background in philosophy and mathematics, a more technical and detailed exploration of the action principle’s connection to Noether’s theorem is indeed warranted. This relationship not only encapsulates one of the most elegant aspects of theoretical physics but also exemplifies the profound interplay between symmetry, invariance, and conservation laws within the fabric of physical theories.

Formal Statement of Noether’s Theorem

Noether’s theorem is formally stated within the context of variational calculus and Lagrangian mechanics. Consider a physical system described by a Lagrangian $L(q_i,{\dot{q}}_i,t)$, where $q_i$ are generalized coordinates, ${\dot{q}}_i$ their time derivatives, and $t$ is time. The action $S$ is defined as the integral of the Lagrangian over time:

$S={\int }_{t_1}^{t_2}L(q_i,{\dot{q}}_i,t)dt$

Noether’s theorem addresses how symmetries of this action under continuous transformations lead to conservation laws. A symmetry transformation is a change in the variables $q_i\to q_i^{\prime }=q_i+\delta q_i$ and $t\to t^{\prime }=t+\delta t$ that leaves the action invariant to first order in the infinitesimals ($\delta q_i,\delta t$). Specifically, for a symmetry, the variation in the action $\delta S$ must vanish:

$\delta S=0$

Noether’s theorem asserts that for each such continuous symmetry, there is a corresponding conserved quantity. Mathematically, if the action is invariant under a continuous group of transformations parameterized by $ϵ$, then there exists a quantity $C$ such that:

$\frac{dC}{dt}=0$

Derivation and Implications

The formal derivation of Noether’s theorem involves taking the variation of the action due to infinitesimal transformations and applying the principle of stationary action, which yields the Euler-Lagrange equations:

$\frac{d}{dt}\left(\frac{\partial L}{\partial {\dot{q}}_i}\right)-\frac{\partial L}{\partial q_i}=0$

The invariance of the action under the symmetry transformation leads to a relationship involving the derivatives of the Lagrangian with respect to the generalized coordinates and their time derivatives. This relationship can be manipulated to produce a quantity $C$ that is shown to be conserved over time.

The profound implication of Noether’s theorem is that it provides a direct link between the symmetries of the action (reflective of underlying physical symmetries) and conserved quantities, which are observable and fundamental properties of the system. This theorem underpins much of modern physics, including:

• Classical Mechanics: Conservation of momentum, energy, and angular momentum are direct consequences of spatial translation, temporal translation, and rotational symmetries, respectively.

• Quantum Mechanics: Noether’s theorem applies to quantum systems via the quantum action principle, with implications for conservation laws in quantum fields.

• Field Theory and Particle Physics: Symmetries related to internal spaces (such as gauge symmetries) lead to conservation laws relevant to charge, color charge, and other properties fundamental to the interactions described by the Standard Model of particle physics.

Nuanced Discussion

The application and significance of Noether’s theorem extend beyond mere conservation laws. It offers a guiding principle for the formulation of physical theories, emphasizing the role of symmetry. The theorem provides a conceptual framework for understanding why certain quantities must be conserved in any physical theory that possesses certain symmetries. This has profound philosophical implications, suggesting that the symmetries of the laws of physics are not mere accidents but are deeply connected to the invariant properties of the universe.

Moreover, Noether’s theorem embodies the unity of mathematics and physics, demonstrating how abstract mathematical principles (symmetries and invariances) directly inform our understanding of the physical world. It also highlights the predictive power of mathematical structures in physics, as identifying a symmetry can lead directly to predictions about conserved quantities and invariant properties of a system.

Conclusion

The connection between the action principle and Noether’s theorem is a cornerstone of theoretical physics, embodying the deep and beautiful relationship between symmetry and conservation laws. This relationship not only has profound implications for our understanding of the universe but also exemplifies the elegance and power of mathematical physics in uncovering the fundamental principles that govern the natural world.