# Conservation Principle

# CONSERVATION PRINCIPLE

Conservation principles tell us that some quantity, quality, or aspect remains constant through change. Such principles already appear in ancient and medieval natural philosophy. In one important strand of Greek cosmology, the rotation of the celestial orbs is eternal and immutable. In optics at least from the time of Euclid (fl. c. 300 BCE), when a ray of light is reflected, the angle of reflection is equal to the angle of incidence. According to some versions of the medieval impetus theory of motion, impetus permanently remains in a projected body (and the associated motion persists) unless the body is subject to outside interference. Such examples abound.

In the seventeenth century, conservation principles began to play a central role in scientific theories. Galileo Galilei, René Descartes, Christian Huygens, Gottfried Leibniz, and Isaac Newton founded their approaches to physics on the principle of inertia—the principle that a body will undergo uniform rectilinear motion unless interfered with. A multitude of other conservation principles gained currency during the seventeenth century—some still with us, some long ago left behind.

Descartes is an interesting example of an author who attempted to derive all of his physical principles from conservation laws (1991 [1644], esp. pt. 2, secs. 36–42). Descartes believed that the principles of his physics could be derived from the God's immutability, supplemented only by very weak assumptions about the existence of change in the world. He claimed, in fact, that we ought to postulate the strongest conservation laws consistent with such change. These laws were that God preserves at all times the total quantity of motion in the world (the quantity of motion of a body being the product of its volume and its speed), that each thing remains in the same state in every respect unless interfered with, and that in collisions the quantity of motion gained by one body is balanced by the quantity of motion lost by the other. The rest of his physics was supposed to follow from these principles alone.

The most remarkable of seventeenth-century analyses of conservation principles is contained in Huygens's essay on elastic collisions (1977). Huygens began by assuming that if two collinearly moving bodies of equal size move toward one another with equal speeds, in the resulting collision they simply exchange velocities. He then showed that it follows from the principle of Galileian relativity—that an experiment has the same outcome whether performed in a laboratory at rest or in a laboratory in uniform rectilinear motion—that whatever the initial velocity of such bodies, the result of a collision is always that velocities are simply exchanged. Huygens went on to analyze collisions between bodies of unequal size, again relying heavily on Galileian relativity. Among the consequences of his analysis were a number of conservation laws for systems of particles interacting only via elastic collisions: that the center of gravity of such a system undergoes uniform rectilinear motion, that the total kinetic energy of such a system is constant in time, and that the relative velocities (*mv* ) of a pair of colliding particles is unchanged by a collision.

In a sense that will be spelled out below, the principle of Galileian relativity is a symmetry principle. So one of the things that Huygens accomplished was to show that from a symmetry principle one could deduce conservation principles. For an extensive class of physical theories—essentially, all of classical (or nonquantum) physics—it is now possible to establish a deep connection between symmetry principles and conservation laws. The balance of the discussion here provides an elementary introduction to the ideas relevant to understanding this connection.

## Symmetry

At the most abstract level, a structure is a set of objects instantiating some set of properties and relations. A symmetry of a given structure is a permutation of the set of objects of the structure that leaves invariant all the properties and relations involved in the structure. For any structure, the identity map on its set of objects is, trivially, a symmetry.

For example, suppose that three points have relative pair-wise distances of three, four, and five units. Then there is no nontrivial symmetry that preserves these distances. But if the points instead form the vertices of an equilateral triangle, there will be several nontrivial symmetries, such as any transformation that interchanges two vertices while leaving the third fixed. We will be interested here in dynamic symmetries. As an intuitively plausible example, ordinary translation and rotation in space should be symmetries of any decent physical theory set in Euclidean space. Note that this does not mean that every translation or rotation will be a symmetry of the states allowed by the theory: The theory might treat the behavior of a finite number of point masses, in which case no configuration of the material points could be invariant under any nontrivial translation or under more than finitely many rotations. Rather, in such a case the invariance of the theory amounts to this: The dynamics of the theory is indifferent to the location or orientation of the system in Euclidean space, in that a translation or rotation of any state allowed by the theory will not change the dynamic evolution predicted by the theory, so long as the evolution of the new state is described relative to coordinate axes that have also been translated or rotated.

Its necessary to make all of this a bit more precise. Specifying a physical theory typically involves specifying a set of physically possible states and a dynamics defined on this space of states. Most often, the states involved will be possible instantaneous states of the system, such as the instantaneous positions and momenta of a set of particles, or the values of some field and of its time derivative at each point of space. These states will be collected together to form a space with some interesting mathematical structure (there is no need to be very specific about this structure at this stage). For convenience, a strict form of determinism will be assumed, under which the dynamics is given by the rule that if the state of the system at a given time is *a*, then its state *t* units of time later will be *b*, which we write as *a* _{t }⃗ *b*. A symmetry of this dynamics, *S*, will be a one-to-one mapping from the state space onto itself that leaves invariant all of the structure defined on this space, including the arrow relation. So *a* _{t }⃗ *b* if and only if *S* (*a* ) _{t }⃗ *S* (*b* ).

## The Hamiltonian Approach

Remarkably, almost all the equations of motion that arise in classical physics can be derived within the mathematical framework of Hamiltonian mechanics.

Consider the Newtonian *n* -body problem (*n* point masses interacting according to Newton's law of gravitation). We construct the phase space for this problem, the space of dynamically possible states of the particles. Choosing a point in this 6*n* -dimensional abstract space amounts to specifying the position and momentum of each of the *n* particles (collision states with two or more particles coinciding in position are ruled out a priori, since the expression for the force of gravitational attraction between coincident particles is ill defined). Now, by the nature of the Newtonian equation of motion (*F* = *ma* ), specifying the positions and momenta of the particles at some initial time suffices to determine their positions and momenta at other times (indeed, at all other times, unless a collision or other singularity occurs). So the dynamic content of the theory takes this form: Specifying a point in the phase space determines a curve in the phase space through that point—the idea being that if the given point represents the state of the system of *n* particles at time *t* = 0, then the curve tells us which points of the phase space represent states of the system at earlier and later times. These curves have the following nice feature: They partition the phase space, in the sense that exactly one curve passes through each point of the phase space (that at least one curve passes through each point follows from the dynamic content of the theory; that no more than one does so is a reflection, roughly speaking, of the determinism of this theory).

At the heart of the Hamiltonian approach lie three insights: (1) The phase space of the system, just in virtue of being a space of possible positions and momenta, carries a natural mathematical structure called a "symplectic form" (a closed nondegenerate two-form). (2) This structure allows the association to each nice real-valued function on the phase space of a family of curves that partitions the phase space. (3) The curves encoding the dynamics of the theory are thus associated with the Hamiltonian for the theory—the function that assigns to each point in phase space the total energy of the corresponding physical state (here the total energy is the sum of the kinetic energy and the gravitational potential energy).

These insights carry over to underwrite a Hamiltonian treatment of a vast assortment of classical (or nonquantum) physical theories. To develop a Hamiltonian treatment, consider the space of initial data for the equations of motion, and take this as the phase space of one's theory, showing that it comes equipped with a natural symplectic form (or generalization thereof) that allows one, in general, to pass from a function on the phase space to a set of curves partitioning the phase space—and in particular to pass from the Hamiltonian function assigning to each state its total energy to the curves on the phase space encoding the dynamic content of the equations of motion of the theory. This strategy works for rigid bodies, systems of moving particles subject to many sorts of constraints, many field theories, and some theories of material continua such as fluids and elastic bodies.

## Symmetries in the Hamiltonian Approach

So under the Hamiltonian approach, a theory consists of a phase space (representing the possible dynamic states of the theory) equipped with a symplectic form (or generalization) and a Hamiltonian function. Below, this symplectic form will be referred to as the geometrical structure of the phase space, although it is important to keep in mind that this structure is different in kind from the sort of metric structure that is normally treated in geometry.

A symmetry of a Hamiltonian theory is a one-to-one mapping from the phase space onto itself that preserves the geometric structure of the phase space and the Hamiltonian. Because these latter two objects are smooth, it follows that symmetries are continuous and differentiable to all orders.

In the *n* -body problem, for instance, all symmetries are smooth maps from the phase space onto itself that correspond to some combination of the following actions: (a) shifting by some fixed amount the positions of all particles in the Euclidean space in which they move; (b) rotating the orientation of the system of particles in Euclidean space by some fixed amount; (c) shifting the temporal origin by some fixed amount (that is, associating each state with the state that normally precedes or follows it by the given amount of time); (d) applying related discrete symmetries, such as a mirror reflection of the positions of the particles or an interchange of negative and positive senses of time.

Because each symmetry of a Hamiltonian theory leaves invariant all of the structure on the phase space that was used to define the dynamics, it also leaves invariant the curves that encode the dynamics—as we should expect from our general account of dynamic symmetries above. (The operation of a Galileian boost is not a symmetry in the present sense. Boosting a system does not leave its Hamiltonian invariant. A boost changes the kinetic energy of each particle and in general alters the total kinetic energy of the system, while leaving the potential energy of the system unchanged. But Galileian boosts do leave invariant the set of dynamic curves of the *n* -body problem.)

## Noether's Theorem

A remarkable consequence of the geometric structure of the phase space is that in any Hamiltonian system, the Hamiltonian is constant along each curve encoding the dynamics. That is, the total energy of the system is a conserved quantity of the dynamics: If one state evolves into another, each has the same total energy.

Are there additional conserved quantities—functions on our phase space that are constant along the curves encoding the dynamics? To find them, consider any one-parameter continuous family of symmetries of a Hamiltonian system closed under composition—such as the family of spatial translations by a varying amount in a given direction in the *n* -body problem. What happens if we allow such a family to act on a point in the phase space of a Hamiltonian system? To find out, we construct a curve in the phase space that describes how each symmetry in our one-parameter family acts on the initial state. By performing this operation for each point in the phase space, we construct partitions the phase space. For the sort of theories that arise in practice, we can then find a nice function on our phase space that the geometric structure associates with this family of curves. From the geometric structure and from the fact that the family of curves in question arises via the action of a family of symmetries that preserve the Hamiltonian, it follows that this function will itself be a constant of motion of the physical theory under consideration—that this new function, associated with our one-parameter family of symmetries, is constant along each of the dynamic curves associated with the Hamiltonian of our theory.

In this way, for any well-behaved Hamiltonian theory, we can construct a conserved quantity corresponding to each one-parameter family of symmetries of the theory. Indeed, we can find as many functionally independent conserved quantities as there are dimensions in the complete family of symmetries of the theory. In the case of the *n* -body problem, we find seven conserved quantities: the Hamiltonian, the components of the total linear momentum of the system, and the components of the total angular momentum of the system. These conserved quantities correspond, respectively, to the invariance of the system under time translations, to its invariance under spatial translations in each direction, and to its invariance under rotation.

These insights derive from work of Emmy Noether in 1918 (1971), though they assume a somewhat different form in her work, since she worked in the Lagrangian framework rather than the Hamiltonian framework.

** See also ** Classical Mechanics, Philosophy of; Philosophy of Physics.

## Bibliography

Brading, Katherine, and Elena Castellani, eds. *Symmetries in Physics: Philosophical Reflections*. Cambridge, U.K.: Cambridge University Press, 2003.

Descartes, René. *Principles of Philosophy* (1644). Translated by Valentine Miller and Reese Miller. Dordrecht, Netherlands: Kluwer, 1991.

Huygens, Christian. "The Motion of Colliding Bodies." Translated by Richard J. Blackwell. *Isis* 68 (1977): 574–597.

Noether, Emmy. "Invariant Variation Problems" (1918). Translated by M. Tavel. *Transport Theory and Statistical Physics* 1 (1971): 184–207.

Singer, Stephanie Frank. *Symmetry in Mechanics: A Gentle, Modern Introduction*. Basel, Switzerland: Birkhäuser, 2001.

Van Fraassen, Bas. *Laws and Symmetry*. Oxford: Oxford University Press, 1989.

Zuckerman, Gregg. "Action Principles and Global Geometry." In *Mathematical Aspects of String Theory*, edited by Shing-Tung Yau. Singapore: World Scientific, 1987.

*Gordon Belot (2005)*

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