conservative

I like Webster's decision to call the Conservative Party 'English History' rather than 'British Politics'. :-)

The Conservative party is the mainstream right-wing party of British politics. They ran the country for eighteen years, from 1979 to 1997, under Margaret Thatcher and John Major. The current leader (May 2001) is William Hague, but he may not retain the position if he suffers a crushing defeat in the forthcoming election. The Conservatives are traditionally a pro-capitalist party, garnering support from big business and successful executives.

They are sometimes alluded to as the Tory party, which was an earlier movement from which they grew.

This is from an American perspective, so the American conservatives are the ones I am addressing here.

I have heard it said that conservatives are people terrified by the idea that somewhere, someone is having more fun than they.

This is true in some cases, but more often, conservatives have an ideal look to a society, and that ideal involves traditional, two parent families, strong communities, and active religious lives by the citizens. They tend to believe in market economies with minimal government welfare (for individuals at least. Corporations are a different matter, and conservatives differ with one another regarding them). This naturally makes them the ideological enemies to those who disagree with the precise nature of this vision. Some people feel that, for example, loving couples who are gay athiests with adopted children should feel perfectly at home in this country, and that the conservatives' vision would ostracize them at best, and make them hide in fear for their lives at worst. They feel that the poor should be helped, even if it means taking taxes from the non-poor to do it.

Some fear that conservatives would force religion on them and their children, and there may be some merit in that view, at least as it pertains to some conservatives. Conservatives are not a monolithic block, however. Some are more religiously oriented, others are more economically motivated. In this country, most people who regard themselves as conservative vote for candidates from the Republican Party. Other conservative parties exist for specific needs, such as the Constitution Party.

I grew up in a liberal household. My parents were born in the 1940s in Mississippi. Conservatives (in their estimation) were an open enemy to their happiness, so, understandibly, they identified with liberals. I think, though, if you speak to most conservatives, you will find a genuine desire that people be happy and secure. Their objectives are generally honorable, as are those of their opponents, actually. It is an unfortunate reality that people tend to stick with those who think similarly to themselves so frequently that they rarely encounter opposing views in an interactive, non-combative manner. When conservatives and liberals sincerely sit down and just talk, rather than argue, they usually find that they agree on a lot of things, and those things they disagree with aren't hateful disagreements. They are honest ones, and generally, aren't open to negotiation because of that honesty. In America, we settle these disagreements with the ballot rather than the bullet. Other places aren't so fortunate.

In vector calculus, the line integral of a vector-valued function on some curve is called the work integral along that curve. The curve can be chosen arbitrarily, but this may change the value of the work integral. If it turns out that the amount of work done is the same regardless of the curve chosen between fixed endpoints, then the function is described as conservative. In physical terms, this means that the effort required to move within a force field F between two points depends on the distance between them, not the route taken: there can be no shortcuts, but there are no 'bad' routes either.
It further transpires that having a conservative function is equivalent to some other interesting properties (based on the various derivatives possible within vector calculus) of vector functions. These shall be illustrated in this writeup; and will hopefully give an insight into some of the key ideas and typical proof methods encountered in first approaching this field of mathematics. For the purpose of clarity functions in R³ (3-dimensional space, approximately the universe we live in) will be used during further discussion, and the symbol '.' shall denote the dot product.

First then, let us state the definition of conservative in a more rigorous fashion. There are two commonly used defintions: I'll use the one that corresponds to the idea described above and then later derive the other formulation.

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Let C be a curve in Rⁿ, parametrised by a function of a variable t such that r:R--> R ⁿ. So, C = { r(t) | t ∈ [ t₀, t_e ] }
Then the work integral of a vector field F: R ⁿ --> R ⁿalong C is given by

              te
∫ F . dr  := ∫ F(r(t)) . dr/dt dt
C             t0

If C₁ and C₂ are arbitrary curves with the same endpoints (that is, r₁(t₀) = r₂(t₀) = x₀ ∈ Rⁿ, and r₁(t_e) = r₂(t_e) = x_e ∈ Rⁿ where r₁ is the parametrisation for C₁ and r₂ is that of C₂) then F is called conservative iff

 ∫  F . dr =  ∫  F . dr
  C1               C2

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From here on, our discussion is restricted to R³, with Ω a simply connected domain

Theorem: if F is conservative, F has a scalar potential.

This is our first interesting result, and we use it to obtain the other defintion of conservative. A scalar potential of F: Ω --> R³ is a function Φ satisfying ∇φ = F. The proof would be an absolute massacre in HTML, but here is the method if not the madness:

• We can define a candidate Φ(x) as precisely the work integral of F along a curve from the origin 0 = (0,0,0) to x= (x,y,z) in Ω; this is a well defined function of x since the value obtained is consistent regardless of the curve chosen, precisely because F is conservative.
• If we consider F = F₁i + F₂j + F₃k (or other appropriate orthogonal curvilinear coordinate system; I adopt i,j,k for clarity here) then we can show that
  ∂Φ / ∂x (x) = F₁(x)
  (To do so, take C:= C₁ U C₂ where C₁ is the straight line from 0 to (0,y,z) whilst C₂ is the straight line from (0,y,z) to x; the integral over C can be split linearly into two integrals, one over each of these curves).
• By Similar methods for ∂Φ / ∂y (x) and ∂Φ / ∂z (x) (constructing C₁ and C₂ such that the first goes to (x,0,z) or (x,y,0) respectively) we can conclude that
  ∂Φ / ∂x (x) = F₁(x) ; ∂Φ / ∂y (x) = F₂(x) ; ∂Φ / ∂z (x) = F₃(x)
• But as F = F₁i + F₂j + F₃k, it follows that F = ∇Φ - our constructed work integral is a scalar potential as desired.

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Theorem: F has a scalar potential iff F is conservative

We have shown that a conservative field has a scalar potential: now we seek to show the converse- having a scalar potential leads to being conservative. Again, a sketch proof is provided. We assume Φ: R³ --> R and let F = ∇Φ.

• By the chain rule, we can show that ∇Φ . dr/dt = d/dt(Φ o r) ( derivative of the composition of Φ with r)
• Hence the defintion of work integral can be re-written as

                te
  ∫ F . dr  := ∫ (d/dt (Φ o r))dt
  C             t0
  

• By the fundamental theorem of calculus, this is simply Φ(r(t_e)) - Φ(r(t₀)). It follows that F is conservative since the value of the work integral can be determined by knowledge of the end points only, so the route taken between them is irrelevant.

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Alternative defintion of conservative

The above result is slightly more powerful than just giving us the property of conservatism. If we consider an arbitrary closed curve in R³ - that is, one where the start- and endpoints are the same, then we observe that r(t_e) = r(t₀). So given that the work integral is obtained by Φ(r(t_e)) - Φ(r(t₀)), it is obvious that this quantity is zero. So a second common definition of conservative is:

F is conservative iff ∫_C F.dr = 0 for all closed curves C.

Anark notes that in applied contexts this definition is typically the one used. If anyone can tell me the HTML/unicode code for the closed integral (circulation) symbol, that'd be much appreciated!

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Theorem: F is conservative iff F is irrotational

Equipped with this new definition, we can easily prove another equivalence, between conservative fields and irrotational ones (having zero curl).

The forward implication is virtually a one-liner: curl F is ∇ ^ F (cross product), but since F is conservative it has a scalar potential, so we obtain curl F = ∇ ^ ∇Φ for some Φ. Multiplying out you get curl F = 0.

The reverse implication is also fairly simple once you are familiar with Stokes' Theorem: by which the integral over a closed curve can be rewritten as a double integral over the surface of which the curve is a boundary. So
∫ _C F . dr = ∫∫_s (∇ ^ F) . ds for a closed C. Since F is irrotational, this is just ∫∫ 0.ds = 0. By our alternative definition, as C is closed, we have F conservative.

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The simply connected condition

Swap points out the importance of the domain Ω being simply connected: consider

"F(x,y,z) = (-y/r², x/r², 0), with r = x² + y². Then curl F = 0, but any line integral that encloses the origin evaluates to 2*Pi (the example comes from complex analysis and the calculus of residues. Thus, this field is not conservative, and there is no scalar potential function either."

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Summary
For a function F in a simply connected domain,
F has a scalar potential iff F is conservative iff F is irrotational

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Based upon my studies of MA20010- Vector Calculus and Partial Differential Equations; a second year module from the University of Bath Mathematics degree; obviously this means I'm not even a graduate yet if you suspect an error the odds are you're right; drop me a /msg!
Thanks to tdent for typo-deathsquad duties, plus Anark and Swap for mathematical feedback.

Conservative, as applied ot one of the two great parties in English politics, was first used by J.W. Croker in an article in the "Quarterly" for January, 1830, and was by Macaulay in the "Edinburgh" for 1832 referred to as a "new cant word." Conservative accordingly began to supersede Tory about the time of the Reform Bill controversies. The plural form of the word has been assumed as a distinctive name by certain political parties in many nations. These parties are sometimes actually, and always avowedly, opposed to changes from old and established forms and practices. In United States history these names have never been in general use, but in Van Buren's administration the name of Conservatives was applied to those Democrats that at the special session of Congress, of September 1837, opposed the establishment of the sub-treasury system. In the Congress that met in December, 1839, they had practically disappeared. The name was also assumed by Southern whites during the reconstruction period following the Civil War, to show their adherence to the old State governments, the abolition of which by Congress they opposed. The name was also used in the North during this period. The Democrats applied it to themselves to draw moderate Republican votes.

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Entry from Everybody's Cyclopedia, 1912.

Con*serv"a*tive (?), a. [Cf. F. conservatif.]

1.

Having power to preserve in a safe of entire state, or from loss, waste, or injury; preservative.

2.

Tending or disposed to maintain existing institutions; opposed to change or innovation.

3.

Of or pertaining to a political party which favors the conservation of existing institutions and forms of government as the Conservative party in england; -- contradistinguished from Liberal and Radical.

We have always been conscientuously attached to what is called the Tory, and which might with more propierty be called the Conservative, party. Quart. Rev. (1830).

Conservative system Mech., a material sustem of such a nature that after the system has undergone any series of changes, and been brought back in any manner to its original state, the whole work done by external agents on the system is equal to the whole work done by the system overcoming external forces.

Clerk Maxwell.

 

© Webster 1913.

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Con*serv"a*tive (?), n.

1.

One who, or that which, preserves from ruin, injury, innovation, or radical change; a preserver; a conserver.

The Holy Spirit is the great conservative of the new life. Jer. Taylor.

2.

One who desires to maintain existing institutions and customs; also, one who holds moderate opinions in politics; -- opposed to revolutionary or radical.

3. Eng. Hist.

A member of the Conservative party.

 

© Webster 1913.